Adaptive deformable mirror- dynamics and modular control

ÔØÚÓÖÑÐÑÖÖÓÖ
ÝÒÑ×ÒÑÓÙÐÖÓÒØÖÓÐ
ÊÅÄÐÐÒÖÓ

ÔØÚÓÖÑÐÑÖÖÓÖ
ÝÒÑ×ÒÑÓÙÐÖÓÒØÖÓÐ
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 8 november 2011 om 15.00 uur
door Rogier Martin Lambert ELLENBROEK,
werktuigbouwkundig ingenieur,
geboren te Heerlen.

Dit proefschrift is goedgekeurd door de promotor:
Prof.dr.ir. M. Verhaegen
Samenstelling promotiecommissie:
Rector Magnificus voorzitter
Prof.dr.ir. M. Verhaegen Technische Universiteit Delft, promotor
Prof.dr.ir. M. Steinbuch Technische Universiteit Eindhoven
Prof.dr.ir. J.M.A. Scherpen Rijksuniversiteit Groningen
Prof.ir. R.M. Schmidt Technische Universiteit Delft
Prof.dr. C. Keller Universiteit Utrecht
Dr.ir. P.R. Fraanje Technische Universiteit Delft
Dr.ir. N.J. Doelman TNO
Prof.dr.ir. J.A. Mulder Technische Universiteit Delft, reservelid
Dr.ir. N.J. Doelman heeft als begeleider in belangrijke mate aan de totstandkoming van het
proefschrift bijgedragen.
This thesis has been completed in partial fulfillment of the requirements of the Dutch Institute
for Systems and Control (DISC) for graduate studies. The research described in this
thesis was supported by the Dutch Innovative Research Project (IOP) Precision Technology.
ISBN 978-90-9026363-2
Copyright c○ 2011 by R.M.L. Ellenbroek
All rights reserved. No part of the material protected by this copyright notice may be reproduced
or utilized in any form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage and retrieval system, without written
permission of the author.
Printed by CPI Wöhrmann Print Service

ÒÛÓÓÖ
Het was al laat in de avond toen de telefoon ging en Roger me vroeg of ik misschien
geïnteresseerd zou zijn om als een tweede promovendus binnen zijn adaptieve optica
project te werken. Er was echter een kleine maar: dit deel van het project zou worden
uitgevoerd aan de technische universiteit van Delft in plaats van die in Eindhoven, waar ik
op dat moment nog werkte. Hij had al een tijd lang zitten opscheppen over zijn project, dus
het was onmogelijk om het aanbod af te wijzen, al moest ik nog wel even wennen aan het
Delft-stad-van-de-koorballen concept. De uiteindelijke keuze was natuurlijk niet aan hem,
dus ik wil hierbij mijn promotor Michel Verhaegen en Pieter Kappelhof (de toenmalige
leider van het project) hartelijk bedanken voor het vertrouwen om mij voor deze positie
aan te nemen. Bovendien wil ik de mensen van het IOP Precisie Technologie hartelijk
bedanken voor het organiseren van de randvoorwaarden waaronder dit project mogelijk
was. Tenslotte zijn er nog een hele sloot mensen aan wie ik voor het bereiken van deze
mijlpaal dank verschuldigd ben en waarvan ik er een paar nog even expliciet wil noemen:
Al snel bleek dat het begeleiden van mij als promovendus niet vanzelf gaat en veel tijd
kost. Ik wil daarom Niek en Michel van harte bedanken voor hun tijd en moeite om mij
bij te sturen en op het rechte pad te houden. Niek, bedankt ook om zelfs na de officiële
eind-datum van het project deze tijd te blijven vinden.
In het project moest er door TNO ook elektronica worden ontwikkeld, hetgeen deels
aan Emdes (later onderdeel van QPI) werd uitbesteed. Ik wil bij deze Paul Keijzer, Ton
Lommen, Wout van de Maden en Gerhard Hein bedanken voor de inzet en samenwerking
die uiteindelijk heeft geleid tot een werkend prototype van een complex systeem.
Verder wil ik Rudolf Mak, Kees van Berkel en William de Bruijn bedanken voor hun
doorzettingsvermogen in het nog efficiënter maken van de ontwikkelde elektronica.
Kitty, bedankt voor je gezelligheid en hulp en dat je tot het bittere eind mijn rots in de
branding van DCSC was. Jelmer, ik weet niet hoe ik je morele ondersteuning, gezelligheid,
etc. als kamergenoot voor 4 jaar in deze lofzang voldoende uit de verf kan laten komen.
Ik ben blij dat we elkaar nu ’bijna buren’ mogen noemen en nog op allerlei manieren meer
van elkaar gaan zien. Karel, bedankt dat je me – in ruil voor je bijdehante antwoorden op
al mijn werk-gerelateerde vragen – bij onze wekelijkste squash sessie zo vaak liet winnen.
De kaasfondue zal ik nooit vergeten en hopelijk komen daar in de toekomst nog allerlei
dingen bij. Sjoerd, je was een onverwachte vriend die op de belangrijkste momenten voor
me klaar stond en toen zover mogelijk weg ging wonen. Vooral die laatste actie blijf ik
jammer vinden.
Roger, is er niks te vertellen dat je niet al weet, maar dat belet me niet om hier een
v

vi Dankwoord
paar woorden neer te zetten. Ondanks de woede-uitbarstingen, de scheld-kanonnades, de
haren-trekkerij, de machtsstrijd, de afgunst en het haantjesgedrag van anderen, zijn we
goeie vrienden gebleven en hebben daar zelfs het ’bijna-familie’ en het ’collega’ gevoel
aan toegevoegd. Ik vind het bijzonder dat je in zoveel aspecten een deel van mijn leven
uitmaakt en ik ben dan ook blij dat we elkaar bij Mapper nog steeds in al die hoedanigheden
tegenkomen.
Pap en mam, voor jullie was mijn vertrek waarschijnlijk best een beetje plotseling en was
mijn promotie project zowel inhoudelijk als praktisch een beetje een ver-van-jullie-bedshow.
Toch heeft jullie warmte en mentale steun er veel aan bijgedragen dat dit boek er
uiteindelijk toch is gekomen.
Martin, dankjewel voor je zonnige gezicht dat alles altijd weer goedmaakt.
En Mirjam, de laatste en dikste knuffels zijn voor jou. Voor alle keren dat ik ’nee’ zei,
gevolgd door een zin met het woord ’proefschrift’. Voor alle keren dat je zei ’e finiscila!’
of ’basta col perfezionismo’. Voor alles en voor altijd: ti amo.

ÓÒØÒØ×
Dankwoord v
Contents x
Summary xi
Samenvatting xiv
Nomenclature xvii
Acronyms xxiv
1 Introduction 1
1.1 Notes on notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Adaptive optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Atmospheric turbulence . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 The wavefront sensor . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 The wavefront corrector . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 Optical configurations . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.5 The control system . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 The wavefront corrector . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 The control system . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Distributed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Distributed control for AO . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Problem formulation and organization of this thesis . . . . . . . . . . . . . 21
1.7 Scientific contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Design requirements and design concept 25
2.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1 Atmospheric turbulence . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.2 The Kolmogorov turbulence model . . . . . . . . . . . . . . . . . 26
2.2 Error budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 The fitting error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 The temporal error . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Error budget division . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Actuator requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Control system and electronics requirements . . . . . . . . . . . . . . . . . 35
vii

viii Contents
2.5 The design concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.1 The mirror facesheet . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.2 The actuator modules . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.3 The control system and electronics . . . . . . . . . . . . . . . . . . 40
2.5.4 The base frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Efficient control for AO: concepts and challenges 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Existing control methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Generic problem statement . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Traditional approach . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 More generic controller designs . . . . . . . . . . . . . . . . . . . 48
3.3 Scaling problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Computational demand . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Practical problems . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Distributed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 Hardware considerations . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.2 Control considerations . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Possible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6.1 Phase reconstruction through analog electronics . . . . . . . . . . . 58
3.6.2 A distributed disturbance model . . . . . . . . . . . . . . . . . . . 58
3.6.3 Iterative distributed phase reconstruction . . . . . . . . . . . . . . 59
3.6.4 Recursive adaptive distributed reconstruction and prediction . . . . 62
3.6.5 Local, identical influence functions . . . . . . . . . . . . . . . . . 66
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Data driven distributed control 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Design approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.1 Parametrization of the distributed controller . . . . . . . . . . . . . 73
4.4.2 Internal model control . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.6.1 Gershgorin’s circle theorem . . . . . . . . . . . . . . . . . . . . . 78
4.7 Identification procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7.1 Optimization criterion and approach . . . . . . . . . . . . . . . . . 79
4.7.2 A two-stage approach . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7.3 Algorithm summary . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Simulation and breadboard results . . . . . . . . . . . . . . . . . . . . . . 85
4.8.1 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . 85
4.8.2 Breadboard data . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.8.3 Artificial data set . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Contents ix
4.9 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 The variable reluctance actuator 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 The single actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 The actuator membrane suspension . . . . . . . . . . . . . . . . . 94
5.2.2 The electromagnetic force . . . . . . . . . . . . . . . . . . . . . . 96
5.2.3 A static actuator model . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.4 A dynamic actuator model . . . . . . . . . . . . . . . . . . . . . . 104
5.2.5 Measurements and validation . . . . . . . . . . . . . . . . . . . . 109
5.2.6 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.7 Lessons learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 The actuator module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.1 Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.2 Power dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6 Electronics 125
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2 Driver electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Communication electronics . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4 Implementation and realization . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.1 Pulse Width Modulation (PWM) implementation . . . . . . . . . . 131
6.4.2 Field Programmable Gate Array (FPGA) implementation . . . . . . 134
6.4.3 The ethernet to Low Voltage Differential Signalling (LVDS) bridge 136
6.5 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 Evaluation of control aspects . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.7 Testing and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.7.1 Communications tests . . . . . . . . . . . . . . . . . . . . . . . . 141
6.7.2 Parasitic resistance measurements . . . . . . . . . . . . . . . . . . 142
6.7.3 Actuator system validation . . . . . . . . . . . . . . . . . . . . . . 142
6.7.4 Nonlinear behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.8 Power dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.8.1 Optimizing the FPGA power efficiency . . . . . . . . . . . . . . . 149
6.8.2 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7 System modeling and characterization 155
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2 Deformable Mirror (DM) integration . . . . . . . . . . . . . . . . . . . . . 156
7.2.1 Integration of the 61 actuator mirror . . . . . . . . . . . . . . . . . 156
7.2.2 Integration of the 427 actuator mirror . . . . . . . . . . . . . . . . 156
7.3 Static system validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

x Contents
7.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3.2 Measurements and results . . . . . . . . . . . . . . . . . . . . . . 164
7.3.3 Power dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.4 Dynamic system validation . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.4.1 Dynamic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.4.2 System identification . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.4.3 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.5 Discrete time control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.5.1 A note on distributed control . . . . . . . . . . . . . . . . . . . . . 190
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8 Conclusions and recommendations 193
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Appendices 200
A Shack-Hartmann spot positioning . . . . . . . . . . . . . . . . . . . . . . . 201
B On using local passivity to enforce global stability . . . . . . . . . . . . . . 203
C Fourier series of a PWM signal . . . . . . . . . . . . . . . . . . . . . . . . 206
D The LVDS protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
E The UDP protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
F Spatial variation of actuator properties . . . . . . . . . . . . . . . . . . . . 213
G Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Bibliography 216
Curriculum vitae 229

ËÙÑÑÖÝ
ÔØÚÓÖÑÐÑÖÖÓÖ
ÝÒÑ×ÒÑÓÙÐÖÓÒØÖÓÐ
The refractive index of air varies a.o. with temperature, humidity, pressure and the CO2
concentration. Due to atmospheric turbulence this refractive index varies both in space and
in time, leading to aberrations in images of light having passed though it. These aberrations
limit the achievable resolution of optical telescopes such that the quality of their images
is no longer diffraction limited. An Adaptive Optics (AO) system is a means to recover
the diffraction limited quality of the images. This can be achieved a.o. by reflecting the
incoming light on a Deformable Mirror (DM) that adapts its shape to the wavefront of this
light such that some norm of the residual wavefront after reflection is minimal.
Such a reflection based AO system consists of three main components: a WaveFront
Sensor (WFS), a DM and a control system. In this thesis novel designs are considered for
the latter two components, primarily aimed at the 8m class of telescopes in visible light.
The WFS is assumed to be of the Shack-Hartmann type and not further investigated. As
astronomers want images of ever fainter celestial objects, larger and larger telescopes are
foreseen to increase both resolution and light gathering power. Therefore, an important
design driver is the extendibility to a larger number of Degrees Of Freedom (DOF) in
combination with a low power consumption to prevent the need for active cooling systems.
The main requirements for the DM and control system are derived for atmospheric
conditions that are typical for telescope sites. The spatial and temporal properties of the
atmosphere are modeled by the spatial and temporal spectra of the Kolmogorov turbulence
model and the frozen flow assumption. The main sources for the residual wavefront
aberrations – important for the DM design – are identified as the fitting error and the
temporal error that are caused by a limited number of actuators and a limited control
bandwidth respectively. A choice for the number of actuators and the control bandwidth is
made that leads to a desired optical quality after correction in terms of a Strehl ratio of 0.85
and for which the specifications are considered feasible for both areas of expertise.
The requirements on the DM actuators are based on a visible wavelengthλ of550nm. This
leads to a required pitch of 3 to 6mm, a total actuator stroke of ±5.6µm, an inter-actuator
stroke of ±0.36µm and a resolution of 5nm. From the resulting actuator density and the
desire for active cooling follows a maximum allowed power dissipation per actuator of
1mW. The corresponding requirement on the control bandwidth is 200Hz, from which
additional requirements are derived for the sampling frequency and the first DM resonance
frequency. From a budgeting of the phase margin both are required to be higher than 1kHz.
The phase delay and latency of the driver and communication electronics at the bandwidth
frequency must be small compared to the sampling time.
When designing controllers for AO systems with increasingly large numbers of DOFs
xi

xii Summary
this will lead to a high computational load that must properly handled. Without efficient
algorithms, the demanded computational power increases quadratically with the number
of degrees of freedom and thus to the fourth power in the telescope’s aperture diameter.
A modularly distributed control system is proposed to extend the control system towards
large numbers DOF without problems with the computational loads. It is shown that
the distributed controller architecture complicates the wavefront reconstruction step that
is commonly present in most AO control systems due to the spatial dynamics of the
Shack-Hartmann WFS used. The performance of a distributed algorithm that combines the
reconstructor with an adaptive prediction algorithm is shown in simulation to be able to
better that of the traditional integrator controller. Moreover, other simulation results show
that under the frozen flow assumption this performance does not deteriorate as the number
of DOF increases.
The distributed algorithm is also investigated in a more general, non-adaptive context to
better understand the limitations caused by the enforced structure and study its feasibility
for AO systems for large telescopes. It is shown which correlations between wavefront
measurement signals can and cannot be exploited by the distributed algorithm and how this
translates into assumptions on the data generating system. An algorithm is proposed to
identify unknown controller coefficients from open-loop measurement data, but despite the
fact that the algorithm is applied off-line its high computational load makes it an unsuitable
candidate for high-DOF systems. But nevertheless it shows the feasibility of a modularly
distributed control architecture for large AO systems under the assumptions of Kolmogorov
statistics and frozen flow and while neglecting the DM system dynamics.
The requirements on the DM system are realized by using electromagnetic reluctance
actuators that connect to a continuous reflective facesheet via a thin rod and are driven by
a 16-bit Pulse Width Modulation (PWM)-based voltage source implemented in Field Programmable
Gate Arrays (FPGAs). The facesheet stretches over all actuators to minimize
the introduced optical aberrations, but all other components are modular. The actuators are
manufactured in modules of 61 in a hexagonal grid and consist of layers to reduce cost and
allow accurate assembly. A driver Printed Circuit Board (PCB) is developed that contains
driver electronics for one actuator module and consists of a PWM generator implemented
in FPGAs and a low-pass filter for each actuator. The PCB is placed behind the actuator
module to preserve the modular concept and the FPGAs also implement an Low Voltage
Differential Signalling (LVDS) based communication protocol that allows to control a DM
system containing up to 32 modules of 61 actuators. An ethernet-to-LVDS communications
bridge is developed such that the system can be connected to a standard computer or laptop.
A mathematical model is derived for the DM system that includes the behavior of the reflective
facesheet, the electromagnetic reluctance actuator and the driver and communication
electronics. The model is verified on several levels: for single actuators driven by a current
source, for actuator modules driven by the PWM voltage sources and on the level of the
complete system. On each level both static and dynamic (i.e. white noise excitation) measurements
are performed that are compared to the model. Differences found between the
single actuator measurements and the model are analyzed based on a sensitivity analysis,
leading to minor improvements of the grid module design and providing a basis for future
improvements. Seven manufactured actuator modules are found to behave in accordance
with the derived model and satisfy the requirements for displacement, force and power con-

Summary xiii
sumption. This proves that the manufacturing and assembly process is robust and allows the
production of reliable actuator modules.
Finally, the developed actuator modules and electronics are integrated with the reflective
facesheet to form a complete DM system. A model for this facesheet is derived based on an
analytical solution to the biharmonic plate equation for the surface shape of the facesheet
due to a point force. The model is used to derive the actuator influence functions and extended
with lumped masses to include the dynamic behavior. From the model, the transfer
functions, impulse response functions and mode shapes are derived. The verification of
the static behavior of the DM system is done using an interferometer. Verification of the
modeled dynamics is performed by applying white noise excitation to the actuators and
measuring both the displacement as well as the velocity response of the mirror facesheet
by means of a laser vibrometer. System identification is performed on the measurement
data using the Multivariable Output-Error State-sPace (MOESP) algorithm to arrive at a
state-space system description from which transfer functions, impulse response functions
and mode shapes are derived that are compared to those of the model. Finally, the system is
evaluated on control aspects, showing that the low damping of the resonances may lead to a
reduction of the achievable bandwidth or to dependence of this bandwidth on the sampling
frequency.

ËÑÒÚØØÒ
ÔØÓÖÑÖÖ×ÔÐ
ÝÒÑÒÑÓÙÐÖÒ×ØÙÖÒ
De brekingsindex van lucht varieert onder andere met temperatuur, luchtvochtigheid,
druk en CO2 concentratie. Door atmosferische turbulentie varieert deze brekingsindex
bovendien in zowel tijd als plaats, hetgeen leidt tot verstoringen in afbeeldingen van licht
dat er doorheen schijnt. Voor optische telescopen beperken de aberraties die deze verstoringen
veroorzaken de haalbare resolutie, zodat de kwaliteit van de afbeeldingen niet langer is
begrensd door diffractie in de telescoop. Een Adaptive Optics (AO) systeem is een middel
om deze diffractie begrensde kwaliteit te herwinnen. Dit kan onder andere worden bereikt
door het invallende licht op een deformeerbare spiegel te laten reflecteren die zijn vorm
aanpast aan het golffront van dit licht zodanig dat het golffront na reflectie minimaal is in
een bepaalde norm.
Een dergelijk op reflectie gebaseerd AO systeem bestaat uit drie hoofdcomponenten: een
golffront sensor, een deformeerbare spiegel en een regelsysteem. In dit proefschrift worden
nieuwe ontwerpen beschouwd voor de laatste twee componenten met 8m klasse telescopen
in zichtbaar licht als richt-applicatie. De golffront sensor wordt niet beschouwd; hiervoor
wordt een sensor van het Shack-Hartmann type aangenomen. Omdat astronomen blijven
hongeren voor afbeeldingen van steeds zwakkere hemellichamen worden er telescopen ontworpen
van steeds grotere afmetingen om zowel de afbeeldingsresolutie als de lichtsterkte te
verhogen. Daarom vormt uitbreidbaarheid naar een groter aantal vrijheidsgraden een belangrijk
uitgangspunt voor de gepresenteerde ontwerpen. Daarnaast is een laag energiegebruik
belangrijk opdat er geen actief koelsysteem nodig is.
De belangrijkste eisen voor de deformeerbare spiegel en het regelsysteem worden afgeleid
voor condities van de atmosfeer zoals die typisch op telescoop locaties voorkomen. De
spatiële en temporele eigenschappen van de atmosfeer worden gemodelleerd door de
spatiële en temporele spectra van het Kolmogorov turbulentie model in combinatie met
de aanname dat de wind de atmosferische aberratie voortbeweegt, maar dat deze in de tijd
stationair is (zgn. frozen flow). De belangrijkste bronnen voor de residuele golffront onvlakheid
die van belang zijn voor het ontwerp van de deformeerbare spiegel zijn de fit-fout
en de temporele fout die respectievelijk worden veroorzaakt door het beperkte aantal actuatoren
en de beperkte regelbandbreedte. Er wordt een keuze voor het aantal actuatoren
en de benodigde regelbandbreedte gemaakt die tot een gewenste optische kwaliteit leidt na
reflectie uitgedrukt in een Strehl-ratio van 0.85 en waarvoor de bijbehorende specificaties
voor beide vakgebieden haalbaar zijn.
De eisen voor het ontwerp worden gebaseerd op licht met een golflengteλ van 550nm. Voor
de correctie van de genoemde verstoringen leidt dit tot een inter-actuator afstand van 3 tot
6mm, een totale actuator slag van ±5.6µm, een inter-actuator slag van ±0.36µm, een resolutie
van 5nm en een maximum dissipatie in warmte van 1mW per actuator. De vereiste
xiv

Samenvatting xv
regelbandbreedte is 200Hz, waaruit de additionele eisen volgen dat zowel de bemonsteringsfrequentie
als de eerste eigenfrequentie van de deformeerbare spiegel beide boven 1kHz
moeten liggen. Echter, bij het ontwerp van regelsystemen voor AO systemen met steeds
meer graden van vrijheid zal de benodigde rekenkracht van de processoren van het regelsysteem
snel toenemen. Zonder efficiënte algoritmen neemt de benodigde rekenkracht toe
met het kwadraat van het aantal vrijheidsgraden en dus met de 4 e macht in de diameter
van de telescoop. Er wordt daarom een modulair gedistribueerde architectuur van het regelsysteem
voorgesteld om het zonder problemen wat betreft rekenkracht te kunnen toepassen
voor grote aantallen vrijheidsgraden. Deze gedistribueerde architectuur bemoeilijkt helaas
de golffront reconstructie stap die onderdeel is van het regel algoritme om te compenseren
voor de spatiële dynamica van de gebruikte Shack-Hartmann golffront sensor. Er wordt
echter in simulatie getoond dat een gedistribueerd algoritme waarin een adaptieve predictor
wordt gecombineerd met een iteratieve reconstructor in staat is om de prestaties van
de traditionele integrerende regelaar te overtreffen. Bovendien laten simulaties zien dat
deze prestatie niet afneemt indien het aantal vrijheidsgraden toeneemt bij gelijkblijvende
rekenkracht per vrijheidsgraad.
Het gedistribueerde algoritme wordt ook in een meer generieke, niet adaptieve context onderzocht
om tot een beter begrip van de beperkingen van de opgelegde structuur te komen.
Er wordt getoond welke correlaties die mogelijk bestaan tussen golffront-metingen door
een gedistribueerd algoritme wel en niet kunnen worden benut om de prestatie te verbeteren
en welke aannames de structuur impliceert voor het data genererende systeem. De onbekende
coëfficiënten van de regelaar worden uit open lus meetdata geschat met behulp van
een specifiek ontwikkeld algoritme. Dit algoritme werkt niet in real-time werkt, maar desondanks
maakt de benodigde rekenkracht het tot een ongeschikte kandidaat voor AO systemen
met een groot aantal vrijheidsgraden. Desalniettemin laat het zien dat een modulair
gedistribueerd regelsysteem tot goede prestaties in staat is indien de atmosfeer zich zoals
aangenomen gedraagt.
De eisen gesteld aan het deformeerbare spiegel systeem worden gerealiseerd met een ontwerp
op basis van elektromagnetische reluctantie actuatoren die via sprieten zijn verbonden
met een reflecterend membraan en worden bekrachtigd door spanningsbronnen op basis van
het Pulse Width Modulation (PWM) principe. Het reflecterende membraan strekt zich over
alle actuatoren uit om zo de optische aberraties die door de spiegel worden geïntroduceerd
te minimaliseren. De actuatoren worden gemaakt in modules met 61 actuatoren geplaatst
in een hexagonaal raster die uit goedkoop te fabriceren lagen bestaan om zo de kosten laag
te houden en accuraat te kunnen assembleren. Er is een Printed Circuit Board (PCB) ontwikkeld
die de aanstuur-elektronica bevat voor een gehele actuator module en voor iedere
actuator een in Field Programmable Gate Arrays (FPGAs) geïmplementeerde PWM generator
en een analoog laag-doorlaat filter bevat. Het PCB wordt achter de actuator module
geplaatst in overeenstemming met het modulaire concept. De FPGAs implementeren naast
de PWM generatoren ook een Low Voltage Differential Signalling (LVDS) gebaseerd communicatie
protocol dat het mogelijk maakt om een deformeerbare spiegel system bestaande
uit 32 modules aan te sturen via een enkele kabel. Om deze aansturing vanuit een standaard
computer of laptop te kunnen doen is bovendien een systeem ontwikkeld dat als brug
fungeert tussen ethernet en LVDS.
Er is een wiskundig model afgeleid dat zowel het statisch als dynamisch gedrag van het
reflecterende membraan, de reluctantie actuator en de aansturings- en communicatie elek-

xvi Samenvatting
tronica van het deformeerbare spiegel systeem beschrijft. Dit model is geverifieerd op verschillende
niveaus: voor losse actuatoren bekrachtigd met een stroombron, voor actuator
modules bekrachtigd met de ontwikkelde PWM spanningsbronnen en op het niveau van het
volledige systeem. Op elk niveau zijn zowel statische als dynamische metingen gedaan die
zijn vergeleken met het model. Gevonden verschillen tussen het gedrag van de losse actuator
en het model worden verklaard door middel van een gevoeligheidsanalyse, hetgeen
leidt tot kleine verbeteringen aan de actuator module en zicht geeft op mogelijke toekomstige
verbeteringen van het ontwerp. Een zevental gerealiseerde actuator modules blijken
zich conform het afgeleide model te gedragen en voldoen aan de eisen zoals die gesteld zijn
wat betreft slag, benodigde kracht en vermogensdissipatie. Dit toont aan dat het maak- en
assemblageproces robust is en de productie van betrouwbare modules mogelijk en haalbaar
is. Zeven ontwikkelde actuator modules worden geïntegreerd met een reflecterend membraan
tot een deformeerbare spiegel met 427 actuatoren. Bij het aanbrengen van een ring
die beschadiging moet voorkomen breekt het membraan helaas, zodat er slechts een spiegel
prototype op basis van een enkele actuator module resteert voor verder onderzoek.
Het reflecterende membraan wordt gemodelleerd op basis van de biharmonische plaatvergelijking,
waarvoor een analytische oplossing bestaat bij belasting met een punt-kracht.
Uit dit model kunnen de invloeds-functies van de actuatoren worden bepaald en door het
model uit te breiden met puntmassa’s die de massa van het membraan representeren biedt
het bovendien inzicht in het dynamische gedrag van het systeem in termen van overdrachtsen
impuls responsie functies en modale vormen. De verificatie van het statische gedrag van
het systeem wordt gedaan op basis van interferometer metingen. Het dynamische gedrag
wordt geverifieerd door de actuatoren aan te sturen met witte ruis signalen en de snelheid
en verplaatsing van het reflecterende membraan te meten met een laser vibrometer. Uit
de excitatie- en meetsignalen wordt met behulp van het Multivariable Output-Error StatesPace
(MOESP) algoritme een state-space model geschat waarvan de overdrachts- en impuls
responsie functies en modale vormen kwalitatief worden vergeleken met die van het model.
Tenslotte worden de regeltechnische aspecten van het systeem geëvalueerd, hetgeen toont
dat de lage demping van de resonanties mogelijk de haalbare regelbandbreedte beperkt,
danwel afhankelijk maakt van de gebruikte bemonsteringsfrequentie.

Symbols
Roman uppercase
ÆÓÑÒÐØÙÖ
Symbol Description Unit
0 vector or matrix whose elements are all zero
1 vector or matrix whose elements are all unity
Aga cross section of the axial air gap [m 2 ]
Agr cross section of the radial air gap [m 2 ]
Am cross section of the actuator membrane suspension [m 2 ]
Aw cross section of the coil winding [m 2 ]
B magnetic field density [T]
Bs magnetic saturation [T]
Bρ influence matrix that links the Pulse Width Modulation (PWM)
voltages to the facesheet deflection at the actuator locations
Bf,w influence matrix that links the PWM voltages to the facesheet
deflection on the measurement grid of the Wyko interferometer
˜Bf,w measured, zero piston, influence matrix that links the PWM
voltages to the facesheet deflection on the measurement grid
of the Wyko interferometer
Bf
influence matrix that links the PWM voltages to the facesheet
deflections at an arbitrary grid of points on the facesheet
C control system
C(s) continuous time controller
C(z) discrete time controller
C1 linear stiffness coefficient [-]
C2 nonlinear stiffness coefficient [-]
Ca
Caf
diagonal matrix whose i th diagonal element is the stiffness ca
of actuator i
stiffness matrix comprehending both the facesheet and actuator
stiffnesses
[m/V]
[m/V]
[m/V]
[m/V]
[N/m]
[N/m]
CFPGA capacitance of the Field Programmable Gate Array (FPGA) [F]
Cl capacitance used in the analog low pass filter [F]
C 2 N Atmospheric turbulence strength [m −2
3 ]
D diameter [m]
Df flexural rigidity [Nm]
Dn index of refraction structure function [-]
Ds diameter of the connection struts [m]
Dt diameter of the telescopes primary mirror [m]
DDM diameter of the Deformable Mirror (DM) [m]
Dφ phase structure function [-]
xvii

xviii Nomenclature
Symbol Description Unit
E Young’s modulus or elastic modulus [N/m 2 ]
Ef Young’s modulus of the mirror facesheet [N/m 2 ]
Em Young’s modulus of the actuator membrane suspension [N/m 2 ]
Es Young’s modulus of the connection strut [N/m 2 ]
Fa actuator force [N]
Fa vector of actuator forces [N]
Fρ net force acting on the facesheet at the actuator location [N]
Fm magnetic force [N]
Fres actuator force resolution [N]
Fs spring force [N]
Fρ
vector of net forces acting on the facesheet at the actuator locations
Fi magnetomotive force in the flux path with index i [A]
G WaveFront Sensor (WFS) system
G WFS geometry matrix [-]
GF Fried geometry matrix characteristic block [-]
GH Hudgin geometry matrix characteristic block [-]
H DM system
H(s) transfer function from voltage to position [m/V]
HI(s) transfer function from current to position [m/A]
Hm(s) transfer function from force to position [m/N]
H ∗ p,Ts(z,θ) discretized transfer function from voltage to position [m/V]
H ∗ v,Ts(z,θ) discretized transfer function from voltage to speed [m/sV]
ˆHp,Ts(z,θ) estimated transfer function from voltage to position [m/V]
ˆHv,Ts(z,θ) estimated transfer function from voltage to speed [m/sV]
ˆH ∗ p,Ts(z,θ) estimated and discretized transfer function from voltage to position
[m/V]
ˆH ∗ v,Ts(z,θ) estimated and discretized transfer function from voltage to
speed
[m/sV]
Hτc Transfer function for the communication latency [-]
HZOH(s) Transfer function of the zero order hold operation [-]
Hbc magnetic field intensity in the coil core [A/m]
Hcm coercivity of the Permanent Magnet (PM) [A/m]
Hga magnetic field intensity in the axial air gap [A/m]
Hgr magnetic field intensity in the radial air gap [A/m]
Hm magnetic field intensity in the PM [A/m]
Hr magnetic field intensity in the core and baseplate [A/m]
I current [A]
I identity matrix
Ia current through the actuator coil [A]
ICl current through the capacitance Cl [A]
If current through the fictitious winding [A]
IRl current through the resistance Rl [A]
J1(·) Bessel function of the first kind [-]
Ja current density in the actuator coil [A/m 2 ]
Ka motor constant [N/A]
Ka diagonal matrix, whose i th diagonal element is the motor constant
ka of actuator i
[N/A]
Km facesheet stiffness matrix [N/m]
[N]

Nomenclature xix
Symbol Description Unit
L length [m]
L0 atmospheric outer scale [m]
L11, L22 self inductance [H]
L12, L21 mutual inductance [H]
La actuator inductance [H]
Ll inductance of the low pass filter [H]
Ls length of the connection strut [m]
Lw length of coil wire [m]
M moment [Nm]
Maf diagonal matrix whose i th diagonal element is the sum of the
moving actuator mass and the lumped facesheet mass at its location
[kg]
N number of windings [-]
Na number of actuators [-]
Nav number of actuators [-]
Nb number of counter bits [-]
Nd demagnetization factor [-]
Nf number of fictitious windings [-]
Nm number of actuator modules [-]
Nn number of distributed controller nodes [-]
Ns number of WFS lenselets [-]
Nw number of pixels used in the Wyko interferometer [-]
P plant to be controlled
P pressure [N/m 2 ]
temporal power spectrum [J/Hz]
Power dissipation [W]
P projection matrix to remove the ’piston’ term [-]
P0 light intensity [lm/m 2 ]
Pa power dissipation in the actuator [W]
Pdyn(fclk) dynamic power dissipation in an FPGA as function of the clock
frequency
[W]
Pe electrical power dissipation [W]
Pload power dissipation due to the load [W]
Psc short circuit power dissipation [W]
Ptot total power dissipation [W]
Qn Hadamard matrix of size nxn [-]
R open-loop control system
R open-loop, minimum variance wavefront reconstruction matrix [-]
Ra electrical resistance of the actuator coil [Ω]
Rc electrical resistance applied for the fine PWM signal [Ω]
Rl electrical resistance of the analog low pass filter [Ω]
ℜbc magnetic reluctance of the part of the baseplate that forms the
core of the actuator coil
[1/H]
ℜc magnetic reluctance of the actuator moving core [1/H]
ℜflc magnetic reluctance of leakage flux path of the coil [1/H]
ℜflm magnetic reluctance of leakage flux path of the PM [1/H]
ℜga magnetic reluctance of the actuator axial air gap [1/H]
ℜgr magnetic reluctance of the actuator radial air gap [1/H]
ℜm magnetic reluctance of the PM [1/H]

xx Nomenclature
Symbol Description Unit
S Strehl ratio [-]
S(s) transfer function from disturbance to residual error [-]
T temperature [K]
Te control loop delay [s]
Ts sampling time [s]
TPWM time periode for the PWM base frequency [s]
Ur three-column matrix containing piston, tip and tilt modes evaluated
on an arbitrary grid
[-]
Uρ three-column matrix containing piston, tip and tilt modes evaluated
on the actuator grid
[-]
V voltage [V]
V matrix of actuator command voltage vectors for identification [V]
Va voltage over the actuator coil [V]
Vcc supply voltage [V]
VC l voltage over the capacitor Cl [V]
vi actuator command voltage vector with indexifor identification [V]
Vm volume of the PM [m 3 ]
Vw coil volume [m 3 ]
VRa voltage over the resistance Ra [V]
W magnetic coenergy [J]
Roman lowercase
Symbol Description Unit
ba mechanical damping in the actuator [Ns/m]
c speed of light in vacuum [m/s]
cD compression factor (=Dt/DDM) [-]
ca actuator stiffness [N/m]
d inter actuator spacing [m]
dt inter actuator spacing projected on the telescope aperture [m]
f frequency of light [Hz]
fc control bandwidth [Hz]
fclk FPGA clock frequency [Hz]
fe undamped mechanical actuator resonance frequency [Hz]
fFPGA FPGA clock frequency [Hz]
fG Greenwood frequency [Hz]
fPWM PWM base frequency [Hz]
fN Nyquist frequency [Hz]
fs sampling frequency [Hz]
g gravitation acceleration [m/s 2 ]
h distance between the core in the undeflected membrane suspension
core and the PM
[m]
h height [m]
hn heat transfer coefficient [W/m 2 ]
l0 atmospheric inner scale [m]
lb magnetic flux path length through the baseplate [m]
lc magnetic flux path length through the moving core [m]
lga magnetic flux path length through the axial air gap [m]

Nomenclature xxi
Symbol Description Unit
lgr magnetic flux path length through the radial air gap [m]
lm magnetic flux path length through the PM [m]
m mass [kg]
mac mass of the moving core in the actuator [kg]
maf mass of the mirror facesheet per actuator [kg]
mf mass of the mirror facesheet [kg]
ms mass of the actuator strut [kg]
n white noise vector
n index of refraction [-]
nair index of refraction of air [-]
r spatial coordinate [m]
r0 Fried parameter [m]
rc communication radius [-]
ri normalized spatial coordinate with indexiin the complex plane [-]
r vector of normalized coordinates in the complex plane [-]
rf mirror facesheet radius [m]
rm actuator membrane suspension radius [m]
s Laplace variable (s = jω) [rad/s]
the number of block-rows used in the Multivariable Output-
Error State-sPace (MOESP) algorithm
[-]
s(t) Open-loop wavefront disturbance measurement vector [rad]
t time [s]
t thickness [m]
tm actuator membrane suspension thickness [m]
tf mirror facesheet thickness [m]
v(h) wind speed at altitudeh [m/s]
v speed of light [m/s]
vw wind speed [m/s]
wx rigid body rotation around the x-axis (tip) [m]
wp rigid body displacement in z-direction (piston) [m]
wy rigid body rotation around the y-axis (tilt) [m]
y(t) Closed-loop wavefront disturbance measurement vector [rad]
za actuator displacement [m]
˙za actuator velocity [m/s]
¨za actuator acceleration [m/s 2 ]
zf facesheet deflection [m]
zf,0 unactuated facesheet deflection [m]
ˆzf measured facesheet deflection [m]
z z-transform variable (z = e jω ) [-]
zia inter actuator stroke [m]
z0 initial axial air gap height [m]
zs suspension membrane deflection [m]
Greek symbols
Symbol Description Unit
α linear coefficient of expansion [m/m/K]
switching activity in an FPGA [-]

xxii Nomenclature
Symbol Description Unit
integrator gain [-]
β integrator leak factor [-]
γw command vector scaling constant [m]
Γv diagonal scaling matrix on command voltages [-]
Γz diagonal scaling matrix on measured displacements [-]
δij Kronecker delta [-]
ǫ(t) vector of wavefront phase at time t [-]
ζ telescope angle w.r.t. Zenith [ ◦ ]
η actuator coupling [-]
θ rotation around the z-axis [rad]
θ angular coordinate [-]
θa angular distance between object and reference star [rad]
Θ angle of the chief ray w.r.t. the optical axis [rad]
κ spatial frequency [1/m]
κx spatial frequency in x direction [1/m]
κy spatial frequency in y direction [1/m]
κz spatial frequency in z direction [1/m]
κf fitting error coefficient [-]
λ wavelength of light [m]
thermal conductivity [W/mK]
flux linkage [Wb]
Λ diagonal scaling matrix on command voltages [-]
µ0 magnetic permeability of vacuum [N/A 2 ]
µr relative magnetic permeability [-]
µrm relative magnetic permeability of the PM [-]
µr b relative magnetic permeability of the baseplate [-]
ν Poisson ratio [-]
νm Poisson ratio of the actuator membrane suspension material [-]
νf Poisson ratio of the mirror facesheet material [-]
ρ material density [kg/m 3 ]
ρm density of the actuator membrane suspension material [kg/m 3 ]
ρa density of air [kg/m 3 ]
ρf density of the mirror facesheet material [kg/m 3 ]
ρi complex coordinate with index i [m]+j[m]
ρs density of the connection strut material [kg/m 3 ]
σ 2 angle wavefront variance due to off axis observation angle [nm 2 ]
σ 2 cal wavefront variance due to calibration errors [nm 2 ]
σ 2 ctrl wavefront variance due to a control error [nm 2 ]
σ 2 delay wavefront variance due to a delay [nm 2 ]
σ 2 fit wavefront variance due to limited number of actuators [nm 2 ]
σ 2 meas wavefront variance due to measurement errors [nm 2 ]
σ 2 n wavefront variance due to measurement noise [nm 2 ]
σ 2 temp wavefront variance due to limited bandwidth [nm 2 ]
σ 2 total total wavefront variance [nm 2 ]
σ(v,i) expected Root Mean Square (RMS) actuator voltage based on a
Von Karmann spatial power spectrum
σ 2 wf wavefront variance [nm 2 ]
τ delay [s]
τe Sensor integration time [s]
[V]

Nomenclature xxiii
Symbol Description Unit
τc communication latency [s]
τUDP delay caused by the User Datagram Protocol (UDP) packet
transfer
[s]
τLV DS delay caused by the Low Voltage Differential Signalling
(LVDS) packet transfer
[s]
φ optical phase [rad]
φ(t) vector containing wavefront phase disturbance values [rad]
φi magnetic flux in a circuit with index i [Wb]
Φ spatial PSD [rad 2 ]
magnetic flux [Vs]
Ωρρ facesheet compliance matrix w.r.t. the actuator grid [m/N]
Ωrρ facesheet compliance matrix mapping forces at the actuator locations
ρ to displacements at an arbitrary gridr
[m/N]
Operators and sets
.
Symbol Description
∇ Laplacian operator (partial derivative)
⊗ Kronecker product
◦ Hadamard (element-wise) product
Tr(·) Trace of the dotted matrix
\ set difference operator
〈·〉 expected value of the dotted expression
Nn(m,C) set of ergodic white noise signals s(t) ∈ R n with mean m and covariance
(matrix) C
n
S(i) Union of the sets S(i), equivalent toS(1) ∪...∪S(n)
i=1
·F
δij
Frobenius norm of
the dotted expression
1 for i = j,
Kronecker delta:
0 for i = j.

Ê Ç Ê ÊÅ ËÁ
Analog to Digital Convertor
Aliased Frequency Response
ËÅ
Function
Æ
Adaptive Optics
Auto-Regressive
ÀÌ
Auto-Regressive Moving
Average
ÈÍ
Application-Specific Integrated
ÅÇË
Circuit
Adaptive Secondary Mirror
Æ
Controller Area Network
Ë
Charge Coupled Device
Canada France Hawaii
ÁË
Telescope
Central Processing Unit
Å
Complementary Metal Oxide
Ç
Semiconductor
Computer Numerical Control
Å
Curvature Sensor
ÄÌ
Digital to Analog Convertor
Dutch Institute for Systems and
ÄÌ
Control
Ê
Deformable Mirror
Degrees Of Freedom
ËÇ
Direct Current
Electrical Discharge Machining
Å
European Extremely Large
Ì
Telescope
ÁÊ
Extremely Large Telescope
ÇÎ
Eigensystem Realization
Algorithm
European Southern
Observatory
Finite Element Model
Field Effect Transistor
Finite Impulse Response
Field Of View
xxiv
ÖÓÒÝÑ×
È Ê ÏÀÅ ÄÇ ÀËÌ Á ÁÈ
Field Programmable Gate
ÁÊ
Array
ÂÏËÌ
Frequency Response Function
ÄÌ
Full Width Half Maximum
ÄÅÁ
Ground Layer Adaptive Optics
ÄÅË
Hubble Space Telescope
Integrated Circuit
ÄÈÎ
Internet Protocol
ÄÌÁ
Infra Red
ÄÎË
James Webb Space Telescope
Large Binocular Telescope
ÄÉ
Linear Matrix Inequality
ÄË
Least Mean Squares
Å
Computer Aided Design
Å
Linear Parameter Varying
ÅÇ
Linear Time-Invariant
Low Voltage Differential
ÅÇÇ
Signalling
Linear Quadratic Gaussian
Least Significant Bit
ÅÅË
Moving Average
Media Access Control
ÅÁÅÇ
Multi-Conjugate Adaptive
ÅÅÌ
Optics
ÅÌ
Multi-Object Adaptive Optics
ÅÇËÈMultivariable
ÅË
Output-Error
ÆÈ
State-sPace
Micro-Electro-Mechanical
ÇÈ
Systems
ÇÏÄ
Multi-Input Multi-Output
Multiple Mirror Telescope
Mean Time Before Failure
Most Significant Bit
Non Common Path Aberration
Optical Path Difference
Overwhelmingly Large

ÈËÁ È È Ë Acronyms
ÈÀ ËÀË
xxv
ÈÅÆ ËÁËÇ
ÈÅ ËÆÊ
Telescope
ÈË ËÎ
Predictor Based Subspace
ÈË ÌÈ
IDentification
ÈË Ì
Ranging
Printed Circuit Board
ÈÌÎ ÌÅÌ
Steepest Descent
Partial Differential Equation
ÈÏÅ ÍÈ
Shack Hartmann sensor
PHYsical layer (ethernet)
ÉÈ ÍÄ
Single-Input Single-Output
Lead Manganese Niobate
ÊÅ ÍË
Signal to Noise Ratio
Permanent Magnet
ÊÄË Î
Singular Value Decomposition
Power Spectral Density
ÊÅË ÎÄÌ
Transmission Control Protocol
Point Spread Function
ÊÌ ÏË
Transfer Function
ÏÀÌ
Thirty Meter Telescope
ÏËË
User Datagram Protocol
Pyramid Sensor
ÇÀ
Ultra Low Expansion
Peak To Valley
ÍÈ
Universal Serial Bus
Pulse Width Modulation
Variance Accounted For
Quadratic Programming
Very Large Telescope
Random Access Memory
WaveFront Sensor
Recursive Least Squares
William Herschel Telescope
Root Mean Square
Wide Sense Stationary
Real-Time Control
Zero Order Hold
ËÁÊSCIntillation Detection and
Xylinx University Program

ÔØÖÓÒ
ÁÒØÖÓÙØÓÒ
The field of astronomy is briefly introduced to show the relevance of Adaptive
Optics (AO) systems. The components of such a system are described and
challenges for future AO systems are discussed. Finally, an overview of this
thesis is given.
1

1
2 1 Introduction
1.1 Notes on notation
Before introducing the subject of adaptive optics, note that the mathematical symbols and
operators that are frequently used throughout this thesis are listed in the nomenclature on
page xvii. Scalar values are generally denoted by italic lower case symbols, whereas vectors
are denoted by boldface symbols. Matrices are denoted by capital, uppercase symbols and
both sets and systems are denoted by calligraphical symbols. The (i,j) th elements of a
matrixAis denoted by a subscript between square brackets: A [i,j] and thei th element of a
vectorxis denotedx [i]. A subscript with round brackets (e.g. a (i)) denotes an enumeration
of the subscripted symbol, whereas subscripted symbols without brackets such asav denote
specific unique symbols.
Further, Ia denotes an identity matrix of size a × a, where the subscript dimension will
be omitted if this is clear from the context. The complex-valued z-transform variable z is
used to indicate discrete-time systems and denotes the forward temporal shift operator. The
corresponding continuous time Laplace operator is denoted by the symbol s. The notation
〈·〉 denotes the statistical expected value of the dotted expression over time t. In case that
the expected value is not taken w.r.t. time this is made clear in the context. The average
value over a specific time range is denoted as:
〈·〉 t1
t0 = 1
t1−t0+1
t1
·(t).
Further, the notation · F denotes the Frobenius norm of the dotted expression and δij
denotes the Kronecker delta s.t.
1 for i = j,
δij =
t=t0
0 for i = j.
The set N n (m,C) comprehends ergodic, normally distributed white noise signals x(t) ∈
R n with mean m = 〈x〉 and covariance C = xx T for which x(t)x T (t−p) = 0 for
p = 0. The union of two setsS1 andS2 will be denoted asS1∪S2 and the union of multiple
setsS (i) fori = 1...m will be denoted:
m
S (i) = S (1) ∪S (2) ∪...∪S (m).
i=1
The set S that consists of all elements in S1 that are not present in the set S2 is denoted
S = S1 \S2.
Further, letG [S1,S2] denote the sub-matrix obtained from a matrixGby extracting all rows
i ∈ S1 and columnsj ∈ S2 thereof in lexicographical order. Similarly,p [S] denotes a vector
obtained by stacking the elementsp [i] fori ∈ S.
1.2 Astronomy
The invention of the telescope is commonly attributed to Lipperhey and Janssen around
1608, although this is a historically debated subject. The first astronomers used transmissive

1.2 Astronomy 3
Flat, undistorted wavefront
Average refractive index
Lower than average refractive index
e.g. higher than average temperature
Higher than average refractive index
e.g. lower than average temperature
Distorted wavefront
t0
t0+t
Figure 1.1: Wavefront distortions caused by variations in the index of refraction of air. Light enters
at the top and passes through hotter and colder than average air bubbles. As a result, the
corresponding rays are retarded and advanced respectively.
lenses with satisfying results on small telescopes, but they soon realized that to obtain more
detailed images a larger aperture diameter was required. The quality of glass castings was
limited, hence the reflectivity of polished metals like copper and tin was used for larger
telescopes. On the other hand, polishing large mirrors was specialized and tedious work
and it took more than 80 years [98] before it was realized that the limited image detail of
1m+ telescopes was not always caused by polishing errors. Although Christian Huygens
proposed around 1656 that the atmosphere was to blame, this was recognized by the
majority of astronomers only at the end of the 19th century. Ever since, the astronomers’
desire for more and more detailed images of the sky has been obstructed by our turbulent
atmosphere.
This introduces quickly changing aberrations to the wavefront of the incoming light that
limit the spatial resolution of the images that can be recorded. The wavefront is the surface
formed by all light particles emitted by an object at the same time, which – in the absence
of disturbances – can be regarded as an expanding sphere. Most celestial objects are so
distant, that when their light arrives at the earth, the wavefront sphere is so large that the
small fraction that will penetrate the atmosphere and fall onto an observing telescope can
be considered flat.
The atmospheric wavefront distortion (unflatness) is the result of variations in the index
of refraction of air. This index represents the speed of light in vacuum relative to that in
air and mainly depends on temperature, but also on humidity and pressure. Since these
parameters vary spatially in the atmosphere, some parts of the wavefront are advanced,
whereas others are retarded (figure 1.1).
With an ideal circular telescope – one whose components do not introduce aberrations
– and without the presence of the atmosphere a point source is shaped by diffraction on its
finite components and is described by the Airy function:
P0(θ) = πD2
4λ 2
⎡
⎣ 2J1
πD|θ|
λ
πD|θ|
λ
Here P0 is the light intensity as function of the angular coordinate θ, λ the wavelength,
D the telescope diameter and J1(·) the Bessel function of the first kind. The Airy function
⎤
⎦
2
1

1
4 1 Introduction
Figure 1.2: The Airy function. Figure 1.3: The Airy function (top view).
is shown graphically in figures 1.2 and 1.3. The first dark ring is at an angular distance of
1.22 λ
D and is called the resolution of the ideal telescope. An astronomical object can be seen
as a number of point sources, each of which spreads according to the Airy function. Mathematically,
the Airy function is an optical transfer function, such that the object’s image is
obtained by convolution of the objects source function with the Airy function. This results
in the image with the least degradation possible and the resolving power of the telescope is
called diffraction limited. In practice, the image is never diffraction limited, but further degraded
due to wavefront distortions introduced by the atmosphere and telescope optics. The
image is now obtained through convolution with a distorted Airy function, which is referred
to with the more general term Point Spread Function (PSF). The PSF is thus determined by
wavefront aberrations and forms the basis for various measures of optical quality:
• Strehl ratio: S(θ) = P(θ)
· 100%. This is the actual central peak intensity (P(θ))
P0(θ)
relative to the diffraction limited central peak intensity of the PSF (P0(θ)).
• The Full Width Half Maximum (FWHM) of the PSF, which is the diameter around
the central peak at which the PSF reaches half the peak value.
• Encircled energy. This measures the energy distribution in the PSF as the energy
fraction contained within a circle of radiusr.
It is only in the last fraction of a second when the light passes through the atmosphere to
arrive at a telescope that its wavefront gets distorted. However, for a long time no solution
to this problem could be found. To obtain the best image of the actual object the wavefront
distortion should either be compensated or the image should be taken before the light hits
the atmosphere. The first ideas towards wavefront compensation were published by Horace
W. Babcock in 1953 [12], which is considered to be the start of the field of Adaptive
Optics (AO). However, no actual compensation was achieved until 20 years later.
Atmospheric wavefront distortions can also be evaded by placing the telescope in space.
Although this was no realistic option for the early astronomers, by now this has been successfully
proven by the clear images of the Hubble Space Telescope (HST). Its successor –
the James Webb Space Telescope (JWST) – will soon be launched, having a much larger primary
mirror. Nevertheless, space telescopes have several important drawbacks: the space
launch is expensive and requires extreme precautions for the fragile telescope; space is a
harsh environment and maintenance is nearly impossible. Consequently, the primary mirror
area of space telescopes is much more difficult to extend than that of their earth based counterparts.
Especially, since AO systems have now reached the level of maturity that allows

1.3 Adaptive optics 5
Object
Science
image
Atmospheric
turbulence
Telescope
Deformable Mirror
Controller
Wavefront
sensor
Figure 1.4: Schematic of the
adaptive optics system in a
ground based telescope
the construction of large telescopes with almost diffraction limited seeing. Recently, even
the power of using several wavefront correctors simultaneously was demonstrated on-sky
using the MAD demonstrator [126].
AO also finds applications in high power laser systems, optical communication and
medicine with each their own specific challenges. This thesis will focus on astronomy,
starting with a description of the system components that are relevant for this application
field in the next section.
1.3 Adaptive optics
A schematic of an astronomical telescope with an AO system is shown in figure (1.4). After
reflecting on the primary and several other mirrors of the telescope, the distorted light reflects
on the wavefront corrector, which is here assumed to be a Deformable Mirror (DM).
In practice several other optical components may be present before and after the corrector,
but eventually a dichroic beam splitter splits the wavefront into two parts. One part proceeds
to the WaveFront Sensor (WFS) and the other to the science camera that records the image.
Based on current and past WFS measurements the Real-Time Control (RTC) calculates new
actuator signals for the DM.
As mentioned, the intended application field for the DM developed within the context of
this thesis is astronomy. In the coming sections the system components relevant for this
field will be described in more detail.
1.3.1 Atmospheric turbulence
As mentioned, the index of refraction of air depends a.o. on its temperature and humidity.
The atmosphere is often represented as a set of different size air bubbles for which these
properties are different, giving each their own refractive index n. Wind carries these air
bubbles over the telescopes aperture, while only very slowly changing these properties. The
1

1
6 1 Introduction
Figure 1.5: Schematic of
a Shack-Hartmann wavefront
sensor.
plane
wavefront
aberrated
wavefront
focal length
CCD
detector
latter is called the frozen flow assumption or Taylor hypothesis. At most telescope sites a
large part of the turbulence is at the lower altitude, where a temperature gradient between air
and ground exists [178]. This is called the ground layer and typically 75% of the distortion
is caused by the lower 2km of the atmosphere.
In 1941 Kolmogorov laid the foundation for currently used atmospheric turbulence models
[112]. Kolmogorov concluded that in a turbulent flow the kinetic energy decreases with
the −5
3 power of the spatial frequency. From this, Tatarski [173] and Fried [63] developed
the standard model for astronomical seeing. In chapter 2 a more detailed description of
atmospheric turbulence and its consequences for the DM design is given.
1.3.2 The wavefront sensor
Wavefront sensors for AO systems come in many types. Examples are Shearing interferometers,
Curvature Sensor (CS) [148], Pyramid Sensor (PS) [145] and Shack Hartmann
sensor (SHS) [160]. All mentioned sensors use an indirect method to measure the actual
wavefront shape (phase) and require a mathematical transformation to derive this information
from the measurements.
The CS uses an array of lenses to focus the wavefront into a multitude of spots. A detector
measures the intensities before and after the focal plane. If there is a local curvature in the
wavefront, the focus position of the spot is changed, leading to variation of the measured
intensities. In the PS the wavefront falls onto many small pyramids. The facets of each
pyramid split the light into a number of beams that are imaged onto a detector. If the
wavefront is flat the result is an equal amount of light in all beams, but otherwise this
distribution changes. Extra resolution can be gained by mechanically moving the pyramids
[145].
For various reasons the SHS is currently the most widely used in AO systems for optical
telescopes and will be considered throughout this thesis. It consist of a two dimensional
array of lenselets of which each lens images incident light onto a Charge Coupled
Device (CCD) detector or quad-cell array (figure 1.5). The position offset of this image
with respect to a reference is a measure for the first spatial derivative of the incoming
wavefront at the location of a lenselet.
Due to this design of the sensor, its dynamics consist of both a spatial and a temporal
displacement

1.3 Adaptive optics 7
ǫ3 ǫ6 ǫ9
ǫ2
ǫ1
y2
y1
ǫ5
ǫ4
y4
y3
ǫ8
ǫ7
Figure 1.6: The Fried geometry. The measured
gradients (arrows) are related to the values of
the four surrounding phase points (circles) via
the definition in (1.1). This definition enables
the unobservable waffle mode indicated by the
checkerboard coloring of the phase points.
ǫ3 ǫ6 ǫ8
ǫ2
ǫ1
y2
y1
ǫ5
ǫ4
y4
y3
ǫ7
Figure 1.7: The Hudgin geometry. The measured
gradients (arrows) are related to the values
of three neighboring phase points (circles)
via the definition in (1.2).
part. The temporal part of the dynamics of the wavefront sensor arises from the working
principle of the underlying CCD camera or quad-cell array. These detectors collect incoming
photons, effectively integrating light energy over the exposure time. The integrating
behavior corresponds to a low-pass filter characteristic. Since these photons must be used
as efficiently as possible, this exposure time is usually equal to the sampling time of the
AO system. This implies an average measurement delay of half a sampling time, to which
read-out and post-processing times must be added, justifying the assumption of a full
sample delay that will be made throughout this thesis.
In practice, the spatial SHS dynamics are modeled via either modal or zonal approaches.
For modal approaches a set of orthogonal, differentiable basis functions is defined such as
the Zernike polynomials [136]. Both the polynomials themselves as well as their spatial
derivatives are analytic functions that can be discretized over the sensor grid. For wavefront
reconstruction, these discretized functions can be used to estimate the modal coefficients for
a given set of gradient measurements. The actual wavefront shape can then be determined
on an arbitrary grid as a weighted sum of the Zernike polynomials discretized on this grid.
For zonal approaches such as the Fried and Hudgin geometries [179], the wavefront gradients
are modeled as a function of the values of a few surrounding phase grid points. Let
the residual wavefront phase distortion that determines the SHS measurements at time t be
denoted by a vector ǫ(t) ∈ R Nǫ . For the Fried geometry it is assumed that the wavefront
phase is discretized over a virtual phase grid as defined in figure 1.6. When including one
full sample measurement delay and measurement noisew (i)(t), each gradient measurement
1

1
8 1 Introduction
vectory (j)(t) ∈ R 2 is expressed in terms of the four surrounding phase values as:
y (j)(t) = 1
⎡
ǫ
1 1 −1 −1 ⎢
2 1 −1 1 −1 ⎣
GF
j
tr(t−1)
ǫ j
br (t−1)
ǫ j
tl (t−1)
ǫ j
bl (t−1)
⎤
⎥
⎦+w
(i)(t), (1.1)
ǫS (t−1)
(j)
where the notationǫ j
tl (t) refers to the residual wavefront disturbance at the phase grid point
that lies top-left (tl) adjacent to sensor grid point j. Similarly, bl, tr and br refer to the
bottom-left, top-right and bottom-right points respectively. The indices of the corresponding
nodes form the four elements of the setS (j). For instance, in figure 1.6 this implies that
S (2) = {6,5,3,2} andS (3) = {8,7,5,4} .
For the Hudgin geometry the spatial gradients are defined as illustrated in figure 1.7. Correspondingly,
the measurementsy (j)(t) are expressed as:
y (j)(t) = 1
⎡
ǫ
1 −1 0
⎣
2 0 1 −1
GH
j
tr(t−1)
ǫ j
tl (t−1)
ǫ j
bl (t−1)
⎤
⎦+w
(i)(t), (1.2)
ǫS (t−1)
(j)
When y (j)(t), ǫ (i)(t) and w (i)(t) are stacked in the vectorsy(t) ∈ R 2Ns , ǫ(t) ∈ R Nǫ and
w(t) ∈ R 2Ns respectively, then for both geometries the measurements can be expressed as:
y(t) = Gǫ(t−1)+w(t).
For the Fried geometry, each 2×Nǫ block-row of the generally tall matrix G ∈ R 2Ns×Nǫ
contains only the four non-zero columns of GF at the columns corresponding to the definition
of ǫS (j) in (1.1). Similarly, for the Hudgin geometry each 2×Nǫ block-row of the
matrix G contains the three non-zero columns of GH at the columns corresponding to the
definition ofǫS (j) in (1.2).
The geometry matrix is rank deficient for both geometries. The Hudgin geometry has one
unseen wavefront shape that yields a zero measurement, which is called the piston mode.
For this mode, all phase values are equal, i.e. ǫ (1)(t) = ... = ǫ (Nǫ)(t).
The Fried geometry has two unseen modes: besides the piston mode it also has the waf-
fle mode. The wavefront shape corresponding to the latter has phase values such that
ǫ j
tl
(t) = ǫj
br
(t) and ǫj
bl
(t) = ǫj tr(t) ∀ j = 1...Ns. For this shape, the values of
diagonally adjacent phase points are thus equal, which corresponds to the checkerboardlike
phase point coloring in figure 1.6. The implications of these unseen modes for control
will be addressed in the following chapters.
1.3.3 The wavefront corrector
The wavefront corrector performs the physical correction of the wavefront. A wide variety
of wavefront correctors exists. The goal of this section is to show the diversity and to point
out the main properties of the different correctors.

1.3 Adaptive optics 9
Probably the oldest wavefront corrector is the segmented mirror. This mirror consists of a
number of small, closely packed mirror segments that can move in one or three Degrees
Of Freedom. In the first case the individual mirror elements can only move up and down
(piston) along the optical axis. In the second case each mirror segment can rotate over
two orthogonal axes of tilt as well. Piezoelectric actuators and strain gauges are most
commonly used to move the segments and to provide position feedback. One example
is the segmented mirror from ThermoTrex Corporation, which is used on the 4.2 meter
William Herschel Telescope (WHT) [198]. This mirror has 76 mirror segments, each of
which have tip/tilt and piston actuation giving a total of 228 Degrees Of Freedoms (DOFs).
Other examples can be found in [37], [104], [97] and [76]. By having separated segments
there is no cross coupling and mirror parts can be relatively easily replaced. Furthermore,
as will be discussed in the following chapters, the lack of cross coupling between actuators
simplifies the control design. These advantages come at the cost of small gaps between the
segments that act as an optical grating and cause diffraction.
Another type of wavefront corrector is the DM. Most DM’s have continuous facesheets
that are deformed out-of-plane by stacks of piezoelectric actuators placed perpendicularly
under the reflective surface. This type of DM has been under development since 1974 and
was first built for high energy laser systems [179]. At the end of the ’70s these mirrors were
also developed for infrared systems [46, 47]. Current development of this type of mirror
is driven by miniaturization [164], increasing actuator linearity, stroke [154] and position
accuracy, decreasing operating voltages, drift [28, 154] and hysteresis [165]. Piezo stacked
deformable mirrors are made a.o. by Xinetics, CILAS and OKO Technologies. One of
the largest AO-systems with a piezo stacked mirror is in the 10-meter Keck telescopes.
Here a 349-channel piezoelectric mirror from Xinetics is implemented [194]. Besides
piezoelectric materials also Lead Manganese Niobate (PMN) or magnetostrictive actuators
are used.
A separate class of continuous facesheet DM’s are the bimorph mirrors. Unlike the DM’s
with stacked piezo actuators, bimorph mirrors have actuators placed parallel to the reflective
surface. A bimorph mirror usually consists of a glass or metal facesheet that is bonded
to a sheet of piezoelectric ceramic. There’s a conductive electrode in the bond between
the piezoelectric material and the facesheet. On the backside of the ceramic a series of
electrodes is attached. When a voltage is applied between the front and back electrode the
dimensions of the piezoelectric material change and a local radius of curvature is forced
into the mirror. Bimorph mirror were first used in astronomy in the beginning of the ’90s on
the Canada France Hawaii Telescope (CFHT) [149]. One of the largest bimorph mirrors is
a 188-element bimorph mirror, developed by CILAS, and currently used in the AO-system
for the 8.2-meter SUBARU telescope. This mirror is 130mm across, but only the inner
90mm is illuminated [172]. The remaining 40 electrodes outside this diameter are needed
to enforce the proper boundary conditions [140].
In bimorph mirrors the local curvature is proportional with the voltage and the coefficient
of the dielectric tensor and inversely proportional with the square of the thickness. The
maximum voltage is given by the breakdown voltage. This also determines the gap between
the electrodes and thereby sets a limit for the actuator density. Since the mechanical
resonance frequency is mainly determined by the diameter-thickness ratio it is clear that a
trade-off between mirror size, resonance frequency and stroke (curvature) is to be made.
Critical in the design are the bonds between the different layers. Bimorph mirrors suitable
1

1
10 1 Introduction
for high power lasers with integrated cooling have also been developed [7, 156, 190].
Besides piezo stacked and bimorph mirrors a few implementations exist with actuators that
introduce bending moments at the edge of the mirror [67].
To reduce the background emissivity from surfaces added by the AO system the number
of reflective surfaces in astronomical telescopes should be kept to a minimum. This is
especially the case for Infra Red (IR) observations. From this thought the idea for an Adaptive
Secondary Mirror (ASM) was born in the ’90s [155]. In contrast with the previously
discussed correctors, secondary mirrors in a telescope are usually strongly curved, giving
additional challenges in making them adaptive. The first ASM was built for the 6.5m
Multiple Mirror Telescope (MMT) in Arizona in the mid ’90s and has 336 actuators [120].
The static secondary mirror was replaced by a thin deformable zerodur shell with a radius
of curvature of 1795mm. The shell is 1.9mm thick and 640mm in diameter [127]. In the
center of the shell a membrane suppresses the lateral DOF’s. A total of 336 small, radially
magnetized permanent magnets are glued to the backside of the zerodur shell and together
with voice coils that are fixed in a reference plate form actuators that push and pull at the
shell. Capacitive sensors are placed concentrically with the actuators in between the Ultra
Low Expansion (ULE) glass reference structure and the backside of the thin shell. They
provide distance measurements for the local feedback loops. A 30mm thick aluminium
plate with cooling channels is used to drain the produced heat [24, 25, 101, 181, 182].
After the conversion at the MMT, two ASM’s were made for the Large Binocular
Telescope (LBT). The ASMs for both of its telescopes have a radius of curvature of
1974.2mm and measure 911mm across. To reduce the deformation forces and resulting
power dissipation a 1.6mm thick zerodur shell was chosen. Each of these shells have 672
electromagnetic actuators [68].
One of the Very Large Telescope (VLT)’s will also be equipped with an ASM for which
first light is foreseen in 2015. This mirror has a radius of curvature of 4553mm, measures
1120mm across and is equipped with 1170 actuators [9].
The above ASMs exhibit a few drawbacks, one of which is their high complexity. Due to
the lack of mechanical stiffness in the thin shell, hundreds of eigenfrequencies lie below or
around the desired control bandwidth and need to be dealt with by the control system. To
achieve this, each actuator is equipped with a capacitive sensor and associated electronics
and needs a significant amount of computational power for closed-loop control [193].
Research on controlling this thin shell is still ongoing [151]. Since the power consumption
is high (MMT:2kW [120], LBT:2.665kW [18], VLT:1.47kW [9]), active fluidic cooling is
needed for which leakage is known to occur [68]. Furthermore the gap between the thin
shell and the reference structure is only ∼ 50µm, rendering contamination a serious risk
[68]. The assemblies have high masses (MMT:130kg, LBT:250kg, VLT:180kg), resulting
in mechanical resonances at low frequencies. As a result, winds causes the assembly hub
to resonate in its metering structure leading to optical degradation. At the MMT, these
resonances lie at 14Hz for the rotation mode perpendicular to the optical axis and 19Hz
for the mode along the optical axis. Additional measures had to be taken to reduce the
detrimental effects hereof [167].
The last type of continuous facesheet deformable mirror to be discussed here is the
membrane mirror. A very thin (

1.3 Adaptive optics 11
a voltage to the electrostatic electrode actuators it is possible to deform the membrane. In
most cases a bias voltage is applied to all the electrodes, to make the membrane initially
spherical. This way, the membrane can be moved in both directions. Probably the most
widely spread example is the 37 actuator electrostatic deformable mirror from OKO
Technologies [122, 180]. Due to the thickness of the membrane these mirrors are very
fragile. Critical in the design is to avoid possible snap down and avoid dust in the very
narrow gaps. Another actuation method on membrane mirrors can be found in [38, 39].
Here a small magnet is suspended by the membrane and coils are used to exert a force and
deform it. Since no mechanical stiffness exists, scaling to large diameters is not possible
while retaining inter actuator stroke and density as well as dynamic properties.
Micro-Electro-Mechanical Systems (MEMS) devices form another class of small DMs.
With the potential to be fabricated in large quantities and with large numbers of actuators
this seems a promising technique. Nevertheless, most MEMS suffer from limited
(inter)actuator stroke and poor surface quality. MEMS DMs are manufactured by Boston
Micromachines and Iris AO.
Finally, note that not all wavefront correctors are based on reflection. High-order transmission
based correctors are also available [121]. Most of these correctors are based on
liquid crystals and are limited in stroke and dynamic behavior. For astronomy this requires
them to be used in woofer-tweeter configurations, where they are located after low-order
DMs that first correct the large, slow part of the wavefront distortion. Such multi-mirror
configurations will be discussed in the next paragraph.
1.3.4 Optical configurations
Although most AO systems only have a single DM, more advanced configurations have
been designed that use multiple wavefront correctors. Such configurations allow a larger
Field Of View (FOV) and/or the observation of multiple stars simultaneously. The FOV is
the area of the sky that can be observed with a certain resolving power at a single moment
in time (i.e. without rotating the telescope). Using a single DM for wavefront correction the
and the light of a single bright star (natural guide star) for wavefront sensing, the correction
quality will degrade as the object under observation is located further away from the guide
star. The area around the guide star for which the wavefront errors can be successfully
compensated is called the isoplanatic patch and forms the telescope’s useful FOV. When
laser beacons (laser guide stars) [98] are used to supply the light for the wavefront sensor,
the location of the guide star can be chosen, thus reducing the problem of anisoplanatic
errors. However, laser guide stars introduce errors of a different, more complex character
that are outside the scope of this thesis. The multi-mirror configurations that will now be
briefly discussed, share the concept of using not only multiple wavefront correctors, but
also multiple wavefront sensors that are aimed at different guide stars.
A first approach to allow observation of multiple stars simultaneously is Multi-Object
Adaptive Optics (MOAO) [10]. Since the light of each object has a different path through
the turbulent atmosphere, the wavefront disturbance slightly varies per object. In MOAO
the incoming starlight is split in the focal plane into multiple areas that are each corrected
by a separate mirror, such that multiple objects over a wider FOV can be observed with
1

1
12 1 Introduction
AO on the same telescope. The wavefront disturbance information relevant for each area
is extracted from the measurements of multiple wavefront sensors using atmospheric tomography
[10, 146, 196]. By combining the measurements of sufficiently many wavefront
sensors at known locations and orientations, atmospheric tomography allows to reconstruct
the wavefront at any point – up to a certain resolution – in the atmosphere. For MOAO
these points are chosen within the area of the object in the focal plane for which correction
is desired.
Tomography is also used for another approach that leads to an increased FOV, called Multi-
Conjugate Adaptive Optics (MCAO). Here each wavefront corrector is optically conjugated
to a specific atmospheric turbulence layer. The effect of these layers can be corrected
based on information extracted from measurements of multiple wavefront sensor using atmospheric
tomography.
A last approach, which allows the observation of multiple objects while using only a single
corrector is called Ground Layer Adaptive Optics (GLAO). This is based on the previously
mentioned fact that the lowest layer in the earth’s atmosphere (the ground layer) is the most
detrimental (figure 2.2). Correction of this layer leads to a significant increase of the optical
quality over a very wide FOV. Therefore, the single corrector is optically conjugated to the
ground layer and atmospheric tomography on measurements of multiple WFSs provides the
necessary disturbance information.
1.3.5 The control system
A last, but vital part of the AO system is the control system. Based on mathematical formulae,
this processes wavefront sensor measurements in real time to determine suitable
command setpoints for the actuators. The formulae consist of several parts [98, 100]. As
mentioned the WFS sensors are often CCD based and require both an image processing and
a mathematical transformation step to determine the wavefront phase shape. Subsequently,
the formulae include models of the behavior of the wavefront corrector and if desired the
spatial and/or temporal dynamics of the behavior of the (atmospheric) wavefront distortion.
The ultimate goal of the control system is to compensate this distortion, the quality of which
can e.g. be measured using the Strehl ratio (section 1.2). According to the Maréchal approximation
[20, 179], this ratio is inversely proportional to the variance of the wavefront
aberration (section 2.2). Therefore, the goal for the control system can be formulated more
specifically as to minimize this variance. In other words, the AO control problem is to
find the control law formulae that lead to the smallest residual wavefront aberration in a
least squares sense. When assuming the noise and disturbance to be generated from Gaussian,
white noise, this means that the control problem for AO fits the Linear Quadratic
Gaussian (LQG) andH2 optimal control frameworks [100]. This will be further elaborated
in chapter 3.
Implementation
At the time when the first AO system was built, computers were far too slow to suitably
implement a control law and dedicated analog circuits were used instead [98]. But the computational
hardware of AO systems has always been cutting edge, as required by the fast
update rates, high numbers of actuators and sensors and the high cross-coupling required

1.3 Adaptive optics 13
WFS
RTC
corrector
incoming
starlight
beam
splitter corrector
science
camera
Figure 1.8: Schematic of an AO system configures
in open-loop.
incoming
starlight
RTC
beamsplitter
WFS
science
camera
Figure 1.9: Schematic of an AO system configured
in closed-loop.
between controller inputs and outputs. Nowadays, the formulae are implemented on stateof-the-art
digital computing hardware. For instance, the control system for the MAD system
[58] is implemented on four power-PC’s. For other, large AO systems dedicated Field Programmable
Gate Array (FPGA) boards are often used that can perform many calculations
in parallel [70]. The data processors obtain the measurements via a fast, usually digital
communication link from the WFS and then apply the control law to determine the actuator
commands. These are then communicated to Digital to Analog Convertors (DACs) and
applied to the actuators.
Control configurations
AO systems can be configured in both open-loop (figure 1.8) as well as closed-loop (figure
1.9). Both configurations have their advantages and both are used in practice. In the openloop
configuration, the measurements are not influenced by the shape of the corrector and
provide direct information on the wavefront distortion.
However, in case of strong turbulence the wavefront distortion may exceed the range of
the WFS, leading to poor performance. In closed-loop control the sensor measures the
corrected wavefront, which requires a smaller sensor range. Further, as the effect of the
control actions is not observed by the open-loop control system, this must completely rely
on a model that accurately describes the behavior of the wavefront corrector. But for the
same reason this model cannot be calibrated in this configuration. On the other hand, as
long as the controller is stable an inaccurate model cannot lead to instabilities. This is in
contrast to the closed-loop case, in which model mismatch may lead to an unstable system.
But this seems a reasonable price to pay for solving all previously mentioned issues of the
open-loop configuration. Therefore, throughout this thesis the closed-loop control system
of figure 1.9 will be considered.
1

1
14 1 Introduction
1.4 Challenges
At the moment, optical telescopes being built have an aperture diameter of at most 10m.
Examples are the VLTs, the LBT, the Keck and SUBARU telescopes. AO systems for these
telescopes are largely ready, but will initially only skim the surface of the full correction
potential. Much is still to be expected from wavefront correctors with more DOFs, faster
dynamic behavior and better optimized control laws.
Nevertheless, the observation of even fainter and more distant celestial objects requires even
larger apertures. Telescopes are currently being designed with aperture diameters ranging
between 30 and 50m. A consortium in the USA has conceived the Thirty Meter Telescope
(TMT) with an aperture diameter of 30m [171]. European Southern Observatory (ESO)
has conceived a 42m aperture telescope called the European Extremely Large Telescope
(E-ELT) [75], which is in fact a smaller version of the originally planned Overwhelmingly
Large Telescope (OWL) 1 with a diameter of 100m [43]. Further, a 50m aperture telescope
was being designed by a consortium around the Swedish Lund University and called the
EURO-50 [6]. This is now superceded by the E-ELT.
The design of AO systems for such large telescopes involves serious challenges for all parts
of the AO system. In this research project these challenges have been investigated for both
the wavefront corrector and the control system. In the next subsections, these challenges
will be discussed more elaborately.
1.4.1 The wavefront corrector
As will be discussed in the next chapter, a constant optical quality corresponds to a constant
actuator density. Therefore, the number of controllable degrees of freedom of DMs for the
mentioned future large telescopes must be in the order of tens of thousands. The highest
number of actuators currently available for DMs is∼1000 costing around 1ke per actuator.
For several reasons it is not trivial to extend current designs to larger numbers of actuator:
• Extendability. Straightforward extension of many current DM designs leads to an
increase in mass that cannot be matched by stiffness and thus leads to a severe reduction
of the resonance frequencies. Low resonance frequencies reduce the achievable
control bandwidth and thus the achievable wavefront correction performance. Extendability
is not only needed for the mechanics but also for the control system and
electronics involved.
• Scalability. DMs are needed with a wide range of actuator pitch. The first generation
AO systems for the E-ELT will have∼30mm actuator pitch and around 8000 actuators
[103]. Later generations will have an actuator pitch down to 1mm with over a 100.000
actuators. No current design is available that matches these requirements.
• Power dissipation. Most DM designs involve substantial power dissipation. As a
consequence, the temperature of the DM surface will rise with respect to its environment,
leading to detrimental air flow in the path of light. To prevent this, an active
1 N.B. OWL also refers to the bird for its keen night vision.

1.4 Challenges 15
cooling system is required, which adds complexity and the risk of leakage. Moreover,
the fluid flow will introduce vibrations on the nm level that affect the wavefront
correction performance.
• Failure probability. As the number of actuators increases, the probability of defective
actuators also increases. When actuators have a high stiffness, a defect actuator fixes
the DM position at its position. This creates a so-called hard point that will affect a
large fraction of the mirror area and thus degrade its performance. Besides developing
actuators with a high Mean Time Before Failure (MTBF), defective actuators should
thus not cause a significant decrease in the optical surface quality.
• The price per channel. Given the 500Me total budget of the Extremely Large
Telescope (ELT) and the current cost per channel of 1ke, an AO system for these
telescopes with in the order of 100.000 channels will not be affordable.
Extendable and scalable mirror design is needed, in mechanics, electronics and control with
lightweight construction with high resonance frequencies, low power dissipation and soft
and cheap actuators. As a starting point for further requirements an 8m telescope on a
representative astronomical site is chosen. In this research project a design was proposed
that is driven by above-mentioned reasons. This design will be sketched in the next chapter
and is described in more detail in [174]. The focus of this thesis will lie with the electronics
and the control system, whose challenges will be elaborated in the next subsection.
1.4.2 The control system
One of the main challenges with early AO systems was that for the system to be able to
correct for atmospheric wavefront disturbances the shape of the DM needed to be updated
around 1000 times per second. The first functioning AO system was finished in 1974
[98], having a 21 channel DM and a shearing interferometer wavefront sensor measuring
32 slopes simultaneously. Similar to the Shack-Hartmann wavefront sensor described
above, a shearing sensor also does not provide direct information of the wavefront phase,
but via a spatial transformation. Inversion of this transformation (reconstruction) and
subsequent calculation of suitable actuator command signals are computationally costly
operations, which would have taken more than a day on a contemporary computer [98].
Instead, an analog electronic circuit was designed in which measurements were introduced
as controlled currents, yielding the actuator commands as measurable voltages. Although
this controller structure was very inflexible, it could perform the reconstruction step within
microseconds, which is fast even for today’s standards.
Other approaches towards fast update rates involved the use of a simple photodiode as a
WFS. This was placed in a focal plane to measure the light intensity at the center of the
guide star image, which is a measure for the Strehl ratio. The controller would then quickly
superpose a series of shapes to the wavefront corrector and measure the corresponding
effects in light intensity. A suitable set of actuator commands was then calculated from
the results using relatively simple computations. However, the length of the series of
shapes and measurements required by this method is equal to the number of actuators of
the corrector. As this number increases, so must the number of measurements and to keep
the sampling rate constant the measurements need to be performed faster. The limits are
1

1
16 1 Introduction
here not determined by the intensity measurements, but by the dynamics of the wavefront
corrector, which limit the speed at which it can change shape.
A final AO system design aimed at reducing controller complexity is the combination of
a CS with a bimorph mirror. As mentioned in section 1.3.3, the actuators of this type of
mirror introduce local curvature to the surface. Consequently, the static mapping from
actuator commands to measurements is almost identity, which means that a computationally
expensive reconstruction step can be avoided in the control law [113, 179]. However, as
discussed in the same section, the construction principle of bimorph mirrors does not allow
the number of actuators to be increased without sacrificing stroke or dynamic behavior
(first resonance frequency). This renders these mirrors unsuitable for future large telescopes.
Currently, Shack-Hartmann sensors are the most widely used in AO systems and digital
control systems have become sufficiently fast to do all required computations. However,
for the future large telescopes being designed, the latter will not be trivial to maintain.
Digital processors may continue to increase in computational power, but the last few years
the gain in power has come mainly from parallel (multi-core) architectures instead of
increased numbers of sequential computations per time unit. Without efficient algorithms,
the required computational power for AO increases approximately with the square of the
number of actuators and thus to the fourth power in the telescope aperture. This is plotted
in figure 3.5 on page 51 based on a desired Strehl ratio of 0.87[-] and a sampling rate of
1kHz. It shows that without efficient numerical algorithms an AO system for the 42m
E-ELT with over 100.000 actuators would require almost 10.000 processors capable of
10 giga-flops. Starting point of this graph is the traditional reconstructor plus integrator
control law [98, 179]. It is yet unclear and difficult to predict how the computational
demand of recently proposed, more general optimal control laws will scale. This will
require representative WFS measurement data sets from actual large telescopes that are yet
unavailable.
The traditional control approach will require a careful design of both hard- and software
to achieve an efficient parallel computer system. In [53, 72, 187] control algorithms
are shown with a computational complexity of O(N 3/2
a ). But even such algorithms will
require many processors to compute the setpoints for the 100.000 actuators at a rate of 1kHz.
Besides computational problems, increasing the number of actuators yields many
practical problems [21]. The actuation principle is usually based on electricity and requires
each actuator to have at least two connection wires. In case of 100.000 actuators, this leads
to 200.000 wires and thus a large probability of defects, disturbances, etc. To keep the
lengths of these wires to a minimum and obtain a straightforward multi-processor hardware
architecture, a modular, distributed control system is proposed.
In this distributed control system each actuator or small group of actuators is driven
by a separate hardware module that has direct communication links to a few neighboring
modules. Each module receives a small fraction of the measurements available from the
wavefront sensor and all modules are identical in hardware, but may differ in software.
This allows cost-efficient production of the modules and enables both the straightforward
construction of a control system hardware for large AO systems as well as quick replacement
of defective modules.

1.5 Distributed control 17
However, by assigning computational power per actuator, the total power increases only
linearly with the number of actuators (figure 3.5), which is not sufficient for the currently
available algorithms. Moreover, as will be discussed in detail in chapter 3, these algorithms
are not suitable for the distributed architecture. This requires research into new algorithms
whose prime design driver is the distributed structure. The performance achieved using
such algorithms may be subject to the choice for specific properties of the structure, but
should approximate that of traditional, centralized architectures. Such properties may be
what neighbors the modules can communicate with, what information they exchange and
which measurements they receive. A suitable choice for these properties requires insight
into their effect on the AO system’s performance.
Since the AO case forms only a specific part of the spectrum of distributed control, a short
literature review on this topic will be given in the next section.
1.5 Distributed control
In the usual, centralized setting, a controller receives all measurements available of the
sensors of a system as its inputs and uses these to compute commands for all actuators
of the plant. The opposite of this approach is called decentralized and is characterized
by the fact that each actuator is controlled by a separate controller. In this setting it is
commonly assumed that all controllers are uniquely associated with a sensor, such that
the only coupling between the controllers is through the plant. A distributed controller
is a combination of these extremes in the sense that each actuator has its associated
controller, but now neighboring actuators are able to communicate. They are thus not only
connected through the plant, but also through communication channels. In the absence of
communication delays, a centralized controller is thus equivalent to a distributed controller
for which direct communication is allowed between all pairs of controllers. In that case
any centralized control law can be implemented without modification. These concepts are
illustrated in figure 1.10.
The reason why to use a distributed control system is always driven by requirements
of an application. If a centralized control system is possible within these requirements,
the complexity and limitations introduced by a distributed structure will not outweigh its
benefits. Nevertheless, for quite some time already the distributed control field is receiving
a lot of research attention. This is closely linked to the rise of application fields where a
centralized controller is not possible (1) or where benefits outweigh the added complexity
(2). In such application fields the system to be controlled often consists of multiple
interacting subsystems that are to some extent physically separated and can all include
an implementation of a local controller. In application fields of the first kind it is either
not practically possible to send the measurements of all sensors in the system to a central
location or to send control commands to all actuators in the system from a central location.
Reasons may be latency or unreliability of communication due to e.g. distances, problems
with cabling because of the large numbers of sensors and actuators, etc.
For application fields of the second kind, the distributed control benefits that outweigh the
added complexity are often flexibility and/or scalability of the system. When the system
itself is subject to change – e.g. the number of subsystems varies – a centralized controller
1

1
18 1 Introduction
Centralized Distributed Decentralized
P
C
P
C1
C2
Ci
Cn-1
Cn
P
C1
Ci
Cn-1
Figure 1.10: The schematic representation of the centralized, distributed and decentralized control
concepts.
must be dimensioned both in computational power and in communication capabilities to
deal with the largest number of subsystems that is to be expected . For a distributed control
implementation the addition of subsystems implicitly leads to an increase in computational
power and communication capabilities and thus provides flexibility. Potentially, this also
leads to a scalable system – provided that the behavior and interaction of the subsystems
allow for the same system performance with the same controller structure.
Examples of application fields where distributed control is developing a role are vehicle
platooning [137] and automated highways, formation flying of aircrafts [200], spacecrafts
[176] and satellites [128], the process industrial plants [8], optical telescope control systems
[107], inflatable structures in space, paper machines [169], power networks [134], etc.
For vehicle platooning or automatic highway systems the communication infrastructure
that would allow all cars to send position and velocity information to a central server
would be very complex and expensive and the computational power this server would
require to process this information would be enormous. Since vehicles interact only locally,
a distributed control system in which local vehicle controllers communicate only with
nearby vehicles would greatly simplify the system implementation. Moreover, it would
significantly reduce its cost, whereas the resulting system has now become flexible and
scalable w.r.t. the number of vehicles.
The problem in formation flying of satellites is that for some applications the communication
between a central server and all satellites cannot be guaranteed, regardless of the
location of the server. Assuming that satellites will always be in range of at least one
other, neighboring satellite, a distributed control architecture could solve this problem.
An additional benefit is a reduction of the required communication range and thus power
C2
Cn

1.5 Distributed control 19
usage.
A application field where distributed control has always played a role is the control of
power networks. Due to the recent rise of small, local power generators (wind-mills, solar
cells, etc.), regulating the frequency and phase of the potential as well as the power through
the network links has become much more complex. A centralized control system is here
not an option for political reasons.
Finally, distributed control finds application in small, highly specialized areas such as alignment
systems for the tiles that form the primary mirrors of future large optical telescopes.
Sensors measure the alignment of each tile in several DOFs w.r.t. their neighbors, based
on which local controllers provide setpoints for the local actuators to achieve global shape
control.
The general approach towards controller synthesis is through the minimization of a
certain cost function. To enforce the so obtained controller to have a distributed structure,
constraints have to be applied, which for general plants renders the optimization problem
non-convex. This implies that it is no longer known whether there is a single optimal
solution or whether there a multiple or how good a candidate solution is compared to
it/them. The basis on which the currently available results on distributed control are
founded is the exploiting of structure present in the system to be controlled. The explicit
constraints that have to be applied to general plants can be relaxed by making a priori
assumptions on this structure, in some cases leading to efficient synthesis algorithms and
convex problems.
For instance results have been shown for distributed systems satisfying certain spatial
invariance properties [14, 41, 79, 129, 168]. In [13, 14] this spatial invariance is used in a
Fourier domain approach, reducing the optimization to a family of problems over spatial
frequency. It is moreover shown that the spatial invariance property is inherited by the
controller that is optimal w.r.t. generalLp induced norm performance criteria.
In simulations, this approach has been applied for the alignment control of hexagonal
segments of the primary mirror of a future large telescope [107]. First, a spatially invariant
system model is here proposed for which a spatially invariant controller is derived with
known quadratic performance upper bounds. This is then truncated and applied to a 19
segment mirror model. Although it is mentioned that stability may be lost in this truncation,
no approach is given to recover this.
In [168], a distributed controller is sought for shape control of large two dimensional arrays.
A distributed proportional+integrating (PI) controller is proposed, where both sets of gains
apply not only to local errors, but also to those of neighbors. An optimization problem
is then formulated for these gains in terms of robust stability and frequency dependent
norms on the shape error and control effort transfer functions. The assumption of spatial
invariance then allows a transformation to the Fourier domain, where the optimal controller
parameters can be found using linear programming. This approach is further extended in
[79] to include a derivative controller action and a method to handle the effects of finite
boundaries including stability issues.
In [129] the system to be controlled is considered to be an interconnection of a number of
identical subsystems. The interconnected systems are not necessarily spatially invariant
and can have an arbitrary interconnection graph topology. An approach is presented to
synthesize feedback controllers for this class of systems that retain the distributed structure.
The synthesis problem is formulated as a multi-objective optimization problem with Linear
1

1
20 1 Introduction
Matrix Inequality (LMI) constraints and system norms (e.g. H2 and H∞) as performance
indices. Although this formulation involves some conservatism, the resulting optimization
problem can be solved very efficiently. Moreover, the proposed approach is demonstrated
– including simulation results – for classical distributed control applications such as the
paper machine and satellite formation flying.
A relatively new approach towards the design of distributed controllers is to apply game
theory and consider the nodes and communication links as players that strive towards a
common goal [147]. Synthesis of an optimal distributed controller is then replaced by an
iterative process, where the local controllers optimize their local cost functions until the
point that none can be improved by adjusting local parameters alone. This point is called
the Nash equilibrium, which in this approach replaces the global optimum (if this exists at
all). In section 3.6.3 of this thesis a wavefront reconstruction algorithm will be presented
that is based on overlapping, local optimization problems with the same limitation.
A significant part of the distributed controller synthesis problem is also present in optimal
distributed filtering – i.e. distributed Kalman filtering. Also here, for specific systems
satisfying spatial invariance properties, the Fourier domain approach has been shown to be
of value [99]. For more general cases the approach based on weighted averaging proposed
in [4] may be used. This approach has been experimentally validated for position tracking
of remotely controlled robots in [5]. However, a drawback of this approach is the high
computational demand arising as the system state increases. Other approaches towards
distributed Kalman filtering may be found in [11, 138, 144, 166].
Despite all progress and achievements, actual implementations of the above approaches are
scarce. This may be explained by the fact that the successful synthesis approaches require
plants to possess such particular structures that distributed control as a field remains a niche.
More general assumptions on the plant structure for distributed controller synthesis to be
assimilated by industry. Therefore, future research is planned to extend such methods with
Linear Parameter Varying (LPV) techniques, where certain plant parameters are allowed
to vary spatially. Besides by its limitations on structure, the need for distributed control is
attenuated by the fact that both the computational power of commercial processors as well
as the communication capabilities of commercial devices is still growing steadily. This
makes that centralized solutions remain feasible even for large systems.
1.5.1 Distributed control for AO
A specific problem that arises when applying available distributed control approaches to the
case of AO, is that the typical AO system also has dynamic coupling through the sensor
and not only through the plant. As mentioned, a SHS offers only indirect measurements –
spatial gradients – of the desired quantity: wavefront phase. As will be shown in chapter 3,
this forms the main challenge of the distributed control problem for AO. Moreover, these
measurements are not collocated with the actuators. In fact, it is shown in appendix A that
the best quality of SHS measurement is obtained when the spots are placed in between the
actuators. As a result it becomes less trivial to associate sensors to specific controllers. In the
approaches presented in chapters 3 and 4 of this thesis, the sensors are therefore assigned to
multiple controllers. Chapter 3 also contains a more extensive survey on distributed control
literature for AO, although the available literature for this application is limited.

1.6 Problem formulation and organization of this thesis 21
1.6 Problem formulation and organization of this thesis
Existing large en future even larger telescopes can only be utilized fully, when they are
equipped with AO systems that enhance the telescopes resolution to the diffraction limit.
The development of new DM technology that meets these requirements is therefore essential.
This thesis will focus on the design, testing and control of a new DM that is extendable
and scalable in mechanics, electronics and control. Since this thesis is a result of a joint
research project there is an accompanied thesis, by Roger Hamelinck [174], on the comprehensive
design of the new AO system.
This thesis is organized as follows. In the next chapter the design requirements for the new
DM will be introduced, based on available knowledge of the wavefront distortion and a desired
optical quality in terms of Strehl ratio. This leads to a conceptual design of the DM
system with requirements for control, communication and driver electronics.
In chapter 3 the problem of controller design for AO systems will be analyzed w.r.t. the increasing
size of optical telescopes. The advantages and limitations of (efficient) algorithms
described in literature will be discussed and problems arising for future large telescopes
will be described. The latter include cabling for many closely placed actuators, but also the
computational power required to implement control laws for many thousands of in- and outputs.
At this point a modular, distributed control system architecture will be proposed. Such
a hardware design can be built for a DM system with any number of actuators. Although
there are no direct scaling problems, its main limitations are the computational power that
increases linearly with the number of actuators and the fixed communication structure. The
applicability of currently available efficient control algorithms for AO will be evaluated for
this architecture, but they are found unsuitable.
On the other hand, several design concepts and DM properties facilitate the use of a modular,
distributed control system. One of these properties is that – when assuming a certain
extent of frozen flow behavior – the disturbance to be suppressed can be well predicted over
a short time horizon using only local information [45, 100]. At the end of the chapter an
adaptive scheme is shown that exploits this fact to perform wavefront reconstruction and
prediction within the distributed framework.
Further, in chapter 4 another approach towards distributed controller design is proposed
that also exploits local predictability of the disturbance. An ideal DM is considered whose
transfer matrix is equal to the identity matrix and a SHS is considered with a full sample
delay. The controller structure is chosen a priori as a network of output interconnected
Auto-Regressive Moving Average (ARMA) filters, whose coefficients are identified from
open-loop measurement data. Several approaches are presented to guarantee the stability of
this open-loop controller via constrained optimization. Application results are shown both
for data obtained from an AO breadboard at TNO Science and Industry as well as for synthetic
data generated according to well known turbulence models.
In chapter 5 the variable reluctance actuator design is introduced that forms the heart of the
new DM. A detailed model is derived that comprehends the electromagnetic and mechanical
domains of its behavior. Measurement results are presented of a realized prototype,
which are compared to the model. Model parameters are identified from the measurement
data and compared to first principle estimates. Some differences are found, but in general
the measurements and parameters agree well with the derived model and first principle estimates.
With minor modifications, the single actuator design is then transformed to a design
1

1
22 1 Introduction
of actuator grid modules consisting of 61 hexagonally arranged actuators. Their design
is presented together with measurement results obtained from seven realized prototypes.
The found differences with the single actuators are explained via a sensitivity analysis on
the derived model and the known design changes. The identified actuator properties such
as its stiffness, resonance frequency, motor constant, inductance and viscous damping are
presented for all actuators and their variation is analyzed and discussed. Finally, recommendations
for future design improvements are given based on the mentioned sensitivity
analysis.
In chapter 6 the design, modeling and realization of the driver and communication electronics
for the actuator is considered. First the requirements are derived, leading to a design
that has been realized by QPI (the former EMDES). The driver electronics are realized
in modules containing drivers for 61 actuators. Each driver is chosen as a Pulse Width
Modulation (PWM) voltage source together with a second order analog low-pass filter.
Three FPGAs implement the PWM generators for all 61 actuators together with the serial
Low Voltage Differential Signalling (LVDS) communication protocol. The actuator model
is extended with the electronic driver circuit and a delay to describe the serial communication
link. The model thus describes the single actuator behavior from the its setpoint to its
position, velocity and several electrical quantities. It is validated using measurements on
the seven realized actuator modules and several model parameters will again be estimated
from the results.
In chapter 7 the actuator modules are connected to the reflective membrane to form a full
AO system. A model of this full system is derived by coupling a number of single actuator
instances through a model of the reflective membrane. This model is then analyzed w.r.t.
influence functions, resonance frequencies and mode shapes, impulse response and transfer
functions. Further, it is validated using measurements on a 61 actuator DM prototype. Static
measurements (i.e. influence function and flatness) are obtained using interferometers and
dynamic measurements using white noise excitation and a laser vibrometer. An LTI model
is identified from the measurement data and compared to the analytically derived model.
The thesis finishes by stating general conclusions and recommendations for future research.
1.7 Scientific contributions
R. Ellenbroek, M. Verhaegen, R. Hamelinck, N. Doelman, M. Steinbuch, and N. Rosielle.
Distributed control in Adaptive Optics - Deformable mirror and turbulence modeling. In
B. L. Ellerbroek and D. C. B. Calia, editors, Proceedings of SPIE: Astronomical telescopes
and instrumentation - Advances in Adaptive Optics, volume 6272, May 2006.
R. Hamelinck, N. Rosielle, M. Steinbuch, R. Ellenbroek, M. Verhaegen, and N. Doelman.
Actuator tests for a large deformable membrane mirror. In B. L. Ellerbroek and D. C. B.
Calia, editors, Proceedings of SPIE: Astronomical telescopes and instrumentation -
Advances in Adaptive Optics, volume 6272, May 2006.
R. Hamelinck, R. Ellenbroek, N. Rosielle, M. Steinbuch, M. Verhaegen, and N. Doelman.
Validation of a new adaptive deformable mirror concept. In N. Hubin, C. E. Max, and P. L.
Wizinowich, editors, Proceedings of SPIE: Astronomical telescopes and instrumentation,

1.7 Scientific contributions 23
volume 7015, Marseille, France, June 2008.
R. Ellenbroek and R. Hamelinck. Adaptief deformeerbare spiegel voor telescopen. Precisietechnologie
jaarboek, 16:112–118, 2009.
1

ÔØÖØÛÓ
×ÒÖÕÙÖÑÒØ×Ò×Ò
ÓÒÔØ
The main requirements for the adaptive deformable mirror and control system
are derived for typical atmospheric conditions. The spatial and temporal
properties of the atmosphere are covered by the spatial and temporal spectra
of the Kolmogorov turbulence model and the frozen flow assumption. The
main sources for the residual wavefront aberrations are identified. The fitting
error, caused by a limited number of actuators and the temporal error, caused
by a limited control bandwidth, are considered to be the most important for the
mirror design. A balanced choice for the number of actuators and the control
bandwidth is made for a desired optical quality after correction. Then the actuator
requirements are defined, such as the pitch, total stroke and inter-actuator
stroke, resolution and power dissipation. Requirements are derived for the
control system and the electronics. Finally, the full Deformable Mirror (DM)
system design concept is presented, consisting of the thin mirror facesheet, the
mirror-actuator connection, the actuators, the control system, the electronics
and the base frame.
Sections 2.1, 2.2, 2.3 and 2.5 are joint work with Roger Hamelinck [PhD]
25

2
26 2 Design requirements and design concept
2.1 Requirements
The goal is to make a DM that can correct a wavefront of an 8-meter telescope in visible
light, which is aberrated by atmospheric turbulence to the diffraction limit. The mirror’s
main requirements will be derived from the spatial and temporal properties of typical atmospheric
conditions as they exist on astronomical sites such as Cerro Paranal in Chile. These
conditions will be shown to determine the number of actuators, the (inter) actuator stroke
and the control bandwidth. Further, the mirror should have low roughness and high reflection
for the wavelengths utilized and be functional in a temperature range between−10 ◦ C
and30 ◦ C [85].
The mirror surface may not heat up more than 1K relative to the environment to prevent
the deformable mirror itself to become a significant heat source. Finally, the number of
sensors and actuators in the Adaptive Optics (AO) system will be of such order of magnitude
that efficient control algorithms are required to prevent problems in the realization
of suitable computation hardware. Known efficient control algorithms such as proposed in
[53, 72, 129, 188] exploit the structure present in a system to obtain efficient implementations.
For AO applications, such algorithms exploit sparsity or spatial invariance of the
Deformable Mirrors (DMs) influence matrix and generally comprehend its temporal dynamics
only in terms of a number of samples delay. The DM to be designed should behave
accordingly up to a sampling time scale defined in section 2.4.
2.1.1 Atmospheric turbulence
In section 1.2 it is explained that refractive index variations of the atmosphere cause wavefront
aberrations. Based on the work of Edlén [49] several contributions have been made
to describe the dependence of the refractive index nair on temperature, pressure, humidity
and CO2-concentration [19, 32, 108, 139, 141]. Many different formulations exist, which
are often aimed at specific wavelength of interest. Because of the weak dependence on the
relative humidity (for vertical propagation through the atmosphere) and CO2-concentration,
these are often neglected [98]. The dependence of the refraction index on pressure and
temperature is given by [36]:
nair = 1+7.76·10 −5P
T
1+ 7.52·10−3
λ2
Where P is the pressure in millibars, T the temperature in K and λ the wavelength in
microns. As a result of the change in the refractive index some parts of the initially flat
wavefront are advanced and some parts of the wavefront are retarded.
2.1.2 The Kolmogorov turbulence model
The work of Kolmogorov in 1941 [112] formed the basis for currently used atmospheric
turbulence models . Kolmogorov concluded that in a turbulent flow the kinetic energy is fed
into the system at the outer scaleL0 and decreases till it is dissipated in heat at the smallest,
inner scale l0. The outer scale corresponds to the radius of the largest air bubbles and the
inner scale to that of the smallest. Outside the outer scale the isotropic behavior of the

2.1 Requirements 27
Solar energy
outer scale
Wind shear
convection
inner scale
Figure 2.1: Schematic of Kolmogorov turbulence.
Energy is fed into the system at the outer
scale and cascades till dissipated in heat at the
inner scale.
Figure 2.2: A typical profile measured with
a SCIDAR instrument at Mt. Graham (profile
taken from S.E.Egner [50]).
atmosphere is violated and inside the inner scale viscous effects are dominant and kinetic
energy is dissipated in heat. This is schematically shown in figure 2.1.
Spatial model of atmospheric turbulence
Kolmogorov described the random movement of the wind with statistical quantities by
means of structure functions. Structure functions describe the mean squared difference between
two randomly fluctuating values. With the assumption that the atmosphere is locally
homogeneous, isotropic and incompressible he concluded from a dimensional analysis that
the kinetic energy decreases with the spatial frequency to the power− 5
3 . Tatarski [173] related
Kolmogorov’s velocity structure function to the index of refraction structure function
Dn(h,r) given by:
Dn(h,r) = 〈|n(h,r ′ )−n(h,r ′ +r)| 2 〉
= C 2 2
N (h)r 3, for l0 ≪ r ≪ L0
where〈〉 denotes the variance of the enclosed expression at heighthand distancer. C2 N (h)
is used to take into account the atmospheric turbulence contributions from all altitudes above
the telescope. figure 2.2 gives a typical C2 N (h) profile. From this refractive index structure
function profile it becomes clear that the ground layer and the high wind speed at the jet
stream at about 10 km height strongly contribute to the wavefront aberrations.
To quantify the effect of variations in index of refractions in terms of wavefront phase,
another structure function is used: the phase structure functionDφ(r). For the values of the
phase φ at any two points in the wavefront that are separated by a distance r this structure
function is given by [98]:
Dφ(r) = 〈|φ(r ′ ,t)−φ(r ′ +r,t)| 2 〉,
2 ∞
2π 1 5
= 2.91 r 3
λ cos(ζ) 0
C 2 N(h)dh,
2

2
28 2 Design requirements and design concept
5
3 r
= 6.88
r0
whereζ is the angle with zenith andr0 is the Fried parameter defined as:
r0 =
0.423
2 2π 1
λ cos(ζ)
∞
0
C 2 N(h)dh
− 3
5
.
(2.1)
The Fried parameter r0 is the characteristic spatial scale, which for λ=550nm typically
ranges between 5 and 20cm [98]. The Fried parameter corresponds to the aperture diameter
Dt of a telescope for which the varianceσ 2 wf of the wavefront aberrations is roughly 1 rad2 .
This variance can be expressed as [136]:
σ 2 wf
= 1.03
Dt
r0
5
3
(2.2)
Other important statistics are described by the spatial Power Spectral Density (PSD),
whichis a measure for the relative contribution of aberrations with spatial frequency
κ = κ2 x +κ2y +κ2 z to the total wavefront distortion. For the Kolmogorov turbulence
model this is given by [136]:
Φ(κ,h) = 0.033C 2 N(h)κ −11
3 ,
Φ(κ) = 0.023r −5
3
0 κ−11 3 (assuming isotropy). (2.3)
This spatial PSD is often truncated at the outer and inner scale of the turbulence in which
the Kolmogorov model is valid. This is mostly done using the Von Karmann model:
Φ(κ) = 0.023r−5 3
0
(κ2 +κ2 o) 11
2 κ
exp−
6 κi
(2.4)
whereκo = 2π/L0 corresponds to the boundary set by the outer scaleL0 andκi = 5.92/l0
corresponding to the lower boundary set by the inner scale l0. The outer boundary is in the
order of tens of meters [34] and the inner scale is in the order of tens of millimeters [48, 118].
The outer scale constrains the lower order wavefront distortions. Since these are dominant,
the outer scale also determines the total stroke requirements for the actuators in adaptive
mirrors. Knowledge of the outer scale at a certain telescope location for Extremely Large
Telescopes (ELTs) is therefore of great importance. For intensity variations (scintillation)
the inner scale is more relevant. In figure 2.3 the Kolmogorov PSD defined by (2.3) and the
Von Karmann PSD defined by (2.4) is shown.
Temporal model of atmospheric turbulence
In analogy with the refractive index structure function a temporal structure functionDφ(δt)
can be defined between two wavefront phase values separated in time byδt:
Dφ(δt) = 〈|φ(r,t)−φ(r,t+δt)| 2 〉,

2.1 Requirements 29
Φ [m −3 ]
10 3
10 1
10 −1
10 −3
10 −5
10 −7
10 −9
10 −11
10 −2
Outer scale Inner scale
10 0
Spatial frequency [m −1 ]
Kolmogorov PSD
Von Karmann PSD
10 2
Figure 2.3: The spatial PSDs of
the wavefront aberrations for the Kolmogorov
and Von Karmann turbulence
models.
where〈·〉 denotes the variance of the enclosed expression over space (r) and time (t). Under
the assumption that the wavefront aberrations are fixed and turbulence layers at altitude h
are moving with a wind speedv(h) over the telescope aperture – the frozen flow assumption
– the temporal structure function can be expressed in the spatial frequencyκ as [33]:
∞ 8 −3 1 κ
Dφ(v,κ) ∝ C
v(h) v(h)
2 N (h)dh.
0
This function integrates the effect of all turbulence layers. When this integration is performed
for a single turbulent layer at altitudehof thicknessδh traveling with a wind speed
vw, the temporal power spectrumP of the phase valueφobserved at a certain point in space
can be expressed in terms of the temporal frequencyf as:
P(f,h) ∝ C 2 N (h)δh
vw
f
vw
−8/3
.
This −8/3 power law is often used in the context of controller design for AO [98, 179],
where integrator structures approximate the−8/3 power law by -2.
In the previous paragraph, the characteristic spatial scale r0 was introduced to quantify
the spatial variance of atmospheric turbulence. A similar value exists that describes the
characteristic timescale for changes in wavefront aberrations [153]: the coherence time τ0.
Various definitions exist [26, 179], but let it here be defined as the time for wind to carry
frozen flow turbulence over an aperture of size r0. Based on the mentioned assumptions,
this would imply that the wind speed is indicative of the coherence time τ0. This is in fact
the case, even though the validity of the frozen flow assumption is questionable: it is e.g.
shown in [40, 153] that the so-called boiling effect plays a major role in the evolution of
phase errors on the timescales of practical interest. Let the coherence time τ0 be expressed
through its inverse, the Greenwood frequencyfG [80]:
fG = 1
τ0
= 2.31λ −6
5
1
cos(ζ)
∞
0
C 2 N(h)v 5
3
5
3(h)dh .
For a single turbulence layer with constant wind speedvw the Greenwood frequency can be
approximated as:
fG = 0.43 vw
. (2.5)
r0
2

2
30 2 Design requirements and design concept
For representative values of the wind velocity vw = 10m/s and the Fried parameter r0 =
0.166m the Greenwood frequency is approximately 25Hz. Since the Greenwood frequency
is a measure for the rate of change of the wavefront distortion, it is related to the required
control bandwidth of an AO system.
2.2 Error budget
The atmospheric conditions and the desired optical quality after correction are the main
design drivers for the AO system. They determine the number of actuators and the control
bandwidth. The optical quality is often expressed by a Strehl-ratio S. This ratio can be
related to the varianceσ 2 of the wavefront measured in radians using the extended Maréchal
approximation [20, 98, 179]:
S ≈ e −σ2
This approximation is valid up to σ = 2rad [98]. For the design of the DM the practical
limit to the diffraction limited level is set at a Strehl ratio of 0.85. This leads to a total error
budget of σ ≈ 2π
16 rad, which forλ=550nm corresponds to 550/16 ≈ 34nm. Assuming that
all error sources are independent, the total variance can be approximated as the sum of the
variances corresponding to the main contributing sources:
σ 2 = σ 2 fit +σ 2 temp +σ 2 meas +σ 2 delay +σ 2 angle +σ 2 cal. (2.6)
σ 2 ctrl
The fitting errorσfit arises from the limited number Degrees Of Freedom (DOF) of the DM
and thus the limited number of spatial frequencies that it can correct. The temporal error
σtemp is due to the limited control bandwidth of the AO-system.
If the light source used for wavefront sensing (i.e. the reference star), is not the same as
the object for which the correction is used (the science object), a so called anisoplanatic
error is made. The variance of this error is related to the angle θa by which the two objects
are separated as σ2 angle ∝ θ5/3 a . Further, σ2 meas covers all the measurement errors
(e.g. measurement noise in the wavefront sensor) and σ2 delay the errors due to delays in
the wavefront sensor and the controller. As will be discussed in section 2.4, a closed-loop
controller influences not only the temporal, but also the measurement and delay related errors,
hence in (2.6) the combination of these sources is related to the controller and denoted
consists of all calibration errors. Calibration is needed for the correction
σ2 ctrl . Finally,σ2 cal
of static aberrations that are not seen by the wavefront sensor and are called Non Common
Path Aberrations (NCPAs) [157]. A good review of the main errors in an AO system can
be found in [98]. Since a large part of the total error budget is consumed byσ2 fit andσ2 temp
which both can be influenced by the DM and controller design, the other error sources will
further be neglected. The fitting and temporal errors will be considered in the next two
paragraphs to derive requirements for inter-actuator stroke and control bandwidth.

2.2 Error budget 31
2.2.1 The fitting error
The variance of the fitting error can be approximated by [98]:
σ 2 fit = κf
dt
r0
5
3
, (2.7)
where dt is the inter actuator distance projected onto the primary aperture and r0 the Fried
parameter. The fitting error coefficientκf depends on the type of mirror that is used:
⎧
⎪⎨ 1.26 for segmented mirrors with only piston correction,
κf = 0.18 for segmented mirrors with tip, tilt and piston correction,
⎪⎩
0.28 for membrane mirrors.
In [122] it is shown that for piston, continuous face-sheet and membrane mirrors for an
equal number of actuator the correction quality does not significantly depend on the actuator
geometry as long as the actuator distribution is fairly homogenous. This means that (2.7)
gives an estimate for the fitting error variance to be expected for a specific DM on a telescope
with known diameter at a site with a certainr0. Althoughκf given above is the smallest for
segmented mirrors with tip, tilt and piston correction, this type of mirror has three actuators
per segment whereas the inter actuator distance dt is assumed to be the segment size. For
a more fair comparison, let the fitting error be expressed in terms of the total number of
actuatorsNa, which can be achieved by writing the inter actuator spacingdt as a function of
Na. For piston and membrane type mirrors, the inter actuator spacing can be approximated
as dt ≈ Dt/2 π/Na, whereas for segmented mirrors with piston, tip and tilt correction
the number of actuators must be scaled by three, yielding dt ≈ Dt/2 3π/Na. After
substitution into (2.7),σ2 fit can thus alternatively be expressed as:
σ 2 fit = κf,NaDt
π/Na
r0
5/3
where
⎧
⎪⎨ 0.63 for segmented mirrors with only piston correction,
κf,Na = 0.23 for segmented mirrors with tip, tilt and piston correction,
⎪⎩
0.14 for membrane mirrors.
This implies that for the same number of actuators, the fitting error is the smallest for a
membrane type mirror.
2.2.2 The temporal error
Although in practice the temporal error depends on all components of the AO system as
well as on actual atmospheric conditions, Greenwood [80] showed that the variance of the
temporal error can be related to the Greenwood frequencyfG as:
,
σ 2 5
3
fG
temp = k , (2.8)
fc
2

2
32 2 Design requirements and design concept
wherefc is the control bandwidth andk a scaling constant. For the ideal – though unrealistic
– case that the controller fully suppresses the wavefront disturbance up to the bandwidthfc
and does not affect higher frequencies, the scaling constantk is equal to 0.191. For a more
realistic integrator type controller it is equal to 1, which means that (2.8) gives an estimate
for the temporal error to be expected for a given type of controller and a given temporal
behavior of the wavefront disturbance.
However, the derivation of this relation is based on many assumptions. Starting point is a
wavefront disturbance with a Kolmogorov spectrum and a frozen flow behavior, which is
corrected by a DM system that is able to track a command signal up to the bandwidth fc.
The temporal error is then defined as the servo tracking error of the DM with respect to
the assumed type of wavefront disturbance. This means that the estimate of the temporal
error variance in (2.8) does not take into account the ability of a (closed-loop) control law to
reduce the detrimental effects of measurement noise or DM dynamics. It does not consider
the dynamics of the wavefront sensor or delays in the disturbance signal to track. In AO
literature, the latter is considered as a separate effect on the eventual performance and is
quantified as the varianceσ2 delay between wavefronts measuredτ seconds apart [64]:
σ 2 delay = 28.44(fGτ) 5
3 .
By considering control system delays as a separate source of errors, it is not taken into
account that the control system can exploit spatio-temporal correlations of the wavefront
distortion to make accurate short term predictions to compensate delays [100]. However,
as the latter strongly depends on the atmospheric turbulence conditions, (2.8) will further
be used for the estimation of the expected error. Since the wavefront sensor is regarded
as a given part of the AO system and delays in a controller affect its already considered
bandwidth, the effect of delays will further be neglected as a separate source of errors.
2.2.3 Error budget division
If the main atmospheric parameters (r0 andfG) are known for a specific telescope location
and only the fitting and temporal errors are considered, the actuator spacing dt and control
bandwidth fc can be related to a desired Strehl ratio (figure 2.4). When also the diameter
Dt of a telescope is known, the number of actuatorsNa can be calculated in approximation
as Na = π
4 (Dt/dt) 2 . For an 8-meter telescope (Dt = 8m), figure 2.5 shows the Strehl
ratios for the number of actuators Na and control bandwidth fc based on fG = 25Hz and
r0 = 0.166m (λ=550nm). Observe from figures 2.4 and 2.5 that the same Strehl ratio can
be achieved by different combinations of control bandwidthfc and number of actuatorsNa.
According to figure 2.5 the effect of increasing the number of actuators is limited when
this it not matched by an increase in control bandwidth and vice versa. A combination of
actuator count and control bandwidth should be chosen for which the fitting and temporal
errors are approximately equal. For a desired Strehl ratio of0.85 this leads to a combination
of 5000 actuators and 200Hz control bandwidth, which is marked by a white star in figures
2.4 and 2.5. The corresponding Root Mean Square (RMS) fitting and temporal errors are
σfit = √ 0.28(Dt/ 4Na/π/r0) 5/6 = 0.34rad and σtemp = fG/fc = 0.17rad, which
forλ =550nm corresponds to 30nm and 15nm respectively.

2.3 Actuator requirements 33
Figure 2.4: The Strehl ratio as function of the
relative actuator densitydt/r0 and control bandwidthfG/fc.
2.3 Actuator requirements
Figure 2.5: The Strehl ratio as function of the
number of actuators Na and control bandwidth
fc based on Dt = 8m, r0 = 0.166m and fG =
25Hz.
Before stating the requirements for the actuators, it should be noted that the Optical Path
Difference (OPD) of the light is the double of the facesheet displacement. This is explained
in figure 2.6 and implies that the magnitude of the mirror deflection required to correct
a wavefront needs to be half the magnitude of the wavefront unflatness. For the nearly
diffraction limited correction of an 8 meter telescope in the visible part of the light spectrum
the main requirements for the actuators are as follows:
• Mirror diameter and actuator spacing
Given a number of actuators Na, the actuator spacing depends on the diameter of
the DM. The location of a DM is not restricted to a single position in the optical
path of a telescope system. Because of the dynamic requirements and the ease of
manufacturing, usually a flat surface with a smaller diameter is chosen. The lower
limit is set by the Smith-Lagrange invariant. This optical invariant is explained in
figure 2.7 and states that at all cross-sections in the optical path the product DΘ is
constant. Herein is D the illuminated diameter or the envelope of all rays and Θ the
angle between the optical axis and the chief (outer) ray of the beam. At the primary
Mirror shape
Incoming distorted wavefront
Outgoing corrected wavefront
Figure 2.6: The magnitude of the mirror deflection
required to correct a wavefront needs only
be half the magnitude of the wavefront unflatness
because the deflection distance of the mirror is
traveled twice. The summed lengths of all pairs
of black and grey arrows are equal.
2

2
34 2 Design requirements and design concept
Dt
D ′ = Dt/10
focus
Θ Θ ′
ΘDt = Θ ′ D ′ → Θ ′ = ΘDt/D ′ = 10Θ
Figure 2.7: The optical Smith-Lagrange invariant, which states that at all cross-sections in the optical
path the product DΘ is constant. D is herein the illuminated diameter, the envelope of
all rays and Θ the angle between the optical axis and the chief (outer) ray. When D
increases, Θ must decrease and vice versa.
mirror of this telescope the invariant is equal to DtΘ, where Θ equals the Field Of
View (FOV) and at apertures further along the optical path the angleΘ ′ will become
Θ ′ = ΘDt/D ′ . To keep Θ ′ within realizable values for e.g. the 8m Very Large
Telescope (VLT) with a half degree FOV and the 42m European Extremely Large
Telescope (E-ELT) with 10 degree FOV, a realistic lower bound for the DM diameter
lies in the order of 500mm. This will be chosen as a starting point in the design. With
the 5000 actuators this defines a 500·10 −3 / 4·5000/π ≈ 6mm actuator spacing.
• Total actuator stroke
The total actuator stroke can be derived using (2.2) describing the RMS unflatness of
the wavefront. With Dt = 8m and r0 = 0.166m (λ=550nm) this gives σ 2 stroke =
σ 2 wf = 1.03(Dt/r0) 5/3 = 657rad 2 . The square root of this variance relates to the
RMS actuator position, whereas in fact the Peak To Valley (PTV) value is sought
that forms the total actuator stroke. For AO applications the RMS and PTV values
are often related via a scaling factor 5, yielding a total actuator stroke of 5· √ 657 =
128rad. Considering the reflection doubling the OPD (figure 2.6) forλ = 550nm, this
corresponds to a required actuator stroke ofλ/2π ·128/2 = ±5.6µm. In addition to
this stroke a fewµm are added to be able to deal with misalignment of the DM in the
optical system.
• Inter actuator stroke
The inter actuator stroke can be calculated using the structure function in (2.1) describing
the mean square difference between two wavefront phase values separated
by a distance r. Substitution of r = dt and using r0 = 0.166m (λ=550nm) then
yields the required mean square inter-actuator stroke as σ 2 ia = 6.88(dt/r0) 5/3 =
2.8·10 −2 rad 2 . Using the factor 5 between the RMS and PTV strokes, the latter becomes
5 · √ 2.8 = 8.3rad. Due to the reflection doubling the OPD (figure 2.6) for
λ = 550nm this corresponds to a required inter-actuator stroke of±0.36µm.
• Actuator resolution
The actuator resolution should be well below the error budget as derived in section

2.4 Control system and electronics requirements 35
2.2 of 2π
16 rad (for λ = 550nm this is 34nm RMS). The design value for the actuator
displacement resolution is therefore set significantly smaller at 5nm.
• Power dissipation
To avoid the need of active cooling, all energy dissipated in heat should be convected
from the mirror surface by natural convection. The temperature difference between
the mirror surface and the surrounding air of 1K is usually allowed. A typical value
for the heat transfer coefficienthn is 1 < hn < 40W/m 2 [17]. Using hn = 12W/m 2
and 5000 actuators on the ∅500mm DM, this allows for ≈ 0.5mW per actuator.
Assuming that only half of the heat dissipated in the actuators is transferred to the
mirror surface, the maximum heat dissipation per actuator is set to≈ 1mW.
Dependence on telescope diameterDt
Observe that according to (2.7) and (2.8) an increase of the telescope diameter Dt only
affects the fitting error varianceσ 2 fit and not the temporal error varianceσ2 temp. To maintain
the desired Strehl ratio, the actuator spacing dt must therefore remain constant and the
number of actuators Na must increase with D 2 t . For the E-ELT this would result in 52 ·
5000 = 625000 actuators.
However, (2.7) and (2.8) do not consider any beneficial effects that a scale increase may have
on the achievable controller error σctrl. For instance, a larger number of correlated sensor
inputs may lead to better short term predictions and a lower sensitivity to measurement
noise. It is therefore likely that the actual number of actuators required at the E-ELT for the
same Strehl ratio is smaller.
2.4 Control system and electronics requirements
The goal of the control system is to calculate suitable actuator commands based on wavefront
sensor measurements. The AO system performance ultimately depends on the accuracy
with which the control system can match the mirror shape to half that of the actual
wavefront disturbance. In section 2.2.3 a desired control bandwidth of 200Hz was specified.
To be able to achieve this using a PID (Proportional Integrator Derivative) type controller,
the delays due to sampling with a Zero Order Hold (ZOH), the sensor and the communication
to the DM system must be limited to leave sufficient phase margin. Although this
reasoning does not directly apply to Multi-Input Multi-Output (MIMO) systems in general,
the foreseen DM system can with its first resonance frequency of around 1kHz be well
diagonalized up to the bandwidth – i.e. decoupled into a number of Single-Input Single-
Output (SISO) systems – by the inverse of the DM influence matrix. This is discussed in
more detail in chapter 3.
The phase margin PM is equal to 180 ◦ minus the total phase budget that can be divided
into four parts: the plant, the controller, the sensor and other delays in the loop. When the
first resonance frequency of the DM lies above 1kHz, the phase delay of the plant around
the 200Hz bandwidth will insignificant, but let the delay due to digital communication between
the controller and the DM system be budgeted to 1/10 th of a sample time. Further,
the sampling with ZOH and the wavefront sensor both introduce half a sample delay and let
the controller computation delay be budgeted to one sample. The total loop delay is then
2

2
36 2 Design requirements and design concept
Figure 2.8: Influence of sampling time Ts and
exposure time Te. The black line represents
|S(2πjf)| 2 and the grey line the disturbance
spectrum P(f) that has a horizontal asymptote
on the measurement noise levelσn.
Magnitude 2 [-]
1
− 8
3
σ 2 n
Ts
Te
fc
P
Frequency [Hz]
2.1 sample times, which means that the phase marginPM can be expressed as:
PM = 180 ◦ −2.1 200Hz
360 ◦ +γc,
where fs is the sampling frequency and γc is the phase added by the controller. For a PID
controller with a 2 nd order roll-off filter the latter can be at most 45 ◦ , whereas for a PI
controller with a 1 st order roll-off this reduces to −30 ◦ and for a pure integrator to −90 ◦ .
This means that for a 10 ◦ phase margin PM the sampling frequency fs must be above
700Hz for a PID type controller, above 1kHz for a PI type controller and above 2kHz for
the pure integrator commonly used for AO.
On the other hand, the sampling frequency cannot exceed the frame rate of the CCD
camera of the wavefront sensor. For current devices 1kHz is a realistic rate and for state-ofthe-art
devices this can even be slightly higher. However, as the exposure time is reduced the
measurement noise becomes more significant. The variance σ2 n of the measurement noise
of a Shack Hartmann sensor (SHS) consists of several components that are related to the
exposure time Te in various ways. For instance, photon noise is attenuated by increasing
Te, whereas dark current and read-out noise are attenuated by decreasing it [105, 106, 135].
When measurement noise becomes significant for Te ≤ Ts and its variance is added to the
temporal error variance in (2.8), this sum may not strictly decrease with the sampling frequencyfs.
This can be illustrated using figure 2.8, which sketches the disturbance spectrum
P(f), white measurement noise with varianceσ2 n and a disturbance rejection characteristic
or sensitivity function|S(s)| 2 of the control system, where s is the Laplace variable. Note
that S(s) is thus the transfer function between disturbance and residual error and that the
notationS(s) implicitly assumes the AO system to be Linear Time-Invariant (LTI).
The external disturbance acting on the control loop does not only include the wavefront
disturbance with temporal spectrum P(f), but also the measurement noise with variance
σ2 n . In contrast to the servo tracking point of view that forms the basis for the temporal
error variance discussed in section 2.2.2, a realistic disturbance suppression characteristic
|S(2πjf)| is here used that includes the effect of loop delays and obeys the Bode-sensitivity
integral. This integral implies that the disturbance rejection at low frequencies leads to amplification
at higher frequencies.
When the filter S(s) is applied to a disturbance signal with temporal spectrumP(f)+σ 2 n ,
fs
|S| 2

2.4 Control system and electronics requirements 37
the output (i.e. the residual error) spectrum can be expressed as |S(2πjf)| 2 (P(f)+σ 2 n).
Using Parseval’s theorem, the control error variance σ2 ctrl introduced in (2.6) on page 30
can then be expressed as:
σ 2 ctrl =
0
∞
|S(2πjf)| 2 P(f)+σ 2 n df.
Now let this be applied to figure 2.8. Accordingly, a decrease of the sampling time Ts
may lead to an increased bandwidth fc, but also to an increased disturbance amplification
at high frequencies and since 0 < Te ≤ Ts also to higher measurement noise σn. As a
result, the error variance σctrl may not diminish by a decrease of Ts. The same error may
be achieved using various choices forTs andTe.
As mentioned above, the Te and Ts form control loop delays, as do communication delays
and computation time. In contrast to the exposure time, a reduction of the communication
delays or computation time will always be beneficial to performance. However,
communication speeds have limits and more computational power will result in limited
performance gain at significant costs. A detailed specification of Te and Ts is complicated
by the fact that the measurement noise σn and read-out time are highly device specific,
whereas the WaveFront Sensor (WFS) is not included in the AO system to be designed.
Moreover, model based controller designs are able to predict (to some extent) future
wavefront disturbances and so compensate for loop delays. In an optimal controller design
the effects of measurement noise are also minimized with respect to some cost function.
The noise residual then becomes dependent also on the DM dynamics and the accuracies of
the models used.
Due to these a priori unknowns, the temporal error variance in (2.8) will further be used to
express the worst case error. An indicative sampling time of Ts = 1ms will be assumed,
equal to the exposure timeTe.
As discussed in [65, 66], the best performance is obtained when the number of measurement
positions of the wavefront sensor is proportional to the number of actuators. In the
supplied references the location of the actuators with respect to the wavefront sensors is
not explicitly analyzed, whereas it is known from control theory that performance may
degrade when actuators and sensors are not collocated. On the other hand, the gradient
measurement concept of a SHS versus the deflection based DM actuation already clouds
the notion of collocation.
Nevertheless, for a SHS with two measurements per lenselet, it will be assumed that the
number of measurements Ns is approximately equal to twice the number of actuators
Na. The total processing power of the control system must then be sufficient to evaluate
the command update equations from ∼ 10000 measurements to 5000 command signals
within the sampling time Ts of 1ms. This also involves the processing of the CCD of the
SHS image to obtain the actual gradient measurements [106, 175]. This will be further
elaborated in chapter 3.
The final part of the control system is formed by the electronics hardware. The displacement
of each actuator is changed by a current through the actuator coil, which will be
generated by dedicated electronics that must have sufficient accuracy to meet the specified
5nm actuator position accuracy. Further, the dynamics introduced by these driver electron-
2

2
38 2 Design requirements and design concept
Deformable facesheet
mirror - actuator connection
Actuator grid
Actuator grid - base frame connection
Deformable facesheet
Actuatorgrid with actuator modules
Base frame
Base frame
Figure 2.9: Schematic of the adaptive deformable mirror design.
ics must not reduce the lowest eigenfrequency or rise time of the system that may lead to a
lower achievable control bandwidth.
2.5 The design concept
The design concept for the adaptive deformable mirror that meets the requirements, as listed
in the previous sections, is schematically given in figure 2.9. The design concept is based
on [81]. In the design a few layers are distinguished, which will be discussed in more detail:
• the mirror facesheet,
• the actuator grid,
• the base frame.
The first layer consists of the thin reflective facesheet, which is the deformable element. The
facesheet is continuous and stretches out over the whole mirror. In the underlying layer - the
actuator grid - low voltage electro-magnetic push-pull actuators are located. The actuator
grid consists of a number of identical actuator modules. Each actuator is connected via
a strut to the mirror facesheet. The mirror facesheet, the mirror-actuator connection and
the actuator modules form a thin structure with low out-of-plane stiffness so a third layer
is added, the base frame, to provide a stable and stiff reference plane for the actuators.
This base frame is a mechanically stable and thermally decoupled structure. Besides these
distinguished layers a control system and electronics is present which is also described
briefly.
2.5.1 The mirror facesheet
A membrane-type mirror is chosen because of its low moving mass and low out-of-plane
stiffness. This results in low actuator forces and the best wavefront correction for a given

2.5 The design concept 39
number of actuators (section 2.2.1). Because of the low out-of-plane stiffness of the thin
facesheet, the inter-actuator coupling and the width of the influence functions can be kept
small. This is desirable for currently available efficient control algorithms [53, 72, 188]
and facilitates the implementation of a distributed control system (chapter 3). Finally, the
limited thickness leads to a short thermal time constant that allows for quick adaptation to
changing environmental temperatures.
The mirror-actuator connection
The connection between the actuators and the mirror facesheet is made by struts. Via these
struts, the actuators impose the out-of-plane displacements on the facesheet. The struts
constrain one DOF since their bending stiffness is significantly lower than the local bending
stiffness of the facesheet. Because the struts leave the φ and ϕ rotation free, the bending
stiffness of the facesheet can form a smooth surface through the imposed z-positions, as is
shown schematically in figure 2.11. The piston-effect, shown schematically in figure 2.10 is
thereby avoided. As a result no higher order aberrations are introduced into the wavefront.
The struts neither constrain the x- and y-positions of the mirror facesheet. Differences
in the thermal expansion coefficient of the facesheet material and the actuators and/or a
temperature difference between them is therefore possible without unwanted deformation
of the mirror surface. The struts are glued with small droplets to both the mirror facesheet
as the actuators.
The mirror facesheet and mirror-actuator connection is discussed in detail in [82–85]
and in [174].
2.5.2 The actuator modules
Because a large number of actuators is needed it is attractive to produce actuator arrays
with a layer based construction instead of single actuators, where each of them is positioned
with respect to its neighbors. Therefore, a standard actuator module with 61 low voltage
electromagnetic actuators in hexagonal arrangement is designed. More than 80 of such
actuator modules are needed for a DM with 5000 actuators. figure 2.12 shows the mirror
facesheet with the mirror-actuator connections and the working principle of the actuators
Actuator side
Actuator - facesheet connections with high bending stiffness
Figure 2.10: A connection, between the actuators
and the facesheet, with high bending stiffness
constrains the localφ- andϕ-rotation in the
mirror surface, causing local flattening.
Actuator side
Actuator - facesheet connections with low bending stiffness
Figure 2.11: A connection, between the actuators
and the facesheet, with low bending stiffness
results in a smooth surface and allows lateral expansion
between the facesheet and the actuators.
2

2
40 2 Design requirements and design concept
in schematic. The actuators are of the variable reluctance type and consist of a closed
magnetic circuit in which a Permanent Magnet (PM) provides static magnetic force on a
ferromagnetic core that is suspended in a membrane. This attraction force is influenced by
a current through a coil, which is situated around the PM to provide movement of the core.
figure 2.12 shows that with the direction of the current the attractive force of the PM is either
increased or decreased, allowing movement in both directions.
The efficient actuators are free from mechanical hysteresis, friction and play and therefore
have a high positioning resolution with high reproducibility. The stiffness of the actuator
is determined by the membrane suspension and the magnetic circuit. There exists a large
design freedom for both. The stiffness of the actuators is chosen such that, if one should
fail, no hard point will form in the mirror surface.
The coil wires are soldered to a flex foil. This flex foil is connected to a Printed Circuit
Board (PCB) with dedicated electronics for 61 actuators. Each actuator module is connected
to the base frame via three A-frames . The actuator grid is scalable, so the actuator pitch
can be chosen freely and extendable since many modules can form large grids of actuators.
The actuator module design is described in [88–91] and in chapter 5 and in more detail in
[174].
2.5.3 The control system and electronics
To make the control system and the electronics of the AO system extendable without a full
redesign, a modular structure is foreseen for these components. Each grid of 61 actuators
is given a dedicated electronics module to supply each actuator with its current. These
modules include 61 Pulse Width Modulation (PWM) drivers implemented using Field Programmable
Gate Arrays (FPGAs) and 61 2 nd order analog low-pass filters. For the DM
prototypes, the FPGAs receive their setpoint updates from a PC via a custom designed,
multi-drop Low Voltage Differential Signalling (LVDS) communication link (chapter 6). A
distributed control system is to be implemented in the FPGAs of the electronics modules
chapter 3 and chapter 4. These modules communicate with a limited number of neighboring
modules instead of the complete set, corresponding to a centralized controller.
This architecture has consequences for the controller design, which is discussed in detail in
permanent magnet
reflective deformable facesheet
membrane suspension
strut
N
S
coil
ferromagnetic core
N
S
base plate
PM PM - Coil PM + Coil
Figure 2.12: Three actuators shown in schematic.
N
S
undeformed
deformed

2.5 The design concept 41
chapters 3 and 4.
2.5.4 The base frame
To support the 80 actuator modules, a light and stiff and thermally stable base frame has
been designed. The diameter of this support structure is 500mm, its height 150mm and
its mass 5kg with a first mechanical resonance frequency of 1kHz. This base frame is a
welded hexagonal box with ribs made of 2mm thick aluminium plates. The cover of the box
is a 25mm thick aluminium honeycomb plate, which supports the actuator modules. Aluminium
is chosen because of its good thermal properties. Since the box is well ventilated,
it will adapt quickly to changes in the ambient temperature and hereby expand and contract
homogeneously. Finally, the box can contain the electronics for the mirror modules.
2

ÔØÖØÖ
ÒØ ÓÒØÖÓÐÓÖÇ
ÓÒÔØ×Ò ÐÐÒ×
As the number of degrees of freedom of the deformable mirror and the wavefront
sensor increase, scaling problems arise for the number of computations
required by the control system. Without efficient algorithms, the demanded
computational power increases quadratically with the number of degrees of
freedom and thus to the fourth power in the telescope’s aperture diameter. Further,
practical problems arise for cabling and driver electronics for the actuators.
A modular, distributed control system is proposed to solve these problems,
but this approach raises new challenges that will be discussed and possible
solutions will be proposed.
43

3
44 3 Efficient control for AO: concepts and challenges
3.1 Introduction
In this chapter, the problems that will arise for the control system when the number of degrees
of freedom of the Adaptive Optics (AO) system increases will be discussed. Therefore,
first the general control problem will be formulated and several commonly used approaches
for controller design will be discussed.
It will be shown in section 3.3 that the computational demand of these approaches scales
with the fourth power in the diameter of the telescope’s aperture. This is not acceptable
for future large telescopes and has lead to lots of research into efficient algorithms (e.g.
[53, 72, 124]) of which a few will be discussed in more detail. However, these algorithms
are tailored for a specific control law, whereas recent reports indicate that a significant performance
increase can be achieved using designs that are both temporally and spatially
optimal (e.g. using the H2 framework [100, 123]). This is mainly due to the incorporation
of more detailed models of the atmospheric disturbance. Therefore, in this thesis such models
will form an important part of the controller design.
Besides computational demand, other more practical issues arise when the AO system is
extended to more actuators and sensors: design of the electronics, wiring, communication,
etc. It will be discussed how these issues are simplified if a distributed structure is chosen
for the control system that will be elaborated in section 3.4. However, such a structure has
major consequences for the controller design. These consequences and problems will be
discussed and several ideas will be presented that may offer (partial) solutions. These ideas
also form the basis for the approaches described in the following chapters of this thesis.
3.2 Existing control methods
In this section the general AO control problem will be stated and followed by several approaches
towards a solution. Subsequently, these methods will be analyzed with respect to
their computational demands as the diameter of the telescope for which the AO system is
designed increases.
3.2.1 Generic problem statement
The aim of the AO system is to compensate the local variations in the wavefront of
the incoming light that are caused by atmospheric turbulence. The observed wavefront
distortion can be influenced with a wavefront corrector such as the deformable mirror
designed within this project. As mentioned in chapter 2, the performance of an AO system
can be measured using the Strehl ratio. According to the Maréchal approximation, this ratio
is related to the variance of the wavefront distortion. It is assumed that the wavefront is flat
before it enters the earth’s atmosphere such that the general aim for the AO control system
is to minimize the wavefront variance. Note here that the image quality is not affected by
the global average of the wavefront (the piston mode). For controller design, the wavefront
entering the earth’s atmosphere is therefore defined to be zero.
As mentioned in chapter 2, the control system will have a closed-loop configuration,
which means that the sensors register the effects of the control actions. The control loop

3.2 Existing control methods 45
φ
φ u u y
H
C
n
G
φ e
Figure 3.1: Schematic of a
generic AO system in closedloop
configuration.
can be described in general terms using operators to represent the dynamic systems and
functions of timetto represent the signals to avoid representation or implementation details.
Let the wavefront phase disturbance of the light entering the telescope at time instant t be
denoted by the signalφ(t). This is influenced by the wavefront corrector that introduces an
additional wavefrontφ u(t) that is related to the actuator commandsu(t) as:
φ u (t) = H{u(t)}, (3.1)
where the operator H represents the wavefront corrector. The combined wavefront error
φ e(t) that forms the input to the wavefront sensor then becomes:
φ e(t) = φ(t)+φ u(t). (3.2)
Let the wavefront sensor be represented by the operator G and its measurements y(t) be
subject to additive measurement noisen(t) such that it can be expressed as:
y(t) = G{φ e(t)}+n(t). (3.3)
The measurementy(t) is fed back to the controllerC with outputu(t). With a slight misuse
of notation, let 〈φ e (t)〉 denote the average or piston term of the wavefront that does not
affect the quality of the resulting image. The controller must then minimize the piston-free
wavefront errorφ e(t)−〈φ t(t)〉 in a minimum variance sense:
C = argmin
C
〈φ e(t)−〈φ e(t)〉2〉, (3.4)
subject to (3.1), (3.2) and (3.3). This control loop is shown schematically in figure 3.1. This
controller synthesis problem can be approached in various ways. Each one mainly differs
in the choices for the models of the wavefront corrector systemHand the wavefront sensor
G and by the assumptions on the disturbance signalsφ(t) andn(t). In the next subsections,
these choices and assumptions will be elaborated on for several important design approaches
described in literature.
3.2.2 Traditional approach
Traditionally, the plant P to be controlled is taken as the combination of the wavefront
corrector system H and the wavefront sensor G [55]. The control loop elements are considered
to be discrete time systems, expressed using the dimensionless time variable t and
z-transform variablez. The plant P is described as a static gain matrixP with an arbitrary
numberd ∈ N of samples delay, s.t. the measurement can be expressed as:
y(t) = Gφ(t)+Pu(t−d)+n(t).
3

3
46 3 Efficient control for AO: concepts and challenges
Figure 3.2: Schematic of a
generic AO system in openloop
configuration with controllerR.
φ
G
The controllerC is given a predefined discrete time structure that consists of integrators with
gainαand leak factorβ and an inverse plant model described by a static matrixP # :
n
s
R
u
H
C(z) : u(t) = −α(I−βz −1 I) −1 P # y(t), (3.5)
whereTs the sampling time. Although this equation suggests that the gainsα andβ must be
equal for all integrators, this is not necessarily the case. Due to the wavefront sensing principle,
the plant matrix P is usually singular and its inverse does not exist. In practice, the
inverse model P # is therefore obtained as a pseudo-inverse to circumvent this. Nevertheless,
theP # term in the controller will decouple the plant dynamics up to the unseen modes
of the WaveFront Sensor (WFS) and allows separate Single-Input Single-Output (SISO)
controllers – in this case integrators – to be designed for each actuator loop. Integrators provide
a high gain at low-frequencies and thus a high disturbance suppression. The gain and
leak factor α and β can be tuned for maximum control bandwidth – and thus disturbance
rejection performance – while satisfying certain stability margins.
This controller structure is commonly used in practice and provides respectable performance.
Partly this is because the static plant assumption is often not far from the truth,
but also because an integrator has the right magnitude profile over frequency to suppress
the atmospheric wavefront disturbance. This can be explained in a SISO setting by first
recalling that for stochastic disturbance rejection the residual error of the optimal controller
is white noise. This means that the product of the disturbance model and the sensitivity
function of the control loop has a magnitude that is constant over frequency. When the
control law is an integrator and the Deformable Mirror (DM) a static gain, the sensitivity
function has a +1 asymptote for frequencies up to the control bandwidth, which depends on
the integrator gain. As elaborated in chapter 2, the atmospheric turbulence is often modeled
by assuming a Kolmogorov [98] spatial spectrum and the Taylor hypothesis (frozen flow)
to provide the temporal spectrum. According to these assumptions, the temporal power
spectrumΦ(f) is related to the temporal frequencyf as:
Φ(f) ∝ f −8 3.
This −8/3 exponent is close to −2, which means that for frequencies up to the control
bandwidth the residual error spectrum described by the product Φ(f)|S(f)| 2 becomes
approximately constant (’white’).
Nevertheless, an integrator only compensates for the temporal behavior of the atmospheric
wavefront disturbance. A spatial model is usually included by designing the inverse
plant modelP # – often referred to as the reconstructor – as the open-loop, minimum vari-
ance reconstructorR:
ˆR
2
= argmin φ(t)−HRs(t)F , (3.6)
R
φ u
φ e

3.2 Existing control methods 47
wheres(t) is the open-loop measurement defined as (figure 3.2):
s(t) = Gφ(t)+n(t) (3.7)
Note that in literature additional regularization and weighting matrices are applied for specific
cases that will not be considered here. This optimal open-loop reconstructorR replaces
P # in the closed-loop control law in (3.5), which means that the effect of closing the control
loop on the spatial spectrum of the disturbance is not accounted for. The pseudo open-loop
control concept presented in [54, 187] compensates for this using an approach based on the
internal model control principle (section 4.4.2).
To obtain ˆ R, it is assumed that the behavior of both the wavefront corrector and the wavefront
sensor can be described by the static matrices H and G respectively. Then rewrite
(3.6) in terms of the vectorialr = vec(R) while omitting references to timetfor brevity:
φ−(s ˆr = argmin
r
T ⊗H)r 2
,
F
= argmin φ
r
T φ−2φ T (s T ⊗H)r+r T (ss T ⊗H T
H)r ,
= argmin φ
r
T
φ −2vec T H T φs T r+r T ss T ⊗H T H r,
= vec (H T H) −1 H T φs T
T−1
ss , (3.8)
ˆR
where⊗denotes the Kronecker product [23] and use is made of the identity vec(ADB) =
(BT ⊗ A)vec(D). The last step follows from a completion of squares argument. By regarding
the plant as two separate processes instead of only their product, statistical knowledge
of the wavefront disturbance can thus be exploited. By using the sensor model
s(t) = Gφ(t−d)+n(t), the covariance matrices can be rewritten to:
T
φs
= φ(t−d)φ T (t−d)G T
= φφ T
G T ,
T
ss = G φ(t−d)φ(t−d) T G T + n(t)n T (t)
= G φφ T
G T + nn T .
The covariance matrix
φφ T
can be calculated from the Kolmogorov or Von Karman
spatial spectra [95] and nnT is often taken as a diagonal matrix whose elements are
determined from e.g. sensor and/or guide star properties.
It is important to notice that the optimal static reconstructor in (3.8) consists of two parts:
T
φs ss T−1
ˆR = H T H −1 H T
part 2
part 1
. (3.9)
Part one is the reconstruction of the wavefront phase from the WFS measurements and part
two is the projection of the wavefront onto the actuator space. These form the solutions to
two sequential least-squares problems formulated as:
ˆφ s(t) = arg min φ(t)−φ s(t)
φ (t) s 2 F, (part 1) (3.10)
û(t) = argmin
u(t) ˆ φ s(t)−Hu(t) 2 F, (part 2)
3

3
48 3 Efficient control for AO: concepts and challenges
where ˆ φ s(t) is the minimum variance reconstruction of the actuator wavefront phase disturbanceφ(t)
given the open-loop WFS measurement vectors(t).
Using the internal model control principle (section 4.4.2), the equivalent closed-loop controller
can be derived from the obtained open-loop controller ˆ R as:
C = C(z) = −(I− ˆ RGHz −1 H) −1ˆ R. (3.11)
When ˆ
R is based on the covariance matrices φφ T
= I and nnT = 0 then the
equivalent closed-loop controller reduces to the traditional integrator structure in 3.5 with
α = β = 1 . In practice the optimal open-loop reconstructor from 3.9 is often used to
replace theP # matrix in 3.11
However, as mentioned this method is only optimal under several specific conditions
[123]. Besides the DM being static, the spatial and temporal parts of the wavefront disturbance
behavior should be independent. Moreover, the temporal behavior of the latter should
be well described by a random walk (integrated white noise) process. Since none of these
conditions are in general fully met, recent work has been aimed at finding more accurate
plant and disturbance models to be applied within e.g. theH2 optimal control framework.
3.2.3 More generic controller designs
The H2 optimal control framework allows the direct synthesis of an Linear Time-Invariant
(LTI), dynamic, closed-loop controller C = C(z) based on dynamic models for the components
in the control loop. These models comprehend the turbulence, the corrector and
the sensor and together form the so called generalized plant. Results have been reported by
[16, 100, 123, 150] showing a significant improvement in optical quality when compared to
that achieved with the traditional controller design. Moreover, this advantage is shown to
increase for higher Greenwood frequencies [100].
Where [123] only suggests to use a dynamic disturbance model, in [16, 150] a physical
model is actually used that is based on Kolmogorov statistics and Zernike polynomials [136]
in combination with Taylor’s frozen flow assumption and a wavefront corrector modeled as
a static matrix gain with a delay. In [100] a data driven approach is proposed in which an
LTI disturbance model is estimated from measurement data alone and the wavefront corrector
is modeled as a two-tap Finite Impulse Response (FIR) filter.
Despite their differences, all these approaches yield closed-loop controllers C = C(z) of
the form:
C(z) :
x(t+1)
=
u(t)
Ac Kc
Cc Dc
x(t)
, (3.12)
y(t)
where x(t) ∈ R Nx is the state vector whose length Nx is the sum of the lengths of the
states of all models considered in the control loop.
In practice the behavior of the atmospheric disturbance varies slowly over time, which
means that the process that generatesφ(t) is not fully stationary. For best performance, the
disturbance model should therefore also vary over time, which can be accomplished using
adaptive control schemes [56, 117]. Such schemes are available to update the covariance
matrices of the traditional control law and also to update the coefficients of more generic
control laws [44].

3.3 Scaling problems 49
3.3 Scaling problems
As discussed in chapter 2, there should be a balance between the spatial bandwidth of the
wavefront corrector that is determined by the number of actuators Na and the temporal
bandwidth of the AO system that is also determined by the control system. The spatial bandwidth
determines the fitting error whose variance is inversely proportional to Na, whereas
the latter determines the temporal error. This depends on the disturbance behavior and is
independent of Na. Thus, to keep the balance for a telescope with a larger aperture area,
Na must be chosen linearly proportional to this area such that the actuator density remains
constant. Similarly, the spatial density of the WFS grid must remain constant when the
aperture area is enlarged. In this section, the consequences hereof will be discussed for both
the hard- and software of the control system.
3.3.1 Computational demand
When the numbers of actuators Na and sensors Ns are linearly proportional to the telescope’s
aperture area, the number of in- and outputs of the control system scale quadratically
with the aperture diameterDt.
At each sampling time, the control system has to perform two sequential tasks. Firstly, when
assuming the WFS to be of the Shack-Hartmann type [98] – it has to process the Charge
Coupled Device (CCD) images of the WFS to find the gradients. Subsequently, it has to
update the setpoints of the actuators of the wavefront corrector according to a given control
law.
Centroiding
Many algorithms have been devised to determine the spatial gradients y(t) from the CCD
image of a Shack-Hartmann sensor [27, 106, 142, 175]. However, the centroiding or center
of mass method [98] is probably the best known. The number of computations of this
algorithm is linearly proportional to the number of spots of the WFS. It can be split into
parallel problems for each sensor spot and even be implemented in the CCD hardware.
Therefore, this part of the computational demand will not be further considered.
The traditional control law
The second task of the control system is to implement the control law. The controller receives
the measured gradients as input and has to calculate the new setpoints for the wavefront
corrector. For the traditional and optimal static controller structures, this consists of
the application of the inverse model P # or the reconstruction matrix R and the temporal,
SISO integrators.
Even though the plant matrix P is for most corrector technologies very sparse – i.e. each
actuator influences only the WFS readings within its vicinity – all elements of the (pseudo-)
inverse P # are usually non-zero. The same holds for the optimal reconstruction matrix
R in (3.9). Thus, if implemented as a direct matrix multiplication, the number of computations
Nc required by the matrix-vector product between P # or R and the measurement
vectory(t) is equal to the productNaNs, where Ns is the number of measurements of the
3

3
50 3 Efficient control for AO: concepts and challenges
Figure 3.3: Sparsity of the matrix G T for a
Fried geometry and a sensor grid of 15 × 15
lenselets.
Figure 3.4: Sparsity of the matrix G T G for a
Fried geometry and a sensor grid of 15 × 15
lenselets.
WFS. Both Na and Ns are linearly proportional to the aperture area Aa of the telescope
and quadratically in its diameterDt:
Na ∝ Ns ∝ Aa ∝ D 2 t .
The number of computations required for application of the integrator scales only linearly
with Na and does not change the order of magnitude of the total number of computations.
The number of computations Nc required for the traditional control law is thus related to
the number of actuatorsNa and the telescope aperture diameterDt as (figure 3.5):
Nc ∝ N 2 a ∝ D 4 t.
This means that to increase the aperture diameter with a factor two, the number of processors
would have to increase at least a factor 16, which is expensive and leads to many
practical problems. Therefore, lots of research effort have been spent at reducing the computational
demand of the application ofP # orRas in (3.9).
This can be done by exploiting structure and sparsity of the involved matrices [53, 73, 74,
187–189]. As mentioned, for most corrector technologies, each actuator has only a local effect
on the wavefront induced by the corrector, which renders the influence matrixHsparse.
Further, as illustrated in figures 3.3 and 3.4 the geometry matrix G that relates the actual
wavefront error to the WFS measurements is sparse for zonal relations such as Fried or
Hudgin (chapter 1). Finally, when considering a Kolmogorov spatial spectrum, the inverse
of the covariance matrix
φφ T
can be approximated using discrete approximations of the
Laplacian operator [53, 187]. The approximation lies in the fact that the −11/3 exponent
of the Kolmogorov spatial spectrum is changed into−4. Using sparse-plus-low-rank matrix
techniques, this allows for an efficient implementation of the application of the reconstruction
matrixRthat scales approximately asN 3/2
s [53].

3.3 Scaling problems 51
Computations per second
10 16
10 14
10 12
10 10
10 8
10 6
10 1
10 2
Primary mirror diameter [m]
0.4 0.6 1 2 4 6 10 20 40
Classic integrator
Optimal control
Distributed control
10 3
Number of actuators
10 4
10 5
10 6
10 4
10 2
10 0
Number of processors
Figure 3.5: The estimated
number of computations or
processors required for the unoptimized
implementation of
various control algorithms including
the centroiding step
versus the telescope’s aperture
diameterDt and the number
of actuators Na of the
wavefront corrector.
The latter referenced method is aimed at obtaining an efficient algorithm to find the exact
solution to the problem posed in (3.6) using sparse approximations of the matrices involved
in the solution in (3.9). Another class of iterative algorithms is aimed at finding an approximation
of this solution with a known accuracy. These methods are usually based on Krylov
subspace methods [152], in particular on the conjugate gradient method [73, 74, 188, 189].
These methods can a.o. be used to solve linear problems of the formAx = b for the vector
x of unknowns and the known matrix A and vector b. The reconstructor in (3.9) can be
written into this form. The solution vector x (m) at iteration m is updated each iteration in
the directiond (m) with a step size α (m) as:
x (m+1) = x (m) +α (m)d (m), where α (m) = rT (m) r (m)
d T (m) Ad (m)
(3.13)
and r (m) = Ax (m) − b is the residual vector. The method requires the search directions
d (m) to be A-orthogonal or conjugate, i.e. d T (m) Ad (n) = 0 for m = n. This makes sure
that the exact solution is reached after Nm iterations, where Nm is the dimension of the
square matrixA. The search directions are built from the residual vectors as:
d (m+1) = r (m+1) +β (m+1)d (m), where β (m+1) = rT (m+1) r (m+1)
rT (m) r . (3.14)
(m)
These choices for α (m) and β (m+1) imply that the conjugacy condition is satisfied.
However, in practice this is prone to numerical round-off errors.
The number of computations of the conjugate gradient method scales as Nalog(Na) times
the number of iterations required for convergence. The latter number depends on the spread
of the eigenvalues of A, but in general the computation time of this method is of the same
order as that of the direct matrix product. It mainly offers a storage reduction as the usually
dense inverse matrixA −1 does not need to be stored.
To improve the convergence speed of the algorithm for the specific problem of
wavefront reconstruction for AO, this method can be extended with a variety of precon-
3

3
52 3 Efficient control for AO: concepts and challenges
ditioners [73, 188]. Using the preconditioner M, the system of equations is modified to
M −1 Ax = M −1 b. The matrix M −1 should approximate A −1 and calculation of the
productM −1 v wherev is an arbitrary vector should require few computations. This allows
shaping of the eigenvalues ofM −1 A for better convergence properties. The preconditioner
can be inserted into the conjugate gradient algorithm in a matrix-free way, i.e. neitherM −1
itself nor the product M −1 A needs to be explicitly evaluated. At each iteration, only the
product ofM −1 with the residual vectorr (m+1) needs to be calculated.
In practice, the multi-grid [73, 189] and the Fourier domain preconditioners [188] yield the
most efficient solutions. The first is much related to the hierarchic reconstructor described
in [124]. It is also used for solving discretized Partial Differential Equation (PDE) problems
such as Finite Element Model (FEM) problems and steady state diffusion equations and
exploits the mesh structure of a problem. The residual vector is resampled on different
hierarchic levels of mesh density. A so-called V-cycle is then often performed, starting
from the finest mesh, through all intermediate levels to the coarsest mesh and back again.
Transition from one level to another involves interpolation and restriction operations
that are alternated with smoothing operations. These smoothing operations can be one or
several iterations of another – computationally cheap – iterative solver such as Gauss-Seidel
or successive over-relaxation (SOR). This way, the wavefront reconstruction problem is
solved on hierarchically alternating levels of spatial resolution. However, the computational
efficiency depends highly on the specific choices for the interpolation, restriction and
smoothing operations. For only the wavefront reconstruction step, an order-N computational
demand – i.e. Nc ∝ Ns ∝ D 2 t – has been reported [72, 189]. It is further shown that
bilinear and even bicubic DM influence functions can be used to to solve the second step of
mapping the wavefront to actuator commands with the same method [189].
Since the 2D Fourier transforms of the sensor geometry matrixG, the covariance matrix
φφ T
and the influence matrix H can often be well approximated by sparse – or even
diagonal – matrices, a Fourier domain preconditioner is a viable alternative to multi-grid
[188, 196]. Here the preconditionerM−1 consists of three operations:
M −1 = F −1 ˜ M −1 F,
whereF represents the 2D Fourier transform and ˜ M −1 a sparse Fourier domain operation.
It is also possible to transform the whole problem into the Fourier domain, but this yields
no further reduction in computational demands [188]. The Nslog(Ns) cost of the Fourier
transforms dominates the computational demand of this preconditioner.
Besides the sparse matrix and iterative approaches, several other efficient algorithms
are reported in literature. Examples are the direct Fourier domain approach in [143]
and the hierarchic approach in [124]. The computational demands of Fourier domain
approaches are dominated by the Nc ∝ Nslog(Ns) cost of the Fourier transform. For
the hierarchic approach this cost is claimed to be Nc ∝ Ns, but similar to the direct
P #
approach described earlier no covariance matrix φφ T
is considered and also the
corrector influence matrixHis taken as an identity matrix.
It can be concluded that for the traditional control law with a static reconstructor R,

3.3 Scaling problems 53
many efficient algorithms are available. However, linear scaling of the computational demand
(i.e. Nc ∝ Ns) is only reported when specific assumptions or approximations are
made on G, H and φφ T
[72, 124]. This may still be acceptable for large telescopes,
since the involved operations can be well parallelized and the computation speed of computation
hardware still increases. The fact that the traditional control law in general does not
provide the best achievable performance is then taken for granted.
The H2 optimal control law
In its most generic form, the computational demand of the H2 optimal controller in (3.12)
involves the products of all four system matrices with the corresponding vectors. This
demand can be quantified as (figure 3.5):
Nc = N 2 x +NxNs+NxNa +NaNs,
whereNx denotes the number of states. It is possible to reduce this computational demand
using a state transformation into for example the output normal [186] or block companion
form [161]. However, this approach does not reduce the order of magnitude of the demand
and therefore offers no solution to the scaling problem. Moreover, such transformations
usually have a detrimental effect on the numerical conditioning of the system matrices and
thus on the controller’s performance.
The computational demand depends highly on the number of states Nx, which makes the
relation between Nx and the aperture diameter of special importance to be able to relate
the computational demand to the aperture size. However,H2 optimal controller design for
AO is only recent work and there is little literature available on possibilities for efficient
implementations; neither for the approaches based on physical modeling of the wavefront
disturbance [16, 123, 150] nor for the data driven ones [100]. An important unknown is the
relation between the number of sensors and the required number of states for a atmospheric
disturbance model identified from measurement data to maintain a sufficient level of accuracy.
For the analysis at hand it will therefore be assumed that the number of states Nx is
linearly proportional to the number of actuators.
Consequently, the computational demand of implementations of theH2 optimal control law
is also quadratically proportional to the number of actuators Na and to the fourth power
with the aperture diameterDt. The same scaling problems will thus have to be faced as for
the traditional approach. However, since optimal control for AO is more recent work, few
approaches are documented in literature.
The system matrices are usually derived from the solution of an algebraic Riccati equation.
This makes it very difficult to enforce a structure for the resulting system matrices that
can be exploited for an efficient implementation. For the data driven approach described
in [100] the structure of the optimal controller is given more explicitly in terms of the disturbance
model, the corrector model and the sensor geometry. This approach allows more
freedom to enforce structure to be exploited for implementation efficiency. It is shown in
[100] that similar to the traditional control law the H2 optimal control law contains a least
squares inversion of the sensor geometry matrixG. Depending on the model chosen for the
corrector this is also the case for the influence matrixH. Both steps can be efficiently done
using methods described for the traditional control laws. Further, the model set to describe
the atmospheric disturbance can be chosen as desired, although this will affect its quality.
3

3
54 3 Efficient control for AO: concepts and challenges
These model structures will be more extensively discussed throughout this thesis in section
3.6 and chapter 4.
3.3.2 Practical problems
Besides computational problems there are many problems of a more practical nature that
arise when the number of actuators is increased [21]. The actuators usually require an analog
control signal that is provided by driver electronics. In case of thousands of actuators,
it is desirable to keep the interconnections between drivers and actuators short to limit mechanical
defects, cross-talk, electromagnetic interference, etc. This can be realized using
driver modules that drive a single actuator or a small group and are positioned close to
them. Fast, digital communication links can then be used between the control system and
the drivers for setpoint updates.
However, digital communication links require additional electronics and introduce latency
that reduces the achievable performance of the AO system and should if possible be avoided.
This has given rise to the concept of distributed control that will be discussed in the next
section.
3.4 Distributed control
The distributed control concept used throughout this thesis is based on the desire for a
control system that is modularly extendible both in hardware and in software (algorithms).
Extendible in hardware means that the hardware consists of modules that all contain
processing power and interface with each other, with the DM actuators and with the WFS.
This implies that neither the computational demand of each module, nor the number
of connections to other modules, nor the number of data words to be communicated
by each module may depend on the total number of DM actuators. Each module may
receive only a fixed number of measurements from the WFS, which – as the system is
extended – becomes a smaller and smaller fraction of the total size of y(t). It can only
obtain other measurements indirectly via communication with other modules. A controller
implementation architecture is chosen for this concept in which each actuator is controlled
by its own controller – implemented on a local hardware module – which can communicate
with a the controllers of a few neighboring actuators.
When considering the frozen flow characteristic of the atmospheric disturbance, the
measurements that are the most relevant to a local controller – e.g. for prediction – are
those that correspond to a location in the wavefront nearby its actuator. In particular the
nearby upstream measurements provide accurate information for short-term prediction, but
the flow directions of the disturbance may vary and are not a priori known. Therefore, no
directionality will be used in the selection of available sensors. It will be assumed that each
controller has access to all measurements taken within a small, fixed radius from its actuator.
The influence of the actuators on the wavefront correction is assumed to be the identity
operation: the vector of actuator commands equals the wavefront correction. The assumption
of a Fried geometry then leads to the actuator/sensor layout in figure 3.6. In practice
the spatial dynamics of a DM is often approximated by a static influence matrix. Although

3.4 Distributed control 55
Figure 3.6: Schematic of the
proposed distributed control
concept. The boxes represent
actuators with their designated
controllers and the
dots mark the measurement locations
of the WFS.
an arbitrary matrix may significantly affect the performance of a distributed system (e.g.
consider the case that each distributed controller node is wired to a random actuator), for
most DMs each actuator has a local effect on the wavefront. Distributed controllers can
then coordinate their actions, such that performance loss may be limited. However, this is
considered an important subject of further research.
In figure 3.6 each box represents a controller with its corresponding DM actuator and each
dot the location of gradients measured by the WFS. The solid arrows indicate undirected
communication that is possible between the central controller node and those within a radius
indicated by the solid circle. Similarly, the dashed arrows indicate the sensors whose
measurements are available to the central node. These sensors lie within a radius indicated
by the dashed circle. These radii may be different, as long as they are independent of the
number of degrees of freedom of the AO system. This method of selecting nodes and sensors
for direct communication will be used throughout this thesis for reasons of simplicity.
3.4.1 Hardware considerations
The proposed concept does not automatically imply that each controller is implemented
in its own control hardware. For the new DM designed within this project, each actuator
module containing 61 actuators is driven by a single driver board. These boards could be
augmented by computation hardware for a similarly clustered implementation of 61 local
controllers.
Depending on the communication requirements of the local controllers, the boards must
have communication links with surrounding boards. When the DM design would be extended
with another module of 61 actuators, this module would only need to be given its
own electronics board with its communication links to its neighbors.
This allows the hardware of each electronics board to be identical, which reduces production
costs for large systems. The modules can be used for correctors with various numbers
of actuators and can be easily replaced in case of failure. They include the required driving
electronics to prevent the need for additional communication and be positioned close to the
actuators.
A final element that has so far not been considered is how each control module receives its
3

3
56 3 Efficient control for AO: concepts and challenges
WFS measurements. Within this research project this sensor has been considered a given
part of the system and its properties have not been investigated. However, an extendible
scenario could be foreseen in which each CCD or quad-cell chip of the WFS is read out by
an electronics module that calculates the wavefront gradients and uses a fast, one-to-many
type of communication link to send the results to all relevant controller modules.
3.4.2 Control considerations
A distributed hardware architecture has major consequences for the synthesis and performance
of a control law that must be thoroughly evaluated. Since each actuator has a controller
with associated processing power, the available processing power scales linearly with
the number of actuators (figure 3.5). As mentioned, algorithms with linearly scaling computational
cost are available for wavefront reconstruction, but it will be shown in the sequel
of this section that these algorithms are not readily suitable for the distributed architecture.
Moreover, it has been shown by e.g. [16, 100, 150] that anH2 optimal control design offers
major performance advantages. They are mainly due to the incorporated disturbance model
that provides accurate predictions based on past measurements. As the number of measurements
increases for large telescopes, this accuracy and thus the optimal control advantage
are expected to further increase.
Therefore, in this thesis the distributed controller design will be aimed at approaching the
H2 optimal performance. Ideally, the control nodes together implement a control lawC that
minimizes the criterion in (3.4) subject to the constraint of a distributed structure. This constraint
will inevitably reduce the control performance in comparison to an unconstrained,
centralized controller. As will be investigated in the next chapter, this reduction depends on
the WFS measurements the nodes receive, on the number of neighbors that each controller
can communicate with and on the information that is communicated. However, the effect
that sampling frequency has on the performance trade-off will not be considered. Sampling
frequency determines the integration time of the WFS and thus the Signal to Noise
Ratio (SNR) of the measurements. It is conceivable that in a distributed setting the sampling
frequency that gives optimal performance in terms of correction quality is different
than in the centralized setting.
In the sequel of this section, an initial investigation will be presented of the consequences of
the distributed architecture for the implementation of existing controller designs. The main
challenges will be shown together with possible solutions that will be further elaborated in
the sequel of this thesis.
3.5 Challenges
As mentioned, the proposed distributed architecture restricts the total available processing
power to a multiple of the number of actuators. This implies that the access of local controllers
must also be restricted to a sub-set of the measurements in y(t). The significance
of this can be shown for the traditional control law and will also be elaborated in chapter 4.
As mentioned in section 3.2.2, the traditional control law consists of two parts: the
reconstructor and the integrator. Since the integrators are independent SISO controllers,
this part directly fits the distributed architecture. However, for the reconstructor part this is

3.5 Challenges 57
entirely different. The open-loop reconstructed command for actuator i is a weighted sum
of all WFS measurements. These weights form the i th row of the reconstruction matrix
R and for the solution in (3.9) all elements are non-zero. This implies that each actuator
requires access to all measurements, which is not allowed for the distributed architecture.
In fact, since the available computational power is linearly proportional to the number of
actuators, algorithms with a higher scaling of computational demand are not suitable. They
may be implemented for a certain number of actuators, but require a redesign of the control
hardware if this number increases.
Although there are only a few wavefront reconstruction algorithms with a claimed linear
scaling of computational demand [72, 124], it is interesting to briefly consider several
principles of these algorithms in more detail. The conjugate gradient methods used
in [72, 73, 188] involve the step distance calculations for α and β given in (3.13) and
(3.14). Both expressions contain in-products over the entire residual vector that cannot
be evaluated on the distributed architecture. Also the multi-grid and Fourier domain
preconditioners of these methods involve operations that do not fit its structure. The
interpolation and restriction operations of the first require a hierarchic communication
structure, whereas the distributed communication structure has only a single-level. The
recursive, fast implementation of the Fourier transform requires a similar structure.
On the other hand, in [124] an approach is presented that does actually fit the distributed
architecture. Local controllers are designed that receive only a few WFS measurements
from a small area surrounding the actuator. Global performance is partly recovered by using
the previous phase estimates ˆ φ(t−1) that are locally available, which is possible because
the wavefront distortion is strongly correlated in both space and time. However, according
to the authors this comes at the cost of a significantly smaller rejection of disturbances
with low spatial frequency that can be recovered by using again a hierarchic architecture.
Without changing to a hierarchic structure, this concept of using past, local phase estimates
to recover global performance is considered within a more general framework in the next
chapter.
When considering the distributed control problem within theH2 optimal control framework
then the structural constraints imposed on the sought controller lead to a non-convex
optimization problem. Results have been shown for distributed systems satisfying certain
spatial invariance properties [14, 41, 79, 129], but no generic results are available for distributed
H2 optimal control. This suggests that for efficient synthesis of distributed controllers,
it is important to design the AO system components appropriately.
However, not all AO system components can be freely designed. Particularly for this
project, the WFS is considered to be given as a Shack-Hartmann sensor. This sensor leads to
the wavefront reconstruction step discussed for the traditional control approach, for which
no scalable distributed implementations are currently available.
Further, controller synthesis for an AO system is a disturbance rejection problem. Although
the structure for the disturbance model can be chosen as desired, the disturbance itself can
obviously not be influenced. It is possible to choose structured model sets with amenable
properties for distributed control, but this may come at the cost of performance. This forms
one of the possible concepts to overcome the problems for distributed control that will be
discussed in the remainder of this chapter.
3

3
58 3 Efficient control for AO: concepts and challenges
3.6 Possible solutions
Given the mentioned problems that arise from the choice for a distributed hardware architecture
of the control system, several (partial) solutions will now be introduced and discussed
of which some are already suggested in literature.
3.6.1 Phase reconstruction through analog electronics
One of the first functional AO systems used a shearing interferometer [98] to measure
the wavefront disturbance. Similar to the Shack Hartmann WFS this sensor needs postprocessing
to estimate the phase disturbance. At that time, computing power was almost
non-existent and phase reconstruction could not be done numerically.
The sensor geometry of the shearing interferometer corresponds to the Hudgin geometry
[98] and yields a very sparse, fixed geometry matrix G. This geometry can be translated
to a simple electronic resistor network, where node voltages correspond to wavefront phase
and resistor currents to gradient measurements. Phase reconstruction – i.e. the application
of G # to the vector of gradient measurements – can be performed using this circuit
by applying the measured gradients as currents using current sources and measuring the
node voltages [96]. The effect of measurement noise can be included by applying a suitable
amount of noise to the currents and adding shunt capacitances between the nodes and
ground.
The settling speed of the network depends on the design of the current sources and the shunt
capacitances and was reported to be in the order of ms. Although this settling time depends
on the network size and thus the number of measurementsNs, studies have shown that this
principle can be extended to systems with 10000 Degrees Of Freedoms (DOFs) [98] and
can even incorporated into the CCD chip of the WFS.
For this the patented principle must be adapted to the Fried geometry. However, it does
not allow knowledge of phase statistics to be included, but e.g. for the data driven optimal
control approach in [100] this is not required. The controller derived therein involves only
the unweighted pseudo-inverseG # .
3.6.2 A distributed disturbance model
When considering the frozen flow properties that are generally attributed to the disturbance,
it can be concluded that the information most relevant for predicting it at a discrete point is
found in a small area surrounding that point. This suggests that the reduction in prediction
accuracy between an unstructured model and a distributed model structure will be limited.
In literature this has been verified by [45]. Here an FIR type predictor is used to predict the
local open-loop sensor readings (i)(t) fori = 1...Ns of the form:
ˆs (i)(t) =
A(i,j)(z)−1 s (j)(t),
j∈Ni
whereA (i,j)(z) are scalar, monic polynomials in the z-transform variablez and the setsNi
contain all indices of measurements within a small patch around sensor i includingiitself.
Optimality of this predictor structure implies an underlying auto-regressive model structure

3.6 Possible solutions 59
for the Wide Sense Stationary (WSS) data generating process:
A (i,j)(z)s (j)(t) = e (i)(t),
j∈Ni
where e (i)(t) are zero-mean, uncorrelated white noise signals. Since in practice the wavefront
disturbance causes all measurements to be somehow correlated, the achievable accuracy
of this model depends on the choice for the setsNi: the larger the areas containing the
locally available sensors are chosen, the higher the prediction accuracy.
In the next sections, the property that the disturbance signals at nearby wavefront positions
are highly correlated will be exploited for the design of distributed wavefront reconstructors
and predictors.
3.6.3 Iterative distributed phase reconstruction
Although the iterative conjugate gradient algorithms discussed in section 3.3.1 are not suitable
for a distributed architecture, this is not the case for all iterative solvers. Consider the
partial reconstruction problem from open-loop gradient measurements s(t) to phase ˆ φ u (t)
as posed in (3.10) on page 47. If the covariance matrices in (3.9) are taken as φs T = G T
and ss T = G T G and a Fried or Hudgin geometry is used to constructG, then the update
equations of the Jacobi and Steepest Descent (SD) algorithms [152, 197] fit the distributed
communication structure. This is indicated by the sparsity structures of the matrices G T
andG T G shown in figures 3.3 and 3.4.
Using iteration indexmand step size α, the update equation and residual vectorr (m)(t) of
the SD algorithm to solve (3.6) can be written as:
ˆφ (m+1)(t) = ˆ φ (m)(t)−αr (m)(t), (3.15)
r (m)(t) = G T G ˆ φ (m)(t)−G T s(t). (3.16)
Similar to the conjugate gradient algorithm in (3.13), for fastest convergence the step sizeα
should be determined at each iteration as [152]:
α (m) = rT (m) r (m)
rT (m) Ar .
(m)
Since the involved in-products do not fit the distributed framework, α can alternatively be
chosen a priori fixed. When assuming a zonal sensor geometry (i.e. the gradient measurements
are expressed in terms of adjacent phase points),Ghas a sparse, distributed structure.
Consequently, each row i of the productG T G will have only a few non-zero elements per
row, located at columnsj, where phase pointj is located nearbyi. The update equation in
(3.15) thus fits a distributed communication structure and the number of computations per
node per iteration is fixed.
However, the number of iterations mc required for convergence is not fixed. Using (3.15),
the propagation of the residual vector in (3.16) can be expressed as:
r (m+1) = Wr (m), where W = I−αG T G.
The iterations are usually stopped when the Frobenius norm of the residual drops below a
certain fraction ǫ of that of the initial residual r (0)(t) = G T G ˆ φ (0)(t) − G T s(t), where
3

3
60 3 Efficient control for AO: concepts and challenges
ˆφ (0)(t) is the initial guess. The Frobenius norm of the residual after a single iteration must
satisfy r (m+1)(t)F ≤ W·r (m)(t)F , where W denotes the spectral norm of W.
Accordingly, the norm of the residual afterm iterations becomes:
r (m)(t)F ≤ W m r (0)(t)F.
SinceW is symmetric, its spectral normW is the square of its spectral radiusρ(W) and
thus iterations are stopped whenρ(W) 2m < ǫ.
For fastest convergence, the fixedα should thus be chosen that minimizesρ(W). The eigenvalues
λ(W) can be expressed in terms of λ(G T G) as λ(W) = 1 −αλ(G T G). Let the
eigenvalues of the symmetric, positive semi-definite matrixG T G be spread on the positive
real axis between 0 and λ. The zero eigenvalues ofG T G (i.e. its kernel) correspond to the
unseen modes that cannot be reconstructed without additional (e.g. statistical) information.
They are mapped toλ(W) = 1 irrespective ofα, which can also be directly observed from
the update law. The residual in (3.16) is orthogonal to the unseen modes such that by (3.15)
the energy of these modes in ˆ φ (m+1)(t) is unaffected. If the smallest positive eigenvalue of
G T G is denotedλ, then:
1−αλ ≤ λ(W) ≤ 1−αλ.
The α that minimizes max ρ(W) must thus satisfy |1 − αλ| = |1 − αλ| and is found as
αo = 2/(λ+λ). The minimal spectral radius can now be expressed as:
ρ(W) = λ/λ−1
λ/λ+1 ∝ γNφ +δ −1
, (3.17)
γNφ +δ +1
where the second step uses the fact that in case of a Fried geometry [98], the matrixGis a
discrete Laplacian matrix. This yields an eigenvalue spread of the productG T G such that
in approximation the ratioλ/λ is linearly proportional to the number of phase pointsNφ.
When assuming thatr (0)F is invariant to the number of phase pointsNφ, after substitution
of (3.17) intoρ(W) 2m < ǫ and solving form, the required number of iterationsmc is found
as:
mc ∝
logǫ
2log(γNφ +δ −1)−2log(γNφ +δ +1) ,
∝ − 1
4 Nφlogǫ forNφ ≫ 1.
In approximation, the required number of iterations and thus computations increases
linearly inNφ, which is not allowed for the distributed architecture.
However, the required number of iterations mc is not only determined by the convergence
speed of the algorithm. If the initial guess ˆ φ (0)(t) is already close to the solution,
then only a few iterations may be required for convergence. Since the disturbance can be
well approximated as random walk process, it makes sense to use the previous solution as
the initial guess for the current problem, i.e. ˆ φ (0)(t) = ˆ φ (mc)(t−1).
In figure 3.7 the number of steepest descent iterations required for convergence (ǫ = 10 −4 )
is shown that was evaluated using numerical simulations on an artificially created data
set. First, a static phase screen with Kolmogorov spatial statistics was generated using a
midpoint displacement algorithm [95]. Then this was interpolated over a square aperture

3.6 Possible solutions 61
Number of iterations m c
10 2
10 1
10 1
ˆφ (0)(t) = 0
ˆφ (0)(t) = ˆ φ(mc)(t − 1)
10 2
Number of phase points Nφ
Figure 3.7: The average number of SD iterations
required for phase reconstruction withǫ = 10 −4
evaluated on synthetic disturbance data.
10 3
Normalized variance accounted for [%]
100
90
80
70
60
50
40
10 SD iterations, no delay
10 SD iterations, 1 sample delay
30
0 200 400 600 800 1000
Number of phase points [−]
Figure 3.8: Performance in terms of (3.18) using
10 iterations of SD with ˆ φ (0)(t) = ˆ φ (10)(t−1)
for measurement delays d of 0 and 1 samples.
window that was translated over the phase screen, thus simulating a frozen flow and
yielding φ(t) for t = 0...5000. The open-loop measurement data set s(t) was then
obtained through the sensor model in (3.7) where the variance of the white measurement
noise signaln(t) was chosen according to a SNR of 20dB.
As can be seen from the figure, the number of iterations can be reduced by using the
previous reconstruction result as the initial guess for the new sample. However, the number
of iterations still increases with the number of phase points Nφ, which means that this
extension to the SD algorithm provides insufficient efficiency gain to fit the distributed
architecture.
A similar scheme is shown in [192], where the authors propose the use of single iterations
of Jacobi and Richardson [152] per sample for closed-loop wavefront reconstruction
with an ideal DM. They use the traditional integrator controller and show that for small
integrator gains, the approach leads to a smaller noise propagation compared to using
R = G # . However, they do not analyze the performance effects of increasing the number
of DOF of the AO system.
Alternatively this performance trend can be analyzed by keeping the number of SD
iterations constant, while increasing the number of DOF. This has yielded the results in
figure 3.8. This shows the performance obtained using 10 SD iterations and ˆ φ (0) (t) =
ˆφ (10)(t − 1) as evaluated for the same data sets as previously while considering zero and
one sample measurement delay. The performance is measured using the following Variance
Accounted For (VAF) criterion:
VAF =
1−
5000 t=0 s(t)−Gˆ φ (10)(t)2 F
5000 t=0 s(t)2
·100%. (3.18)
F
The VAF values in the figure are normalized to those of a centralized solution to show
the trend in the performance lost due to the chosen communication structure. For the case
without measurement delay this centralized solution is taken as ˆ φ(t) = G # s(t) and for a
3

3
62 3 Efficient control for AO: concepts and challenges
one sample delay this is taken as an FIR prediction:
ˆφ(t) = B1s(t−1)+...+Bns(t−n) (3.19)
with n lags. The coefficient matrices Bi are estimated from the data set as the minimizers
of the cost function:
s(t)−G n
JB = Bks(t−k) .
k=1
The number of FIR lags has been chosen n = 10, yielding the highest VAF values for a
separate validation data set.
Observe from figure 3.8 that for a small number of DOF and no delay the 10 SD iterations
yield almost the same performance as the centralized solution which deteriorates as
the number of DOF increases. However, in all practical situations the measurement delay
will be at least one sample, in which case the performance will significantly drop (figure
3.8). Although the depth of this drop depends on the disturbance behavior, the trend for an
increasing number of DOF is the same as for the zero delay case. The loss of performance
due to the choice for a distributed architecture increases with the number of DOF. A concept
will now be introduced to improved this trend.
3.6.4 Recursive adaptive distributed reconstruction and prediction
The discussed SD and Jacobi algorithms can be further extended to also include knowledge
of the spatio-temporal dynamics of the wavefront disturbance. As suggested both by the
Kolmogorov structure function in (2.1) on page 28 as well as by the frozen flow assumption
discussed in section 2.1.2, this disturbance is highly correlated for nearby points in the
wavefront. Now consider a network of processing nodes – one associated to each phase
point – that performs both wavefront prediction and reconstruction. Each nodei updates its
output ˆ φ (i)(t) using the following candidate update law:
ˆφ (i) (t) =
j∈C (i)
n−1
a ˆ
(i,j) φ(j) (t−1)+
k=0 j∈M (i)
2
F
b (i,j,k)s (j)(t−k). (3.20)
Here the sets C (i) and M (i) contain the indices of the phase points and sensor spots whose
phase values and measurements are available for the prediction of φ (i)(t) respectively.
Further, s (j)(t) is the two-element gradient measurement vector from sensor spot j and
n is the number of temporal lags of the Moving Average (MA) structure. Let the Auto-
Regressive (AR) coefficientsa (i,j) of the filter structure be given a priori in accordance with
a single iteration of the SD or the Jacobi algorithm. For the SD case described in (3.15) and
(3.16), the coefficientsa (i,j) form the (i,j) elements of the matrix A = I−αG T G. This
implies that for α = αo the filter in (3.20) is marginally stable with two poles on the unit
circle corresponding to the unseen modes of the Shack Hartmann sensor (SHS). Moreover,
due to the sparsity structure of G T G (figure 3.4) the sets C (i) can be restricted to consist
of i and the indices of the four phase points diagonally adjacent to i. This set becomes
larger when a larger numberm > 1 of SD iterations is performed within one sample time,

3.6 Possible solutions 63
C (i)
S (j)
j
i
W (i)
M (i)
Figure 3.9: Summary of the introduced
sets of indices.
W(i):gradients that are dependent on
φ (i) and define the local cost function.
M(i):gradients available for the output
update ˆ φ (i) in (3.20).
C(i): phase points used for the output update
ˆ φ (i) in (3.20) as a result of the
SD iteration(s).
S(j):phase points defining the measurements(j)
according to the Fried geometry.
corresponding to A = (I−αG T G) m .
Now let the MA termsb (i,j,k) be estimated as the minimizers of the cost function:
s(t)−G
J = φ(t−d) ˆ
,
j=1
i∈S (j)
2
F
Ns s(j)(t)− =
G ˆ
[j,i] φ(i)(t−d)
2
F
, (3.21)
where G [j,i] ∈ R2×1 denotes the (j,i) th block-element of the matrix G and the set S (j)
contains the indices of the four adjacent phase points that define the two-element gradient
vector s (j)(t). A schematic summary of the used sets is shown in figure 3.9. Further,
d represents the number of samples delay due to the CCD based measurements and any
processing time to obtain ˆ φ(t). Estimation of the coefficients b (i,j,k) can be performed
by partitioning the cost function and using Least Mean Squares (LMS) or Recursive Least
Squares (RLS) algorithms for each subsystem. The adaptiveness of these algorithms also
allows to compensate for the slowly time-varying dynamics of the wavefront disturbance.
Substitution of the update law in (3.20) into the cost function in (3.21) yields:
Ns
J =
j=1
˜s (j)(t)−
2
G [j,m]p (m)(t)
,
m∈S
(j) F
where ˜s (j)(t) = s (j)(t)−
a ˆ
(m,l) φ(l)(t−d−1),
and p (m)(t) =
l∈M (m) k=0
m∈S (j)
G [j,m]
l∈C (m)
n−1
b (m,l,k)s (l)(t−d−k).
This cost function can be partitioned into overlapping, local pieces for each node i ∈
{1...Nn}:
J (i) =
˜s(j)(t)−
G [j,m]p (m)(t) . (3.22)
j∈W (i)
m∈S (j)
2
F
3

3
64 3 Efficient control for AO: concepts and challenges
Each node minimizes its cost function w.r.t. the variables b (i,j,k) for j ∈ M (i) and k =
0...n − 1. Let the sets W (i) be defined as the union of all sets S (l) for l = 1...Ns that
containi. This assures that only those errors are weighted that can be directly influenced by
the local variables.
By partitioning the cost function and letting each node optimize its own piece, the global
optimum is traded for a Nash equilibrium: optimization stops when no node can reduce
its cost function by its own decision variables alone. Such a game theoretic objective is
explicitly used in [147] to find a distributed controller via an iterative process based on
price mechanisms.
The signals p (m)(t) form = i involve coefficientsb (m,l,k) that do not form variables local
to node i. For node i to evaluate its cost function J (i), neighbors must communicate the
value of these signals at each sample time. This also requires communication of the phase
estimates ˆ φ (l)(t−d−1) from neighboring nodesl ∈
j∈M (i)
m∈S (j) C (m).
To arrive at a procedure to estimate the variables b (i,j,k) associated to each node i, let the
local cost functionJ (i) from (3.22) be rewritten to:
˜s(i)(t)−G J (i) = [W(i),L (i)]p (Li)(t) −G [W(i),i]s
˜e (i)(t)
T (i) (t)b
(i) 2
, (3.23)
F
where ˜s (i)(t) stacks all ˜s (j)(t) for j ∈ W (i) in lexicographical order and s (i)(t) stacks all
s (l)(t−d−k) forl ∈ M (i) andk = 0...n−1. The vectorb (m) stacks the corresponding
coefficients b (i,l,k) and the set L (i) is defined as L (i) = (
j∈M S (j)) \ {i}. Using a
(i)
completion of squares argument, the vector ˆ b (i) that minimizes J (i) can then be solved
from:
s (i)(t)γ (i)s T (i) (t)
ˆb(i) = s (i)(t)G T [Wi,i] ˜e
(i)(t) ,
where the scalar γ (i) = G T [W (i),i]G [W(i),i]. Using e.g. the LMS algorithm [3, 119] this
estimation can be performed recursively on-line by updating the parameter vector ˆ b (i) at
each time instantt:
ˆb (i)(t) = I−α (i)γ (i)s (i)(t)s T (i) (t)
ˆb(i)(t−1)+α
(i)s (i)(t)G T [W (i),i] ˜e (i)(t),
where α (i) is the step size for node i. For all nodes these can either be tuned manually or
optimal values can be derived from known statistical properties of the disturbance signals
[3].
An important issue with the proposed method is that neither the local nor the global cost
functions weight the unseen piston mode (section 1.3.2). As a result, the energy corresponding
to this mode will show random walk behavior that can be suppressed by adding a
regularization termZ (i) of the form
Z (i) =
j∈W (i)
ρ ˆ
(i,j) φ(j)(t−d)
to the local cost functions in (3.23). For suitable coefficients ρ (i,j), the regularization affects
piston and waffle components with as little effect as possible on the remainder. The
2
F

3.6 Possible solutions 65
Normalized variance accounted for [%]
98
97
96
95
94
93
SD−LMS with 4 neighbors
SD−LMS with 8 neighbors
92
0 200 400 600 800 1000
Number of phase points [−]
Figure 3.10: Results of the
distributed SD-LMS reconstruction
procedure applied
to the artificial data sets of
section 3.6.3.
waffle mode is specific to the Fried geometry matrix (section 1.3.2) and can also be evaded
by choosing e.g. the Hudgin geometry matrix. The unseen piston mode is fundamental to
the Shack-Hartmann WFS, but incorrect, large piston values do not affect the AO correction
quality and only require an overly large stroke of the wavefront corrector. Nevertheless,
these are trade-offs that require further research.
Figure 3.10 shows the results of performing this procedure for the artificial data set described
in section 3.6.3 using d = 1, n = 3 and with sets M (i) that contain the indices of
four and eight nearest sensor spots. For simplicity, the step sizes α (i) were chosen equal.
Similar to figure 3.8 the results are expressed in terms of a VAF-value that has been normalized
against that for the centralized predictor in (3.19). The VAF-value itself is the normalized
average prediction/reconstruction error that determines the cost function in (3.21):
VAF =
1−
t1
t=t0 s(t)−Gˆ φ(t) 2 F
t1
t=t0 s(t)2 F
·100%.
The data set was evaluated over 5000 samples and the VAF values were computed using
t0 = 3000 and t1 = 5000 to skip the settling time of the LMS algorithm. For these
data sets, the VAF values corresponding to the traditional, static reconstructor R = G #
that does not take the one sample delay into account is approximately 20%, stressing the
importance of a predictor. Observe that for the four neighbor case, the normalized VAF
value remains virtually constant, whereas for eight neighbors it increases with the number
of DOF. This means that for the used synthetic data sets the difference with the centralized
solution actually decreases as the system becomes larger. However, note that these results
will depend highly on the spatio-temporal behavior of the wavefront disturbance. Nevertheless,
the notion that the decrease of the locally available fraction of SHS measurements can
be compensated by exploiting spatio-temporal correlations of the wavefront disturbance is
promising. It will be addressed in more detail and using more general parameterizations of
the local subsystems in the next chapter.
3

3
66 3 Efficient control for AO: concepts and challenges
3.6.5 Local, identical influence functions
Both the described traditional control law and the H2 optimal control law described in
[100] involve an inverse of the influence matrix H of the wavefront corrector to fit actuator
commands to a desired mirror shape. As mentioned, there exists extensive literature
on performing this operation in a computationally efficient manner [53, 188]. However, to
achieve this efficiency, these methods use approximations of the influence matrixHwhose
accuracy could be improved by suitable design of the DM.
For instance, in [53] the influence matrix is approximated as a sparse matrix. It is assumed
that each actuator only affects the wavefront in a small area surrounding it and influence
below a certain threshold is neglected. This approximation thus benefits from narrow influence
functions that makeHmore sparse and the algorithm implementation more efficient.
Other approaches [188] assume the DM influence functions to be spatially invariant. A circulant
structure is attributed such that the influence matrix becomes a spatial filtering kernel
with a sparse Fourier domain representation. This can be efficiently applied using the previously
described Fourier domain techniques. Further, spatial invariance is important for the
more generic distributed controller synthesis methods described in [14, 41, 79, 129, 168].
The spatial invariance approximation becomes more accurate when the variation between
the shapes of different influence functions is reduced. These differences may e.g. be the
result of manufacturing tolerances or variations in material properties (chapters 5 and 6) or
edge effects of the DM (chapter 7).
Further, the above efficient algorithms all assume the DM to be well modeled as a static
gain, which means that their applicability or performance will be compromised if the DM
has significant temporal dynamics. Moreover, the performance of a distributed controller
will deteriorate when the spatial propagation speed of the plant dynamics exceeds that of the
inter-node communications. For quasi-static DMs with local influence functions, this means
that after each control signal update the controller nodes should be able to communicate to
all nodes that can influence their measurements. A requirement for the inter-node communication
distance can thus be reduced by a DM design with narrow influence functions.
3.7 Conclusions
In this chapter the scaling problems that arise when the number of DOF of the AO system
increases were investigated. This has been based on both traditional approaches towards
control design for AO and for more general, optimal approaches. For all methods the computational
complexity is proportional to the number of DOF with an exponent larger than
one, which indicates that for these cases the computational power of the control system
must grow faster than the number of DOF. This leads to huge parallel processor systems for
the new large telescopes such as the European Extremely Large Telescope (E-ELT) and the
Thirty Meter Telescope (TMT). Since these systems must be custom-designed, they will be
very costly. Without a modular architecture, such systems do not befit from the modular AO
system to be realized within this research project.
To arrive at a fully modular system, a control system with a distributed architecture is required.
This was proposed, consisting of computational nodes that each drive a single actuator,
receive a limited set of wavefront sensor measurements and can communicate to a few
neighboring nodes. This structure was chosen for practical reasons and will significantly

3.7 Conclusions 67
affect the achievable performance. This performance loss was investigated based on literature,
but is difficult to quantify because it depends entirely on the behavior of the DM, the
sensor and the atmospheric disturbances. Therefore, several approaches towards distributed
control are proposed, some of which are evaluated in numerical simulations and compared
to centralized solutions.
A first idea is to use an electrical resistor network, similar to the one used for wavefront
reconstruction in the first operational AO system. Such a network can perform basic wavefront
reconstruction very fast, but is sensitive to variation in the properties of its analog
components and does not allow to exploit any wavefront statistics. The latter is possible
with the subsequently proposed distributed disturbance model whose performance has been
verified in literature.
An alternative for the analog resistor-network reconstructor can be an iterative, distributed
wavefront reconstruction algorithm based on the SD algorithm. All computations of this algorithm
are fully distributed, but the number of computations required per computation node
grows with the spatial dimension of the system. This approach does not allow to exploit any
wavefront statistics, but this becomes possible by extending it with LMS based wavefront
prediction. Simulation results of this distributed, adaptive wavefront predictor and reconstructor
showed that its performance actually increases with the number of DOF, but further
research is required into dealing with the unseen waffle-mode of the Shack-Hartmann WFS.
Finally, a number of properties was discussed in the context of a DM system that are often
assumed of the plants in distributed controller design approaches found in literature. A spatial
invariance assumption may for instance be fulfilled by a suitable mechatronic design.
However, the main challenge for modularly distributed control for AO lies with wavefront
reconstruction.
3

ÔØÖÓÙÖ
ØÖÚÒ×ØÖÙØ ÓÒØÖÓÐ
To deal with the increase in the computational demand for Adaptive Optics
(AO) control systems of future large telescopes, a modular, distributed controller
structure is proposed consisting of local Auto-Regressive Moving
Average (ARMA) controllers. A data based controller design approach is presented
that is elaborated for the case of an ideal wavefront corrector. The internal
model control principle is used to transform the closed-loop problem
into an open-loop problem while preserving the distributed structure. A twostage
procedure is proposed to identify the unknown controller coefficients
from open-loop measurement data based on a minimum variance, output error
criterion. Constrained optimization is used to guarantee stability of the
identified open-loop controller using Gershgorin’s circle theorem. Application
results are presented on both experimental and synthetic data.
69

4
70 4 Data driven distributed control
4.1 Introduction
In this chapter a synthesis approach will be proposed that yields a controller that is
suitable for the distributed system architecture proposed in chapters 2 and 3. The hardware
architecture of the control system is assumed to be a network of small controller modules
whose hardware is identical. Here it will be assumed that each module controls a single
actuator, but for an actual implementation this can also be a small group. The modules can
communicate with a small number of neighbors and have only access to a small set of local
WaveFront Sensor (WFS) measurements.
As mentioned, such a distributed hardware architecture has major consequences for the
synthesis and performance of a controller that must be thoroughly evaluated. Firstly,
because the available computational power will only increase linearly with the number of
actuators. But also because the constraints that must be applied to the optimization problem
in (3.4) to restrict the solution to distributed controllers render this problem non-convex.
In the previous chapter the bottlenecks arising for traditional control laws have been
investigated together with several approaches towards solutions. The main challenge for
distributed control has been identified as the wavefront reconstruction step that in its pure
form requires all sensor measurements for the reconstruction of each phase point. Further,
as the optimality of the traditional control law has been shown to be subject to very specific
conditions [56, 100, 123], in this chapter a more general, but distributed control law will be
used. However, the distributed structure is considered to be the design driver and as will be
shown this leads to a loss of performance when compared to centralized solutions.
A distributed, minimum variance controller will be proposed that is data-driven in the sense
that its coefficients will be directly estimated from open-loop disturbance measurement
data. The available measurements and data from neighboring nodes will be chosen such
that they are unchanged by the transformation between the closed and open-loop controllers
using the internal model principle. The particular controller structure leads to the implicit
assumption that the gradient measurements are generated by a network of interconnected,
low order, Wide Sense Stationary (WSS) stochastic processes. Further, a Shack-Hartmann
wavefront sensor is considered with a one sample delay, a Fried geometry and white
measurement noise. Except for the delay, this sensor model corresponds to the one used in
[53, 124]. Finally, for the sake of simplicity an ideal wavefront corrector will be assumed
whose transfer matrix is the identity matrix, such that the actuator command vector is
equal to the introduced wavefront correction. This means that both the temporal dynamics
and the influence the actuators have on each other via the reflective facesheet are ignored,
hence this assumption is too restrictive for the Deformable Mirror (DM) system outlined
in chapter 2. Nevertheless, it is adequate to investigate the feasibility of a distributed
controller structure for an Adaptive Optics (AO) system based on a Shack-Hartmann WFS
that forms a major challenge for a distributed control architecture (chapter 3).
This chapter is outlined as follows: in the next section several preliminaries are introduced,
after which a detailed problem definition is given in section 4.3. In section 4.4 the
approach towards the design of a distributed controller is described. The closed-loop problem
is first translated into the equivalent open-loop problem and then a system identification
problem is formulated to identify the unknown coefficients from open-loop measurement
data. In section 4.7 an algorithm will be proposed to do so. Finally, application results to

4.2 Preliminaries 71
both breadboard measurement data as well as synthetic data will be presented in section 4.8.
4.2 Preliminaries
Throughout this chapter, individual, distributed, discrete time sub-systems will be denoted
through a subscript index, e.g. C (i)(z). The global, interconnected systemC(z) is obtained
by lifting the sub-systems in this index space and applying the interconnection constraints.
The fact that this system is structured is reflected in the bold italic typeface used forC. The
input and output vectors of the global system will be described with structured vectors that
stack the vectors or scalars corresponding to local sub-systems. Such structured vectors are
also denoted in a bold italic typeface as x instead of the regular bold typeface x used for
unstructured vectors.
Throughout this chapter the following symbols will be often used: the total number of nodes
Nn, the number of lenselets Ns in the wavefront sensor array and the order n of the individual
distributed controllers.
For more general definitions of notation the reader is referred to section 1.1 and the nomenclature
from page xvii onwards.
4.3 Problem description
Consider a linear, discrete time, WSS stochastic processT(z) that generates the wavefront
phase disturbanceφ(t) = [φ (1)(t),...,φ (Nn)] T at phase grid points1...Nn and is driven
by an unknown noise signal v(t) ∈ N Nn (0,Cv). The residual wavefront disturbances
ǫ (i)(t) that affect the science image is influenced by the wavefront corrector (e.g. a DM).
As discussed above, it is assumed that the transfer function matrix of the corrector is the
identity matrix. Therefore, there will be no further reference to its dynamics and both its
input and output signals will be called u (i)(t) for i = 1...Nn. Let the residual wavefront
ǫ (i)(t) be defined as:
ǫ (i)(t) = φ (i)(t)−u (i)(t).
The residuals ǫ (i)(t) are observed through a Shack-Hartmann wavefront sensor that is
modeled using the Fried geometry [179], a one sample delay and measurement noise
w (j)(t) ∈ N2(0,C (w,j)) forj = 1...Ns. The Fried geometry has been treated in detail in
section 1.3.2 on page 6 of which several important properties will here be briefly repeated.
Firstly, the Fried geometry defines the closed-loop gradient measurementse (j)(t) ∈ R 2 in
terms of the four surrounding phase values as:
e (j)(t) = GFǫS (j) (t−1)+w (i)(t), (4.1)
where the setsS (j) contain the indices of the four phase points that define the two gradients
e (j)(t) as in (1.1) on page 8. Further, when e (j)(t) for j = 1...Ns and ǫ (i)(t) for
i = 1...Nφ are stacked in the vectorse(t) andǫ(t) respectively, the measurements can be
expressed as e(t) = Gǫ(t−1)+w(t). The matrix G is rank deficient, which causes the
piston and waffle modes to yield a zero measurement. Implications of this for control will
be addressed at a later point in this chapter.
4

4
72 4 Data driven distributed control
Figure 4.1: Schematic of the
design problem of the con-
v
w
φ
+ -
+
e
trollerC(z). T(z) Gz -1
u I
u
C(z)
In this chapter a spatially interconnected network of discrete time controllers C (i)(z)
with outputu (i)(t) is considered that together form the global controllerC(z) and minimize
the output error variance (figure 4.1):
Ĉ(z) = arg min
C(z)
2
e(t)F .
This optimization criterion does not consider a priori knowledge of statistics of the wavefront
phase disturbance or the measurement noise, which is similar to the approach discussed
in chapter 3 whereR(z) is restricted to a static matrixR such that:
ˆR = argmin
R
s(t)−GRs(t) 2 F .
When assuming s(t) ∈ N2Ns(0,I), this has the solution R = G # , where G # is the
unweighted pseudo-inverse of G. The latter does not exist uniquely and in practice the
Singular Value Decomposition (SVD) inverse is used that constrains the unseen modes in
the reconstructed phase vector to zero. When evaluating the performance of the distributed
controller in section 4.8, thisR = G # will be used as a reference.
Further, the local controllers will further be called control nodes. The communication
between these nodes will be limited to their output signals u (i)(t) instead of their full state
information. This limits the inter-node communication load and makes this independent
of the order (state dimension) of the nodes. However, by denying the nodes full access to
their neighbors’ states an additional loss of generality is introduced and a possible loss of
performance.
Let the inputs of node i consist of the measurements e (j)(t) for j ∈ M (i) and the outputs
u (k)(t) of their neighbors k ∈ C (i). In the sequel of this chapter a specific choice will be
made for the setsM (i) andC (i).
4.4 Design approach
The approach taken in this chapter to obtain the distributed controllerC(z) can be sketched
as follows. First, a distributed parametrization is proposed for the controller and a choice is
made for the sets M (i) that specify which measurements are available to each node. Then,
using the internal model principle the closed-loop synthesis problem is translated into an
open-loop problem. The sets C (i) that specify which output estimates are available to each
node are defined such that this transformation does not compromise the goal that the output
of each local controller node is determined entirely on information from spatially nearby
sources.
ǫ

4.4 Design approach 73
rc
4.4.1 Parametrization of the distributed controller
1
1
Figure 4.2: Distributed control
setting. The boxes represent
the controller nodes and
the dots the WFS gradient
measurements. Arrows indicate
the information flow for
the solid black node with its
greyed neighbors and sensors
located within a communication
radius rc.
In AO literature, several distributed controller structures have been described. In [45] and
[52], local Finite Impulse Response (FIR) filters are described that predict open-loop measurement
as a weighted estimate of past, local measurements. As shown for on-sky measurements
[45] and experimental breadboard measurements [52], such local predictors are
still capable of achieving small prediction errors. This is due to the strong local wavefront
correlations both in space – indicated by Kolmogorov or Von Karman statistics – as well
as in time – indicated by the frozen flow assumption. For the first, the phase covariance
between two points in the wavefront decays with the distance between them [95] due to the
definition of the structure function in (2.1). For the second, two adjacent points may even
observe time-shifted versions of the exact same temporal phase fluctuations.
However, in both [45] and [52] the wavefront reconstruction step is considered as a separate
part of the controller. Here this step is considered a critical part of the controller, which
significantly complicates its design. In [124] wavefront reconstruction is also implicitly
considered and the following distributed controller structure is proposed:
C (i)(z) : u (i)(t) =
a (i,j)u (j)(t−1)+
b (i,j)e (j)(t).
j∈C (i)
j∈M (i)
This Auto-Regressive Moving Average (ARMA) controller is similar to the one proposed
in section 3.6.4, except for the presence of only a single Moving Average (MA) term.
The coefficients a (i,j) and b (i,j) correspond to the application at each sampling time
of one or a few iterations of a distributed linear solver, starting from the last estimate
of the previous sample. Therefore, these coefficients are determined from steady state
convergence criteria. An important difference is that the inputs of controller node i
now also consist of the previous outputs of itself and neighboring controller nodes:
i.e. of u (j)(t − 1) for j ∈ C (i). At each time step, this leads to propagation of wavefront
phase information over the network of nodes. This allows partial recovery of the
performance loss arising from gradient measurements being unavailable to local controllers.
In this chapter an approach will be proposed to improve the recovery of the performance
loss, take into account the WFS delay and exploit local knowledge of the behavior of the atmospheric
disturbance. To achieve this, the following class of controllers with a distributed
4

4
74 4 Data driven distributed control
input-output parametrization is proposed:
C (i)(z) :
a (i,j)(z)u (j)(t) =
j∈C (i)
j∈M (i)
k (i,j)(z)e (j)(t), (4.2)
where the polynomialsa (i,j)(z) and k (i,j)(z) are of fixed ordernfor all nodes and defined
as:
n
a (i,j)(z) = δij − a (i,j,l)z −l ,
n−1
k (i,j)(z) =
l=0
l=1
k (i,j,l)z −l .
Since δij = 1 for i = j, the polynomial a (i,i)(z) is monic, whereas the first coefficients
of a (i,j)(z) for i = j are zero. This is assumed for well-posedness of the interconnected
system. Further, the coefficients k (i,j,l) are of size 1×2since each lenselet of the Shack-
Hartmann WFS yields two gradient measurements. This parametrization is also proposed
in [61], where a structured, multi-dimensional Transfer Function (TF) model is used for
system identification. The input-output form circumvents the estimation of a local state
sequence for each node, which is a yet unsolved problem.
Similar to [45, 52], the sets M (i) will be defined through the radius rc ≥ 1 as the indices
j of all measurements e (j)(t) taken at a distance from node i smaller than rc (figure 4.2).
When the communication radius rc is increased until for all nodes i the set M (i) contains
all j = 1...Ns, then the problem reduces to the centralized setting. Otherwise, the sets
M (i) can be smaller for nodes near the border of the grid than for nodes in the center, as
there are fewer sensors present within the communication radius. The same will hold for
the setsC (i) that will be defined in the next section.
4.4.2 Internal model control
The internal model control principle can be used to transform a closed-loop controller into
an equivalent open-loop controller. Since the measurement inputs of an open-loop system
are not affected by its outputs and in this case the forward plant modelGz −1 is a distributed,
static gain, the coefficients of its update equation will be easier to identify.
The closed-loop schematic depicted in figure 4.1 can be equivalently represented by the
schematics in figures 4.3 and 4.4, where the latter more clearly shows the open-loop configuration.
The lower grey area in figure 4.3 now corresponds to the controller C(z), but consists
internally of the open-loop controllerR(z) and the forward plant model. Expressed on a local
scale, the inputss (j)(t) of the open-loop controllerR (i)(z) are equal to the closed-loop
measurementse (j)(t) minus the effects of all controller actions:
s (j)(t) = e (j)(t)+GFu (S(j))(t−1). (4.3)
Substitution of (4.3) fore (j) into the controller parametrization of (4.2) yields the open-loop
controller:
R (i)(z) :
a (i,j)(z)u (j)(t) =
k (i,j)(z)s (j)(t)−
k (i,j)(z)GFu (S(j))(t−1).
j∈C (i)
j∈M (i)
j∈M (i)

4.5 Performance 75
T(z) Gz-1 + -
v
w
φ
+
e
u
C(z)
Gz -1
R(z)
s
+
Figure 4.3: A schematic equivalent to the
closed-loop in figure 4.1.
ǫ
T(z) Gz -1
R(z)
v φ
+
s
w
Figure 4.4: An open-loop schematic equivalent
to figures 4.1 and 4.3.
Observe that this controller has an additional term compared to the parametrization of
C (i)(z). Each u (S(j))(t) in the additional term consists of the outputs of nodes l ∈ S (j).
Due to the definition of the setsS (j) all these nodes are spatially located near nodei. To prevent
any differences between the sets of available inputs of the nodesC (i)(z) andR (i)(z),
the indices of these nodes should be included in C (i). Therefore, let the sets C (i) be defined
accordingly as:
C (i) =
S (j).
j∈M (i)
The open-loop controllerR (i)(z) can then be parameterized more compactly as:
R (i)(z) :
j∈C (i)
where the polynomialsã (i,j)(z) are equal to:
ã (i,j)(z)u (j)(t) =
−1
ã (i,j)(z) = a (i,j)(z)+z
c∈M (i)
j∈M (i)
k (i,j)(z)s (j)(t), (4.4)
k (i,c)(z)G [c,j]
andG [c,j] ∈ R 2×1 denotes the(c,j) block element of the geometry matrixG.
4.5 Performance
+ -
u
ǫ
(4.5)
For H2 optimal controller synthesis, the controller structure follows from the plant model
structure, but in this case the desired distributed controller structure is the design driver.
The consequences of this choice for the controller’s performance will now be investigated
by showing the assumptions that this choice implies for the structure of the plant.
Assuming that the network of open-loop controllersR (i)(z) is optimal in the sense that
it minimizes the output error variance e (j)(t)2 F for all j, then the output errors must be
white noise signals, i.e. e (j)(t) ∈ N(0,Cej). After application of the sensor model, the
control signals u (i)(t) thus form the optimal one step ahead prediction for the open-loop
measurement signals s (j)(t). Consequently, the network of open-loop controllers of (4.4)
4

4
76 4 Data driven distributed control
can be combined with the output equation in (4.3) to form an alternative process that generates
the open-loop measurementss (j)(t) and is driven by the white noise inputse (j)(t). By
considering this innovation form of the predictor, insight into the implicitly assumed structure
of the data generating system is obtained. First recall those two equations and consider
them part of a single system:
⎧
⎨ ã (i,j)(z)u (j)(t) =
j∈C (i)
⎩
k (i,j)(z)s (j)(t)
j∈M (i)
(4.6)
s (j)(t) = GFu (S(j))(t−1)+e (j)(t)
This process resembles a Kalman filter, but in absence of a well defined state this term is
not applicable. When substituting the output equation of (4.6) for s (j)(t) into its ARMA
equation and then using (4.5), this leads to:
⎧
⎨ a (i,j)(z)u (j)(t) =
k (i,j)(z)e (j)(t),
⎩
j∈C (i)
j∈M (i)
s (j)(t) = GFu (S(j))(t−1)+e (j)
This equivalent representation of the data generating system thus has the same distributed
structure as the controller C(z) in (4.2) and is driven by white noise injected at the sensor
locations. However, this form is not realistic as a physical model since the white noise signal
that drives the process generating the phase distortionφ(t) is here coupled with the noise of
the WFS. Nevertheless, if the proposed distributed structure is enforced on the controller,
this can only be the minimizer of the output error variance if the data generating system has
a realization with the same spatially distributed structure.
Conversely, if the controller that minimizes e(t)2
F for the disturbance process T(z)
with sensor model Gz−1 does not have a realization that fits the structural constraints set
for R(z), the signal e(t) will not be white noise. This can be illustrated by examining the
Markov parameters ofR(z).
Substitution of the update equation in (4.4) into itself allowsu (i)(t) to be written as:
u (i)(t) =
j∈C (i)
+
a (i,j)(z)
j∈M (i)
⎛
⎝
m∈C (j)
k (i,j)(z)s (j)(t),
a (j,m)(z)u (m)(t)+
m∈M (j)
⎞
k (j,m)(z)s (m)(t) ⎠
wherea (i,j)(z) = δij −ã (i,j)(z). Repeated application allowsu (i)(t) to be expressed as:
Nt−1
u (i)(t) = ξ (i)(t)+
l=0
j∈F (i,l)
f (i,j,l)s (j)(t−l), (4.7)
where the sets F (i,l) and the Markov parametersf (i,j,l) can be expressed recursively as:
⎧
⎨M
(i) for l = 0,
F (i,l) =
⎩ F (j,l−1) for l ≥ 1, (4.8)
j∈C (i)

4.6 Stability 77
⎧
0 for l < 0,
⎪⎨ k (i,j,l) +
f (i,j,l) =
⎪⎩
n−1
ã (i,p,m)f (p,j,l−m−1) for 0 ≤ l < n,
m=0p∈C
(i)
n−1
ã (i,p,m)f (p,j,l−m−1) for l ≥ n.
m=0p∈C
(i)
Further,ξ (i)(t) expresses the contribution of past outputsu(tp) fortp ≤ t−Nt. Assuming
thatR(z) is stable, this term goes to zero when the used number of tapsNt increases. This
implies that R(z) can be approximated using (4.7) with ξ (i)(t) = 0 and limited Nt, which
forms an FIR description.
Observe from (4.8) that the set F (i,l) of measurements that influenceu (i)(t) grows with the
temporal lagl. Each communication step allows the sensor information to further propagate
over the network of nodes. For the communication structure illustrated in figure 4.2 the
number of controller nodes whose outputs are affected by a measurement initially available
to the central node grows quadratically in time from 9 on the second time step to 16 in the
third and so on. In this manner all measurements s (j)(t) can eventually affect all outputs
u (i)(t+d), but with a delaydthat is proportional to the distance between sensorj and node
i and scaled by the communication radiusrc.
Conversely this means that if the transfer function matrix of the data generating system
in innovation form between the excitation noise e (j)(t) and the wavefront phase ˆ φ (i)(t)
has a delay smaller than d, these fast dynamics cannot be compensated by the distributed
controller. If the excitation signals e (j)(t) corresponding to the centralized case would be
known, the covariance between e (j)(t) and the measurements (m)(t+l) for m ∈ S (i) and
l = 0...∞ would be indicative of the allowed delay and thus the required communication
rangerc.
4.6 Stability
Although it can be safely assumed that the atmospheric disturbance is a stable process, the
distributed open-loop controllerR(z) is estimated from measurement data and is therefore
not automatically stable even if the independently identified controller nodes are stable.
A constraint is therefore sought that can be enforced on local controller nodes while leading
to stability of the interconnected system of controllers. This means that the feedback interconnection
of one controller node with more and more other nodes in the network is stable
regardless of the number of interconnected nodes, which implies that the enforced condition
must be retained in the interconnection. This is e.g. not the case when constraining each
controller node to satisfy the small gain theorem (or bounded real lemma [199]), i.e. having
an infinity norm smaller than one. It is well known that the feedback interconnection of two
stable systems H (1) and H (2) is stable when H (1)∞ · H (2)∞ < 1. But although the
interconnection of two small gain systems is stable, the infinity norm of the interconnected
system is unknown such that the small gain property is not retained in the interconnection
and cannot be used to guarantee stability of a network of controllers.
To overcome this, the notion of passivity could be used [8]. Strict passivity implies asymptotic
stability and is retained in the interconnection of two strictly passive systems. However,
as shown in appendix B passivity is very restrictive for discrete time systems having a zero
4

4
78 4 Data driven distributed control
direct feed-through term. Such terms would lead to ill-posedness of the interconnected system.
An approach that is restrictive, but does not suffer these drawbacks is described in the next
subsection.
4.6.1 Gershgorin’s circle theorem
Stability of a discrete time system essentially requires the zeros of its characteristic polynomial
to be located within the unit disc. For state-space systems this characteristic polynomial
is obtained directly from an eigenvalue analysis of its state transition matrix. Computation
of the poles of the interconnected system in (4.4) will here be done by first lifting
its autonomous dynamics in the spatial coordinate and then considering the block-controller
canonical state-space form [161]. The autonomous part in lifted form can be expressed as:
u(t) = Ã (1)u(t−1)+...+ Ã (n)u(t−n).
where the controller coefficients ã (i,j,l) form the (i,j) elements of the matrices à (l). This
autonomous system can be rewritten into the following block-controller canonical statespace
form:
⎡ ⎤
à (1) ... ... à (n)
⎢ I ... 0 ⎥
x(t+1) = Ax(t) where A = ⎢
⎣
. ⎥
.. ⎦ ,
0 I 0
where the state x(t) = [u(t),...,u(t−n+1)].
Gershgorin’s circle theorem [71] states that each eigenvalue of a matrix A with elements
A [i,j] lies within at least one of the closed discs centered at A [i,i] with radius
r (i) =
i=j |A [i,j]|. Consequently, all eigenvalues of the matrix will lie inside the unit
disc and the system is (marginally) stable if the sum of the absolute values in each row of
the matrix A is smaller than or equal to one. For the second and higher block-rows this is
automatically guaranteed by the identity matrices, but for the first block-row this leads to
the constraint that for eachi = 1...Nn:
j∈C (i) k=1
n
|a (i,j,k)| ≤ 1. (4.9)
How restrictive this constraint is in terms of performance depends on the data generating
system, but it further compresses the class of data generating systems derived in section 4.5.
Equation (4.9) gives rise to an independent set of constraints for each nodei that involves the
sum of absolute values. A constraint of this form|a|+|b| < 1 with two parameters can be
written into 4 simultaneous linear constraints: a+b < 1∩−a+b < 1∩a−b < 1∩−a−b < 1.
One constraint is required for each binomial possibility of parameter signs such that the
total number of constraints can be expressed as N (c,i) = 2 M (a,i), where the numberM (a,i)
of involved parametersa (i,j,k) is equal to n times the cardinality of C (i). As will be shown
at the end of the next section, this constraint concept leads to the identification of the local
controller parameters via local quadratic programming problems.

4.7 Identification procedure 79
4.7 Identification procedure
4.7.1 Optimization criterion and approach
With the controller structure defined, the values of the unknown coefficients need to be
determined. Let the unknown coefficients of the distributed controller R(z) be estimated
from open-loop measurement data s(t) fort = 0...N −1 as the minimizer of:
Ns e
J = (j)(t) 2 F
j=1
N−1
t0
. (4.10)
This cost function is used both for identification and validation. Recall that e (j)(t) is
defined in (4.3) and u (S(j)) in (4.4). In contrast to the traditional cost function described in
section 3.6 this cost function weights the error in terms of WFS gradients and not in terms
of wavefront phase. The latter would require an explicit model of the wavefront phase
disturbance, which is outside the scope of the data driven approach presented here. For
the purpose of identification the time t0 will be chosen as small as possible as determined
by the identification algorithm. For validation it will be chosen large enough to prevent
weighting of initial transient behavior.
When taking into account the update law in (4.4), the definition of e (i)(t) contains a
product between the unknown signal u (S(j))(t) and the unknown coefficients ã (i,j,l). This
renders to optimization problem nonlinear in its unknowns and the coefficients that minimize
(4.10) cannot be expressed explicitly. However, this is a common problem and methods
have been developed to deal with it. Such methods are usually iterative and based on
gradient search concepts [119]. However, the large number of unknown coefficients makes
calculation of the Jacobian and Hessian terms computationally costly operations. This is
also the case for simulations that must be performed for the intermediate controllers on the
measurement data to evaluate the cost function. Since these controllers may be unstable,
computationally expensive stability checks must be performed and – if necessary – the controller
must be stabilized within its structural constraints. Combined with the experience of
the author that convergence is very slow and highly sensitive to initial estimates, an alternative
approach is proposed. Similar to the gradient search algorithm, this algorithm is to
be performed off-line on a global data set and is itself not distributed. Only the coefficients
estimated by the algorithm form a distributed controller.
4.7.2 A two-stage approach
An off-line procedure will now be proposed in which the outputsu (i)(t) and the coefficients
ã (i,j,l) andk (i,u,l) are estimated sequentially. An estimate ofu (i)(t) can be obtained by first
estimating the coefficients of the controller’s FIR approximation described in section 4.5 and
subsequently applying this FIR filter to the measurements. This signal cannot be reproduced
by the distributed controller using the FIR coefficients, because – as was shown in section
4.5 – these do not satisfy the distributed structure. The second identification step therefore
consists of the identification of the coefficients ã (i,j,l) and k (i,u,l) that parameterize the
distributed open-loop controller R (i)(z) defined in (4.4) through linear regression on the
locally available measurements and the output estimates of the first step.
4

4
80 4 Data driven distributed control
Stage 1: estimatingu (i)(t)
By substituting (4.3) into the cost function J in (4.10) and replacing u (S(j))(t) with the
output of the FIR approximation ofR(z) in (4.7), the following cost function is obtained:
Ns
J1 =
where ũ (i)(t) =
j=1
Nt−1
l=0
s(j)(t)−GFũ
(S(j))(t−1)
p∈F (i,l)
f (i,p,l)s (p)(t−l).
2
N−1 F
Nt
, (4.11)
This can be rewritten into the spatially lifted form for the interconnected system as:
J1 = s(t)−G ⎡ ⎤
s(t−1)
2
N−1 ⎢ ⎥
F (0) ... F (Nt−1) ⎣
. ⎦
,
Ψ s(t−Nt) F Nt
s(t−1)
= s(Nt) ... s(N −1)
−GΨ
S
s(Nt −1) ... s(N −2)
2
. (4.12)
F
S
where the only non-zero elements (i,j) of the matrices F (l) for i = 1...Nn and j ∈
F (i,l) are equal to the FIR coefficients f (i,j,l). Using the vectorization operator ”vec” and
Kronecker identities from [23], the cost function in (4.12) can be rewritten to:
J1 = S−GΨS 2 , (4.13a)
F
= vec(S)− S T
⊗G vec(Ψ) 2
, (4.13b)
F
= S 2 2
F − vec
Ne
T (G T SS T )vec(Ψ)+ 1
vec
Ne
T
(Ψ) SS T ⊗G T
G vec(Ψ),
Q
whereNe = 2Ns(N −Nt). For the case when the communication radiusrc is large and a
centralized controller is sought, all elements ofΨand thus vec(Ψ) are free to be estimated.
The Ψ that minimizes J1 can then be expressed in the following form that does no longer
contain a Kronecker product:
ˆΨ = (G T G) −1 G T
T
SS SS T−1 .
Although it can be assumed that then loop measurement signal s(t) is persistently exciting
andSS T has full rank, the inverse(G T G) −1 does not exist due to the singularity ofG.
Without restricting the analysis to the centralized case, the solution can become ill defined
due to the pre-multiplication by G in (4.13a). Consequently, any columnΨ·i of Ψ that lies
in the kernel of G does not influence the cost functionJ1. As discussed in section 4.3, the
kernel ofGconsists of the unseen piston and waffle modes denoted by the vectorsmp and
mw respectively. These vectors can be linearly transformed into two orthogonal vectors

4.7 Identification procedure 81
m (1) and m (2) consisting of only ones and zeros that are complementary and divide the
phase grid into two separate sub-grids (figure 1.6, [143]). Sub-grid i consist of the points
corresponding to the non-zero values in m (i). Assuming a rectangular Fried geometry,
these are such that all diagonally adjacent points belong to the same sub-grid, as indicated
by the black and white backgrounds in figure 1.6. The solution ˆ Ψ thus becomes ill defined
if its sparsity structure allows a termαm (1) orαm (2) forα = 0 to be added to any column
Ψ·i. Since the sub-grids corresponding to m (1) and m (2) are spread over the entire spatial
domain, this occurs only if the temporal lag l is such that the effect of a measurement
s (j)(t−l) has spatially propagated to all outputsu (i)(t) for i = 1...Nn. This means that
Markov matricesF (l) of which all elements are allowed to be non-zero cannot be uniquely
identified regardless of persistence of excitation of the data generating system.
Hard constraints can be applied for the elements of Ψ that must be zero by removing
rows and columns from the vectors and matrices in (4.13b), but this leads to large matrices
with non-trivial structures. The latter becomes even more stringent when the solution is illdefined
and e.g. a pseudo-inverse technique is needed to find one. Instead, soft constraints
can be used, in which case the elements of Ψ that must be zero are explicitly weighted.
These terms are then allowed to be small instead of zero, with their magnitude depending
on the weights used in combination with the correlations present in the measurement signal
s(t). This leads to a new cost function:
˜J1 = J1 + 1
= S 2 F
Ne
vec T (Ψ)Wdvec(Ψ), (4.14)
2
− vec
Ne
T (G T SS T )vec(Ψ)+ 1
vec
Ne
T (Ψ)(Q+Wd) vec(Ψ),
whereWd = diag(vec(W)) is a diagonal weighting matrix, whereW has the same size as
Ψ and its elements are only nonzero, positive when the corresponding element inΨis zero.
The solution vec( ˆ Ψ) can then be solved from:
(Q+W) vec(Ψ) = vec G T
SS , (4.15)
after which the soft-constrained elements that must be zero are forced to zero. By using a
Krylov subspace conjugate gradients solver [77], a pseudo-inverse solution to (4.15) can be
obtained for which the ill defined terms ofΨare constrained to zero.
In this algorithm,Q and Wd only occur in product with a residual vectorrcg = vec(Rcg).
When using the definitions of ofWd andQ(the latter in (4.13)), these products reduce to:
Qrcg = (SS T ⊗G T G)rcg,
= vec(G T GRcgSS T )
and
Wdrcg = diag(vec(W))rcg
,
= vec(W◦Rcg)
where ◦ denotes the Hadamard (element-wise) product. The second reduction steps are
based on Kronecker and Hadamard identities from [23] and [115] respectively. Since these
expressions no longer contain the Kronecker product, this means a large reduction in memory
requirement. Further, as it is assumed thats(t) is generated by an wide sense stationary
stochastic process,SS T should have a block-Toeplitz structure that can be exploited for further
storage reduction. However, even with these optimizations the number of computations
4

4
82 4 Data driven distributed control
required to solve ˆ Ψ from (4.15) increases to the third power in the number of nodes Nn.
Further reductions will require future research, but this procedure is performed off-line and
is not time critical.
The so obtained set of estimates for the Markov coefficients differs from the sought set
that minimizes the cost function J1 for two reasons. Firstly, they were derived based on
the modified, soft constraint cost function ˜ J1. Depending on the correlations in the input
data series and the chosen weighting matrix Wd, this leads to a different solution that approximates
that for the original cost functionJ1 as Wd increases. Secondly, the conjugate
gradient algorithm leads to an estimation error. Although in theory the algorithm converges
on the correct solution afterNq iterations, whereNq is the number of unknowns, numerical
round-off generally leads to significant error and a stopping criterion terminates the iterative
process beforeNq cycles are reached. When iterations are stopped when the Frobenius
norm of the residual vector has been reduced by a certain factor1/ǫ (see section 3.6.3), the
second error can be limited by choosingǫ sufficiently small. However, the total estimation
error and its effect on the eventual performance of the model in terms of prediction error
are not further investigated in this report due to the high computational cost involved in
obtaining the exact, hard-constrained solution for systems of relevant dimensions.
Now the estimates for the Markov parametersf (i,j,l) fori = 1...Nn,j ∈ F (i,l) andl =
0...Nt−1 from ˆ Ψ can be substituted into the FIR approximation in (4.7). When neglecting
the termsξ (i)(t), estimatesû (i)(t) for the output trajectoriesu (i)(t) can be obtained as:
û (i)(t) =
Nt−1
l=0
j∈F (i,l)
Stage 2: estimating the controller coefficients
ˆ f(i,j,l)s (j)(t−l). (4.16)
In the second stage, the unknown coefficientsã (i,j,l) andk (i,u,l) of the distributed controller
as in (4.4) can be estimated. The estimates will be obtained by using the trajectory estimates
û (i)(t) and solving independent linear regression problems fori = 1...Nn.
Letr (i)(t) be the equation error of (4.4) where the optimal output trajectoryu (i)(t) has been
replaced by the estimate û(t):
r (i)(t) =
ã (i,j)(z)u (j)(t)−
k (i,j)(z)s (j)(t).
j∈C (i)
This can be rewritten using matrix notation as:
j∈M (i)
r (i)(t) = û (i)(t)−θ T (i) υ (i)(t), (4.17)
where the column vector υ (i)(t) stacks all û (j)(t − l) and s (m)(t − l + 1) for j ∈ C (i),
m ∈ M (i) and l = 1...n. The column vector θ (i) stacks the corresponding unknown
controller coefficientsã (i,j,l) andk (i,m,l).
Let the estimate ˆ θ (i) of the coefficient vectorθ (i) now be defined as the argument that minimizes
the cost functionJ 2,(i):
J 2,(i) = r (i) 2 F
N−1
t0
, (4.18)

4.7 Identification procedure 83
= û(i)(t0) ... û
(i)(N −1)
Û (i)
−θ T (i)
υ(i)(t0 −1) ... υ (i)(N −2)
Υ (i)
2
F
, (4.19)
where the second step follows by substitution of (4.17). The solution ˆ θ (i) can now be expressed
in the well-known form:
ˆθ (i) =
Υ (i)Υ T −1Û(i)Υ (i) (i). (4.20)
The coefficients a (i,j,l) of the closed-loop controller C(z) can be calculated from ˆã (i,j,l)
and ˆ k (i,u,l) by reverse application of (4.5) on page 75.
Finally, note that Υ (i) stacks only those output and measurement signals to which node i
has direct access at each sampling time. Therefore, each solution ˆ θ (i) can be obtained using
only the information available to node i, making the second identification stage a set of
independent, distributed operations. The number of computations required to solve each
local problem depends only on the controller ordernand the communication radiusrc and
not on the number of nodesNn. Although this suggests possibilities for on-line estimation,
this has no practical relevance due to the centralized nature of the first estimation step.
Enforcing stability
As discussed in section 4.6, stability of the network of open-loop controllers – which
guarantees stability of the closed-loop provided that the plant model is correct – is not trivial
even if all individual controller nodes are stable. Therefore, the above described two-stage
identification algorithm will be extended to comprehend two different approaches to enforce
stability. Besides the stability constraint based on Gershgorin’s circle theorem discussed in
section 4.6.1, a second method will be employed to enforce stability. By adding explicit
weights to the elements of the coefficient matrices à (k) in the second identification stage,
the pole locations of the identified system can be influenced. This is done by augmenting
the cost-function J 2,(i) in (4.19) with the regularization term θ T (i) W′ (i) θ (i), where W ′ (i) is
a diagonal weighting matrix of which only the diagonal elements that correspond to the
unknown coefficients ã (i,j,l) are non-zero. When all non-zero weights are chosen equal, a
line search method can be used to find the weight for which the poles are contained within
the unit circle. This means that in contrast to the Gershgorin method, global stability is
not guaranteed after local optimization, which renders this approach unfit for the future
development of an on-line adaptive distributed control law. Moreover, both the second
identification stage and the stability check become computationally costly for large systems.
The principle based on Gershgorin’s circle theorem as proposed in 4.6.1 is applied as
follows. For eachi = 1...Nn the linear constraints in (4.9) together with the cost function
J 2,(i) in (4.19) lead to a Quadratic Programming (QP) problem that can be expressed as:
ˆθ (i) =argmin
θ (i)
subject to
θ T (i) Υ (i)Υ T (i)θ (i) −2 Û (i)Υ T (i)θ (i)
Ξ(i) 0 θ (i) < 1−λ,
(4.21)
4

4
84 4 Data driven distributed control
where Ξ (i) =∈ R2M (a,i)
×M(a,i) is a binomial matrix consisting of the sign-values ±1. Its
rows form all possible signs for the parametersa (i,j,k) inθ (i) that can be obtained by taking
M (a,i) values from the set {−1,1} with replacement.
The QP problems are solved using Matlab’sÕÙÔÖÓ[130] function and – in principle
– must be performed only once. However, for λ = 0 the stability constraint allows poles
arbitrarily close to the unit circle, which in practice may lead to marginally stable systems.
The value of λ that yields the best performance in terms of (4.10) varies per data set and is
again found using a line search.
4.7.3 Algorithm summary
The derivation of the closed-loop distributed controller C(z) can be summarized into the
following steps:
1. Start with a set of open-loop WFS measurement data of the wavefront disturbance.
2. Estimate the Markov parametersf (i,p,l) of the distributed controllerR(z) as the minimizers
of the cost function (4.14). A conjugate gradient solver is proposed to solve
the resulting system of equations in (4.15) for vec(Ψ). Since the cost-function is
based on soft constraints, Markov parameters that are required to be zero will only
be arbitrarily small and must be fixed to zero after conjugate gradient iteration has
converged.
3. Substitute the identified Markov parameters into (4.16) to obtain an estimate for the
output trajectoriesu (i)(t) of the open-loop controller nodesR (i)(z).
4. Estimate the coefficients ã (i,j,l) and k (i,j,l) of R(z) as the minimizers of the cost
function (4.18). Two approaches are proposed to enforce stability ofR(z):
• Add regularization terms of the formθ T (i) W′ (i) θ (i) to the cost function hence to
the inverted expression in the explicit solution in (4.20).
• Apply linear constraints based on Gershgorin’s circle theorem (section 4.6.1)
and solve the resulting QP problem in (4.21).
5. Obtain the remaining coefficientsa (i,j) of the closed-loop distributed controllerC(z)
from (4.2) using (4.5).
4.7.4 Discussion
Unseen modes and bias errors
As mentioned in section 4.7, the rank deficiency of G is the reason that only a limited
number of Markov parameters F (k) can be uniquely estimated in the first identification
step. Only these coefficients can yield a non-zero contribution to the unseen modes in û(t)
as for the others this is constrained to zero. For the other coefficients, this contribution is
undefined and fixed to zero when solving the estimation problem.
In the centralized case when even the first Markov parameter is not uniquely defined, this
means that the time series in û(t) are not independent, rendering the second identification
step an ill posed problem. A solution can still be obtained using a pseudo-inverse or similar
to [100] an orthogonal projection onû(t) can be applied to remove the dependent signals.
In the distributed case, the ill defined parts of F (l) imply an unknown error on û(t) and

4.8 Simulation and breadboard results 85
therefore an unknown bias on the estimates ˆã (i,j,l) and ˆ k (i,u,l). On the other hand, for
a fixed communication radius rc, the number of Markov parameters that contain no illdefined
columns increases with the number of actuators Na. Assuming that the dynamics
underlying the unseen modes are stable and have a finite impulse response, the bias will
therefore decrease for increasing Na. Alternatively, physical models of the atmospheric
wavefront disturbance can be used in the first identification step. For instance, the temporal
spectra of the unseen modes have been modeled based on Kolmogorov statistics and the
frozen flow assumption in [33] and can be used as additional weighting terms in the cost
functionJ1.
Further, the estimation results will be biased because only a limited numberNt of Markov
parameters is estimated and the cost functions in (4.11) and (4.19) represent only finite
sample approximations of the actual variances. However, these effects can be limited by
choosing the number of coefficient matrices Nt and the number of samples N sufficiently
large.
Order selection
Since neither the true order of the data generating system T(z) is known nor its spatiotemporal
behavior, the choice for Nt and n is not straightforward and may depend on the
chosen communication radiusrc.
The number of tapsNt must be chosen such that the estimateû(t) is consistent. In practice,
it can be obtained by gradually increasing it until this yields only a marginal decrease of the
cost function J1. The number of states per node n can be chosen using the same approach
on the cost functionsJ 2,(i).
4.8 Simulation and breadboard results
In this section, the results of application of the described identification algorithm to several
sets of measurements will be presented. A first set of measurements has been obtained from
an AO breadboard at TNO Science and Industry, which is depicted in figure 4.5. Although
the results on this data demonstrate the validity of the described approach, the number of
spots of the wavefront sensor is too limited to show the effect of scaling up the system
dimensions. Therefore, a second data set with a larger number of sensor spots has been
artificially created.
4.8.1 Performance measures
The performance of the controllers resulting from both identification stages will be measured
as a relative prediction error in terms of the Variance Accounted For (VAF). For the
first stage this will be defined as:
VAF1 =
J1
1−
〈sTs〉 N−1
Nt
·100%, (4.22)
where J1 is the cost function defined in (4.11), but evaluated on a separate validation data
set. For the second stage, the signal ê(t) will be obtained after application of the identified
4

4
86 4 Data driven distributed control
ARMA controller on the validation data set and used to evaluate:
⎛
T N−1
ê ê t0
VAF2 = ⎝1−
〈sTs〉 N−1
⎞
⎠·100%,
t0
wheret0 is chosen large enough to skip the transient errors.
Further, the results will be compared to that of the baseline strategy, which is defined in
section 4.7.1 as the static controller R(z) = G # . Substitution into (4.3) gives the corresponding
prediction error signalê0(t) as:
ê0(t) = s(t)−GG # s(t−1).
Let the corresponding VAF0 value then be defined as:
4.8.2 Breadboard data
T N−1
ê0ê0 1
VAF0 = 1−
〈sTs〉 N−1
·100%.
1
The breadboard (figure 4.5) contains a 630nm laser source whose light is fed through a pinhole
to simulate a point source. A hand-polished, glass disc with a 110mm radius is used as
a turbulence simulator. Its spatial characteristics resemble Kolmogorov turbulence with an
intensity characteristicDt/r0 = 5, whereDt denotes the diameter of the telescope aperture
andr0 the Fried constant [98]. In order to simulate the temporal behavior of the wavefront
disturbance, the disc rotates through the laser beam, whose diameter at this point is 10mm.
After several reflections, the beam reaches a wavefront sensor consisting of an OKO-Tech,
hexagonal Hartmann array with 127 lenselets and an SVS-Vistek CCD camera. As discussed
in chapter 2, a compression factor 16 is foreseen from the telescope aperture onto
a DM surface, such that a 6mm actuator spacing of a DM becomes approximately 96mm
when projected on the telescope’s aperture. When assuming the number of WFS spots to be
equal to the number of actuators – in this case 127 – this means that the telescope diameter
Dt corresponding to the setup is approximately 11·96mm≈ 1m, where 11 is the number
of sensor spots over the diameter.
The disturbance translates over the aperture as a pure frozen flow and the rotation speed of
the turbulence simulator can be varied to simulate different wind speeds. These wind speeds
can be related to Greenwood frequencies using the approximation in (2.5). However, since
rotation speed and sampling time have the same effect (except for measurement noise and
motion blur effects), let the temporal behavior of the disturbance be characterized using the
ratio between Greenwood frequencyfG and sampling frequencyfs. For the obtained measurements,
this ratio lies in the range 0.01 < fG/fs < 0.41. For an AO system for an 8m
telescope with a sampling frequency of 1kHz, this corresponds to wind speeds between ca.
5 and 200m/s.
Identification of the FIR coefficients is done on a set of gradient measurements containing
10000 samples, whereas the performance is evaluated on a separate validation set of 1000
samples. Figure 4.6 shows VAF1 against Nt for various communication radii rc. It can
be observed that even for the minimal communication radius, which allows communication

4.8 Simulation and breadboard results 87
DM
Laser
TT
SC
WFS
Figure 4.5: The AO breadboard setup at TNO Science and Industry. SC is the science camera, TS
the turbulence simulator and TT a tip/tilt steering mirror. The picture is courtesy of K.
Hinnen.
Variance accounted for [%]
100
95
90
85
80
75
70
65
r c = 1
r c = 1.5
r c = 2
r c = 2.5
60
1 3 5 7 9 11 13
Baseline
15
Number of taps N [−]
t
TS
Figure 4.6: Performance of
the structured FIR approximation:
VAF1 from (4.22) as a
function of the number of taps
Nt on a breadboard data set
with a Greenwood to sampling
frequency ratiofG/fs ≈ 0.3.
with only the four directly adjacent nodes, a better phase prediction is obtained than for the
centralized baseline strategy. On the other hand, with only slightly more than 4 rings or in
total 69 illuminated sensor spots, the distributed controller becomes centralized forrc = 11.
As restriction of the radius rc causes the distributed controller to trade spatial correlations
for temporal ones, a decrease inrc can be observed to yield an increase in the relevant number
of tapsNt to achieve a high VAF1 value. However, this restriction has little effect on the
maximum VAF value achieved for a high numberNt of taps.
Figure 4.7 shows the performance results of the distributed controller obtained after the
second identification step for rc ∈ {1,1.5,2} and n = 1...5 obtained by evaluating the
measure VAF2 on the validation data set. Stability was enforced using both the regularization
method described in section 4.7.2 and using the Gershgorin approach of section 4.6.1.
4

4
88 4 Data driven distributed control
Figure 4.7: Performance of
the distributed controller:
VAF2 as a function of the
local controller order n on a
breadboard data set with a
Greenwood to sampling frequency
ratio of fG/fs ≈ 0.3.
The VAF-value of the Gershgorin
approach for rc = 1
varies between 60 and 70%.
Variance accounted for [%]
100
95
90
r c = 1.0 (Regularization)
r c = 1.0 (Gershgorin)
r c = 1.5 (Regularization)
r c = 1.5 (Gershgorin)
r c = 2.0 (Regularization)
r c = 2.0 (Gershgorin)
85
1 2 3
Filter order n [−]
Baseline
4 5
The performance obtained for the first approach is slightly better than for the Gershgorin
approach, but this difference is small. For both approaches the performance increases with
the communication radiusrc and the controller ordern, but for the latter only up to n ≈ 5.
The performance is worse than that obtained using the structured FIR coefficients, which
shows that the proposed combination of the ARMA controller structure and the two-step
identification method does not yield a controller that is able to fully exploit the available
spatio-temporal correlation. Nevertheless, even for rc = 1 and n = 1 the performance
of the identified distributed controllers exceeds that of the baseline method. By increasing
rc, the performance gradually approaches that of the centralized case, which corresponds
to the distributed system for rc ≥ 11. Therefore, the radius rc serves as a means to make
a trade-off between the costs of computation and communication hardware and achievable
performance.
4.8.3 Artificial data set
To show the effect of increasing the number of Shack-Hartmann spots on the performance
of the distributed controller, a data set is required with a higher number of sensor spots.
This data set has been artificially created. First, a static phase screen with Kolmogorov
spatial statistics was generated using a midpoint displacement algorithm [95]. Then this
was interpolated over a square aperture window that was translated over the phase screen
with a speed corresponding to fG/fs ≈ 0.24, thus simulating a frozen flow and yielding
φ(t). The measurement data set s(t) was then obtained through the sensor model in (4.1)
where the variance of the measurement noise signalw(t) was chosen according to a signal
to noise ratio of 20dB.
The performance of the identified controller is again evaluated using VAF1 and VAF2 and
performed on a different data set. For all data sets, a communication radius rc = 2 was
used and the number of coefficient matrices was chosenn = 2. Results are plotted in figure
4.8 for grid sizes between 3×3 and19×19 sensor spots. Note here that in contrast to the

4.9 Conclusions and future work 89
Variance accounted for [%]
100
90
80
70
60
50
40
30
20
10
0
FIR approximation
Distributed ARMA (Regularization)
Distributed ARMA (Gershgorin)
Baseline
50 100 150 200 250 300 350 400
Number of sensor spots [−]
Figure 4.8: The measures
VAF1 and VAF2 for both identification
steps as a function of
Ns, the number of points in
the sensor grid, using rc = 2
and n = 2.
previous results, the performance obtained from the Gershgorin approach is slightly better
than that of the regularization approach. For the centralized case the VAF-value is expected
to increase, since the relative contribution of the badly predictable part of the disturbance
entering the aperture at one edge decreases. This is still the case for the FIR approximation
of the distributed controller, but the performance of the distributed ARMA controllers shows
a slight decrease when the number of sensor spotsNs increases. This decrease becomes less
steep asNs increases.
4.9 Conclusions and future work
Due to the working principle of the Shack-Hartmann wavefront sensor, wavefront reconstruction
forms a critical part of the AO control system. As shown in the previous chapter
this poses a serious challenge for the design of a distributed controller, because local
controllers have access to only a limited number of measurements. In this chapter some
ideas of the previous chapter were extended towards a more general distributed controller
parametrization in the context of a closed-loop control setting in which the WFS observes
the effect of the control actions. Further, an idealized DM is considered that is modeled as
an identity transfer matrix and the WFS is of the Shack-Hartmann type with one sample
delay and white measurement noise. The approach towards a distributed controller is driven
by the desired structure and aimed at determining the performance achievable given this
structure to determine the feasibility of distributed control for AO.
This structure was defined as a network of output interconnected ARMA controllers,
where the connections are chosen based on a communication radius. It was shown that by
assuming this parametrization it is implicitly assumed that the structure of the disturbance
generating system in innovations form is also a network of output interconnected ARMA
systems.
A two-stage approach has been proposed to identify the unknown controller parameters in
which the estimation problem is split into two linear regression problems. In the second
step, stability is enforced using two alternative approaches: regularization and constrained
4

4
90 4 Data driven distributed control
optimization based on Gershgorin’s circle theorem. Both identification steps need to be
solved off-line and the first is a centralized instead of a distributed operation.
Results were presented both on measurement data obtained from an optical breadboard
and on synthetic data. Little performance difference is found between the Gershgorin
and regularization methods to enforce stability. The performance of the distributed
controller is shown to depend on the chosen communication radius, but even for very
small communication radii it is found to exceed that of the baseline strategy. The latter
is defined as centralized reconstruction of the latest vector of measurements using the
pseudo-inverse of the Fried geometry matrix G. Further, the performance decays slightly
with the number of sensor and actuator channels, which is not the case for a structured
FIR approximation, which indicates that this is not a fundamental limitation of the distributed
architecture. Future research is required to analyze such scaling properties in more
detail, preferably using wavefront disturbance measurements from an actual large telescope.
In this chapter the DM system has been assumed to be an ideal corrector by neglecting
both its temporal and spatial dynamics and letting its influence on the wavefront phase
be equal to the actuator commands. This is not the case for the DM system developed
within this project, which means that the presented algorithm cannot be directly applied.
However, in future work it can be investigated whether the principle of exploiting the local
spatio-temporal dynamics of the disturbance signal to perform a global operation such as
wavefront reconstruction can also be applied to the servo problem of calculating the optimal
DM command vector for the predicted wavefront shape. In anticipation of this, the components
of the DM introduced in chapter 2 are described and modeled in the next chapters.
This leads to a complete spatio-temporal model of the DM system that is suitable both for
verification purposes as well as for the design of a (distributed) controller.

ÔØÖÚ
ÌÚÖÐÖÐÙØÒØÙØÓÖ
The design of the electromagnetic reluctance type actuator will be introduced.
The actuator converts a current through a coil into a mechanical force – and
thus a deflection – by means of a magnetic flux that varies with the deflection.
A mathematical model will be derived that describes this relation based on
equations from the magnetic, mechanical and electronic domains. A measurement
setup to test a prototype actuator is presented and measurement results
are shown. The derived model structure is validated and model parameters are
estimated from these results and compared to the model. Differences are analyzed
based on a sensitivity analysis, leading to minor design changes before
manufacturing grid modules of 61 actuators.
The design of these modules is then presented and a second measurement setup
is described and measurement results are shown. From these results the actuator
properties are again identified and their statistical spread is analyzed. The
actuators in seven modules are found to behave in accordance with the derived
model and within the specifications set in chapter 2. This proves that the manufacturing
and assembly process is robust and allows the production of reliable
actuator modules.
Joint work with Roger Hamelinck
91

5
92 5 The variable reluctance actuator
5.1 Introduction
Figure 5.1 shows the schematic of the actuator that is used in the adaptive deformable mirror
and whose design process is discussed in detail in [174]. It consists of a PM that provides
a pretension on a suspension membrane that is connected to the reflective mirror facesheet
via a thin rod. An electric current through the coil at the bottom of the PM modulates the
magnetic force of the PM on a small ferromagnetic core suspended by the membrane and
thus affects the deflection of the mirror facesheet. Figure 5.2 shows the CAD drawing
of the actuator and figure 5.5 shows a photo of a single actuator. An insert which contains
the PM, the coil and forms part of the magnetic circuit is shown on the left of the photo.
The actuator is a so-called variable reluctance type actuator. This is an electromagnetic
actuator in which an electromagnetic force is influenced by the variation of a reluctance in
the path of the magnetic flux. The actuator is chosen as a variable reluctance actuator for
several reasons. Firstly because the variable reluctance design has no moving coil. This
means that no moving wires are required that form parasitic stiffnesses and are sensitive to
break. Moreover, it allows for a low moving mass and a high resonance frequency. Finally,
an indirect advantage is that air gaps – and thus reluctances – can be very small, leading to
a high efficiency.
The magnetic flux from the PM passes through the axial air gap, the ferromagnetic moving
core and a radial air gap to one of the three pole shoes located in the baseplate and finally
back to the PM. Since the mechanical spring is not a part of the magnetic flux path, no
trade-off is required for the selection of the materials for the spring and the moving core.
They can be chosen for mechanical strength and magnetic permeability respectively. In
this specific actuator, the reluctance of the flux path through the axial air gap varies with
the width of this gap, which depends on the position of the moving ferromagnetic core
and thus on the deflection of the membrane. The PM pulls at the moving core, which is
counteracted by the suspension membrane that forms a mechanical spring. This spring
provides a positive stiffness that is matched by the negative stiffness of the PM. Since
both stiffnesses vary with the actual deflection, multiple equilibrium points exists of which
one is stable and yields a residual, positive actuator stiffness. A suitable value for this
stiffness is desired that is low enough to prevent the need for high actuator forces and thus
power dissipation, but high enough to keep the resonance frequency of the Deformable
Mirror (DM) at approximately 1kHz regardless of its size. This minimum resonance
Figure 5.1: Schematic of the variable
reluctance actuator used in the adaptive
deformable mirror.The magnetic flux
from the PM passes through the axial air
gap, the ferromagnetic moving core and
a radial air gap to one of the three pole
shoes located in the baseplate and finally
back to the PM.
connection strut
z
axial airgap ferromagnetic moving core
radial airgap membrane suspension
pole shoe
baseplate
Φ
n s
PM
mirror facesheet
coil

5.1 Introduction 93
Figure 5.2: CAD drawing of the variable reluctance actuator. The membrane suspension, with three
leafsprings and the ferromagnetic core is shown. The magnetic flux from the PM passes
through the axial air gap, through the ferromagnetic moving core and through a radial
air gap the three pole shoes located in the baseplate, back to the PM.
frequency is required to achieve the 200Hz control bandwidth as discussed in chapter 2. It
also means that the DM system has a static gain up the the bandwidth, such that it can be
adequately diagonalized for control using a static decoupling matrix. The corresponding
required on actuator stiffness is discussed in detail in [174], where the optimum value is
derived as 500N/m. This stiffness is low compared to the out-of-plane stiffness of the DM’s
reflective facesheet, such that a malfunctioning actuator does not cause a hard point – i.e. a
point with a fixed deflection – in the reflective surface and thus has a limited effect on the
optical quality of the DM.
The equilibrium position of the moving core can be influenced by an electric current
flowing through the actuator coil that affects the magnetic force acting on it. This current is
provided by driver electronics discussed in chapter 6.
In this chapter, first the relevant actuator parts will be discussed and mathematical equations
for their behavior will be derived. This leads to nonlinear models describing both the
static and dynamic behavior of the actuator. A series of measurements is then performed
on a single actuator prototype whose results are used to validate the derived model. The
sensitivity of the actuator behavior is then analyzed w.r.t. geometric, magnetic and elec-
5

5
94 5 The variable reluctance actuator
Figure 5.3: Dimensions in mm of the tested membrane designs.
tric properties, which is used to improve the actuator design before manufacturing them in
modules. Finally, measurement results will be shown for the actuator module prototypes
and conclusions an recommendations will be formulated. Further information about the
actuator design can be found in [174].
5.2 The single actuator
5.2.1 The actuator membrane suspension
The stiffness of the membrane, in which the moving core is suspended, largely determines
the actuator’s resonance frequencies and thereby the resonance frequencies of the adaptive
Force [mN]
200
150
100
50
0
−50
−100
−150
−200
Measurement
FEM data
−150 −100 −50 0 50 100 150
Displacement [µm]
200
150
100
50
0
−50
−100
−150
−200
Measurement
FEM data
−150 −100 −50 0 50 100 150
Displacement [µm]
Figure 5.4: Comparison of the nonlinear spring characteristic of the suspension designs as shown in
figure 5.3, as calculated with FEM and as measured. The measurements were performed
using the test setup described in [174].

5.2 The single actuator 95
Figure 5.5: Photo of a single actuator. On the
left, the insert is shown that contains the PM and
the coil.
deformable mirror . The magnetic force generated by the PM and the coil acts on the moving
core and pre-tensions the membrane suspension. According to [177], the nonlinear relation
between spring forceFs and membrane deflectionzs can be approximated by:
Fs(zs) = −C1
Emt3 m
r2 Emtm
zs −C2
m r2 z
m
3 s
(5.1)
where tm and rm are the suspension membrane thickness and radius and Em is the
membrane material’s Young’s modulus. The coefficients C1 and C2 depend on the design
and boundary conditions and will be estimated from FEM results and measurements.
The membrane deflection zs follows the sign definition as indicated in figure 5.7 and is
always negative due to the PM pretension. Since the resulting deflection is larger than the
membrane thickness, a linear approximation of the stiffness is no longer valid: not only the
bending stiffness Emt3 m/r2
m , but also the nonlinear stiffness due to in-plane stretching
becomes relevant.
Figure 5.3 shows on the left a design with springs placed radially towards a central
disc. Results from both FEM analysis and measurements depicted in figure 5.4 show that
the radially placed springs cause the suspension to stiffen quickly. This nonlinear effect is
undesirable as this will complicate the design of a control system for the DM and reduce
the achievable optical correction quality. One way to reduce the nonlinearity is to use a
relatively thick membrane such that the linear term in (5.1) remains dominant for larger
deflections zs. Another way to reduce the nonlinearity is to allow rotation of the central
part to reduce the tensional forces. This leads to the design shown on the right in figure 5.3.
Here the springs are placed tangentially. Due to the out-of-plane displacement, bending in
the leaf springs occurs and the central part will rotate with typically 2 ◦ per Newton [2].
Since the glued connection strut has a low rotational stiffness (figure 2.12 on page 40), this
rotation will not lead to an actuator malfunction after assembly of the DM.
Stiffness measurements
To verify the FEM analysis a test set-up was designed to measure the nonlinear stiffness
for different membrane suspensions. This measurement setup is described in [125] and
5

5
96 5 The variable reluctance actuator
in [174]. In the setup the displacement is measured optically with a Philtec D21 sensor
with sub-µm resolution. The force required to enforce the displacement is measured with
a Kistler 9203 piezo sensor, with mN resolution. The membrane suspensions are placed in
containers (∅25x8mm) to be able to handle them and place them in the measurement setup.
In the containers the membranes are clamped at the outer edge. In the setup the suspensions
are subjected to an out-of-plane displacement of ±100-150µm. For the suspensions shown
in figure 5.3 the results of the nonlinear FEM analyses are compared with the measurements
and shown in figure 5.4. A few remarks on the measurements are given below:
• FEM prediction of the stress in the membrane is needed prior to the measurement to
avoid plastic deformation during measurement.
• Around the central position a negative stiffness is observed. This can be explained,
partly from the stress present in the material due to the production of the foil, which is
rolled, and partly from the clamping forces in the container. However, the membrane
suspension will not be used in this position as the PM pre-tensions the membrane
suspension. To deal with the negative stiffness, the membrane is placed a little higher
above the magnet than originally designed. The spring force model in (5.1) is well
able to describe a negative stiffness aroundzs = 0 and only the coefficientC1 needs to
be adapted. By using a single rolled sheet for all actuators, the variation in membrane
material stresses between the actuators is minimized.
• The measurements are made using slow back and forward motion. A small difference
between the two directions is observed, which is attributed partly to hysteresis in the
clamp and partly to charge leakage in the piezo based sensor.
The suspension membranes of the first actuators (e.g. figure 5.5) were made with titanium
rolled sheets. The titanium sheets had limited yield strength (250N/mm 2 ) and yielded
vulnerable actuators. In later designs, sheets of Havar , a non-magnetic, cobalt based,
high-strength alloy, were used. Its yield strength is 1860N/mm 2 and its Young’s modulus
200GPa [1]. The available choice in sheet thickness is limited. The design on the right
in figure 5.4 was made with a 25µm Havar rolled sheet. The constants C1 and C2 were
estimated using a least squares fit as:
5.2.2 The electromagnetic force
C1 = -0.12, C2 = 0.02.
The axial magnetic force acting on the ferromagnetic core will be modeled as a function of
the actuator deflection za (figure 5.7) and the actuator current Ia. First the magnetic flux
density in the axial air gap is determined, followed by a force derivation based on magnetic
coenergy.
The magnetic circuit of the actuator from figure 5.2 is shown schematically in figure 5.6.
The model includes leakage flux paths: one that short-circuits the coil and one that shortcircuits
the PM. As will be shown later, the first one mainly affects the actuator coil inductance,
whereas the latter affects many properties such as motor constant and actuator

5.2 The single actuator 97
φ1
φ3
Va
N
S
za
φ2
ℜb
ℜgr
φ1
ℜc
1
ℜga (za)
NIa
ℜm
-Hcm lm
Figure 5.6: Left: a schematic representation of the variable reluctance actuator from figure 5.2.
Right: the electrical equivalent circuit including two leakage flux paths.
stiffness. The indicated flux paths contain two sources – the PM and the coil – and eight
reluctances: the reluctance ℜm of the PM itself, ℜga(za) of the axial air gap, ℜc of the
ferromagnetic core, ℜgr of the radial air gap, ℜb of the baseplate, ℜbc of the part of the
baseplate that forms the core of the coil and ℜflc and ℜflm of the leakage flux paths that
short-circuit the coil and PM respectively. Based on first principles, the reluctances of the
PM, the axial and radial air gaps and the coil core are expressed as:
lm
ℜm =
µ0µrmAm
,ℜga(za) = z0 +za
,ℜgr = lgr
µ0Aga
φ4
ℜbc
φ2
lc
,ℜbc =
µ0Agr
µ0µrbAm 2
3
φ5
φ3
ℜflm
ℜflc
, (5.2)
where µ0 is the permeability of vacuum and µrm and µrb are the relative permeabilities of
the PM and the baseplate material respectively.
Since the thickness of the pole shoe is larger than the thickness of the ferromagnetic core
plus the displacement range, the reluctance ℜgr of the radial air gap is considered to be
independent of the displacement za. Am and Agr are the cross sectional areas of the flux
paths through the PM and the radial air gap respectively and (z0 + za), lgr and lc are the
axial air gap height and the lengths of the flux paths through the radial air gap and coil core
respectively. A schematic of the actuator with the definitions of za and z0 is depicted in
figure 5.7. The effective lengths and areas of the flux paths through the base plate and the
ferromagnetic core are estimated from the actuator geometry, leading to the reluctancesℜb
andℜc as listed in table 5.1. Their values lie two orders of magnitude below the reluctances
of the air gaps and the PM and will be combined into a single reluctance ℜr = ℜc + ℜb
with a characteristic path lengthlr.
Figure 5.6 indicates five different magnetic fluxesφ1...φ5 with positive directions. Since
the sum of the fluxes towards each node must be equal to zero, these fluxes are related as:
⎧
⎪⎨ φ1 −φ2 +φ3 = 0,
φ2 −φ3 −φ4 +φ5 = 0,
(5.3)
⎪⎩
φ4 −φ5 −φ1 = 0.
The PM is represented as a source with an internal reluctance and the coil as a source of
magnetomotive force. According to Ampre’s law, the magnetomotive forces F1...F3
5

5
98 5 The variable reluctance actuator
can be derived for the three different flux paths indicated in the figure as:
⎧
F1 =
⎪⎨
⎪⎩
Hdl = NIa = Hmlm +Hga(za +z0)+Hrlr +Hgrlgr +Hbclbc,
1
F2 =
Hdl = 0 = Hmlm +Hflmlflm,
2
F3 =
Hdl = NIa = Hflclflc +Hbclbc,
3
where Hm, Hga, Hgr, Hr and Hbc are the magnetic field intensity in the PM, the axial air
gap, the radial air gap, the combined baseplate and moving core and the coil core respectively.
Assuming that all flux conductors represent linear magnetic materials, their flux densitiesB
are related to their magnetic field intensityH via the material’s magnetic permeabilityµ as
B = µH. For the PM, this relation includes an offset:
Bm = µ0µrm(Hm −Hcm), (5.4)
where Hcm is the coercivity of the PM. According to Gauss’s law, flux φ is the integral of
the flux densityB over an areaA:
φ = B ·dA = BA, (5.5)
A
where the latter equality assumes that the flux density is constant over the cross-sectional
areaA.
When substituted into (5.4), this allows the magnetic field intensity of the PM to be
expressed as:
Hm = Bm
µ0µrm
+Hcm =
φ4
µ0µrmAm
+Hcm. (5.6)
Substitution of (5.6), Gauss’s law from (5.5), the definitions of the reluctances from (5.2)
and the linear relations between flux density and magnetic field intensity into the expressions
forF1...F3 then yields:
⎧
⎪⎨ F1 = NIa = ℜmφ4 +Hcmlm +(ℜga(za)+ℜr +ℜgr)φ1 +ℜbcφ2,
⎪⎩
F2 = 0 = ℜmφ4 +Hcmlm +ℜflmφ5,
F3 = NIa = ℜflcφ3 +ℜbcφ2.
Figure 5.7: Definition of the axial air gap height
through the initial gap z0 and the displacement
za. The height h is the axial air gap when
the suspension membrane is not deflected, i.e.
zs = 0. The membrane deflectionzs is related to
the actuator displacementza aszs = za+z0−h,
and is negative for downward membrane deflection.
-zs
h z0
za

5.2 The single actuator 99
From these three relations and the three flux relations in (5.3), five uncoupled expressions
can be solved for the fluxesφ1...φ5 as:
where
φ1(Ia,za) = NIaℜflcℜ2 −Hcmlmℜflmℜ3
˜ℜ(za)
φ2(Ia,za) = NIaℜ(za)−Hcmlmℜflcℜflm
˜ℜ(za)
φ3(Ia,za) = NIa(ℜflmℜm +ℜ1(za)ℜ2)+Hcmlmℜbcℜflm
˜ℜ(za)
φ4(Ia,za) = NIaℜflcℜflm −Hcmlmℜ(za)
˜ℜ(za)
φ5(Ia,za) = NIaℜflcℜm −Hcmlm(ℜbcℜflc +ℜ3ℜ1(za))
˜ℜ(za)
ℜ1(za) = ℜga(za)+ℜr +ℜgr,
ℜ2 = ℜm +ℜflm,
ℜ3 = ℜbc +ℜflc,
ℜ(za) = (ℜflc +ℜ1(za))ℜ2 +ℜflmℜm,
ℜ(za) = (ℜflm +ℜ1(za))ℜ3 +ℜbcℜflc,
˜ℜ(za) = (ℜflmℜm +ℜ2ℜ1(za))ℜ3 +ℜbcℜflcℜ2.
(5.7a)
(5.7b)
(5.7c)
(5.7d)
(5.7e)
Note that the flux φ4(Ia,za) through the PM will be zero when the winding current Ia is
equal to Iacc = −(Hcmlmℜ(za)/(Nℜflmℜflc) or in absence of leakage flux to Iacc =
−Hcmlm/N. For this current, the coil’s magnetic field fully cancels that of the PM. For
the values in table 5.1 and in the absence of leakage flux, this corresponds to a current
Ia =324mA. For the coil’s ∅50µm copper wire this corresponds to an unrealistic current
density of 165A/mm 2 .
From the derived expressions for the fluxes, the operating point of the PM on its B-H curve
is obtained. This operating point indicates how efficiently the volume of the PM is used
to generate a desired flux density. Substitution of the flux φ4(Ia,za) from (5.7d) into the
expression for the magnetic field intensity of the PM in (5.6) provides the magnetic field
intensityHm of the PM as:
Hm = Hcm − NIaℜflmℜflc +Hcmlmℜ(za)
µ0µrmAm ˜ . (5.8)
ℜ(za)
In the unactuated state (i.e. Ia = 0, za = 0), with the use of the values from table 5.1,
this yields Hm =-313kA/m. Subsequent substitution into (5.4) then leads to Bm =0.33T.
The product of flux densityBm and magnetic field intensityHm indicates the available PM
energy per unit volume. Using (5.4), the maximum of this product is found as:
|BmHm| max = µ0µrm(Hm −Hcm)Hm
= µ0µrmH
max 2 cm /4 = 106kJ/m3 .
5

5
100 5 The variable reluctance actuator
The value derived for the actuator is |HmBm| =104kJ/m 3 , which is very close to the
optimum. In fact, this optimum corresponds to the situation whenHm = −Hcm/2 and thus
– when considering (5.8) forIa = 0 – to:
ℜ(za) 1
= .
˜ℜ(za) 2ℜm
In absence of leakage flux (i.e. ℜflc,ℜflm → ∞) this reduces to ℜm = ℜbc + ℜ1 =
ℜbc +ℜgr +ℜr +ℜga(za), which implies that the highest volume efficiency of the PM is
obtained when the internal reluctance ℜm of the PM is exactly equal to the external reluctance
felt by the PM.
The magnetic force on the ferromagnetic core is calculated via flux linkage and magnetic
coenergy [59]. In this procedure the PM is modeled as a fictitious winding with the equivalent
magnetomotive force. This means thatHcmlm is replaced by−NfIf , whereNf is the
number of turns of the fictitious winding and If the fictitious current through it. The flux
linkagesλ of the coil andλf of the fictitious winding are given by [59]:
λ(Ia,za) = Nφ2(Ia,za) = L11(za)Ia +L12(za)If, (5.9)
λf(Ia,za) = Nfφ4(Ia,za) = L21(za)Ia +L22(za)If,
whereL11(za) andL22(za) are the self inductances of the coil and the PM andL12(za) and
L21(za) the corresponding mutual inductances:
L11(za) = N2 ℜ(za)
˜ℜ(za)
L21(za) = NNfℜflcℜflm
˜ℜ(za)
, L12(za) = NNfℜflcℜflm
,
˜ℜ(za)
, L22(za) = N2 fℜ(za) .
˜ℜ(za)
(5.10)
As expected, the two mutual inductances L12(za) and L21(za) are equal. The magnetic
coenergy can be expressed in terms of these (mutual) inductances as [59]:
W(Ia,za) = 1
2 L11(za)I 2 a +L12(za)IaIf + 1
2 L22(za)I 2 f, (5.11)
= 1
2 L11(za)I 2 a
+ L12(za)
Nf
IaHcmlm + L22(za)
2N2 H
f
2 cml2 m ,
1
=
2˜ 2 2
N Iaℜ(za)−2NIaHcmlmℜflmℜflc +H
ℜ(za)
2 cml2 mℜ(za) .
Note that in the second step the magnetomotive forceNfIf of the fictitious winding is again
replaced by the −Hcmlm of the PM. After substitution of the inductances from (5.10) in
the final step, the expression forW(Ia,za) becomes independent ofNf andIf .
The electromagnetic forceFm(Ia,za) is equal to the partial derivative of the coenergy with
respect to the displacementza and can be expressed as:
Fm(Ia,za) = ∂W(Ia,za)
∂za
= −1
2Agaµ0
2 NIaℜflcℜ2 −Hcmlmℜflmℜ3
˜ℜ(za)
(5.12)

5.2 The single actuator 101
5.2.3 A static actuator model
Together, the derived equations for the electromagnetic force Fm(Ia,za) in (5.12) and the
mechanical spring force in (5.1) provide static relations for the behavior of the actuator.
This provides insight in the nonlinear actuator stiffness, the required actuator current and
voltage, its motor constant and power dissipation. However, the derived relations do not
provide insight into dynamic properties such as resonance frequency, damping, inductance,
etc. that affect the achievable controller performance and thus the correction quality of the
Adaptive Optics (AO) system. Therefore, in section 5.2.4 these equations are extended to
also include dynamic behavior.
The static force equilibrium can be expressed as:
Fs(zs)+Fm(Ia,za) = 0. (5.13)
Let the nominal operating point of the actuator be defined as the unactuated equilibrium
point, where the air gap z0 is such that the electromagnetic force Fm(Ia = 0,za = 0)
equals the membrane suspension spring force Fs(zs). The deflection zs of the suspension
membrane can be expressed in terms of theza, z0 andhas defined in figure 5.7 by:
zs = za +z0 −h.
Using this definition withza = 0, the initial gapz0 can be solved from the force equilibrium
in (5.13) through substitution of (5.12) and (5.1):
− 1
2µ0Aga
2 Hcmlmℜflmℜ3
˜ℜ(za)
Emt
−C1
3 m
r2 Emtm
(z0 −h)−C2
r2 (z0 −h)
m
3 = 0, (5.14)
This leads to a fifth order equation in z0 with five solutions. The solution that corresponds
to practice is real-valued. Moreover, it forms a stable equilibrium – i.e. the derivative of
the sum of forces with respect to z0 at the solution for z0 is negative – and finally has a
value within the range 0 < z0 < h. This solution is found numerically after substitution
of the parameters from table 5.1. Note that the values for some parameters in this table are
determined from measurements as described in section 5.2.5. The initial gap z0 found is
z0 =109µm. The spring and electromagnetic forces of (5.14) are plotted in figure 5.8 as a
function ofz0.
5

5
102 5 The variable reluctance actuator
Force [N]
Table 5.1: Electromagnetic and mechanical parameters of the variable reluctance actuator.
0.2
0.15
0.1
0.05
0
Parameter Value Unit Parameter Value Unit
Hcm -540 a kA/m C1 -0.12 c -
lm 0.30 mm C2 0.02 c -
Aga 0.79 mm 2 Em 200 GPa
Agr 0.48 mm 2 tm 25 µm
Am 0.79 mm 2 rm 2.5 mm
ℜga(0) 111 1/µH mac 3.6 mg
ℜm 262 1/µH Ra 39.0 Ω
ℜr 512 1/µH h 230 d µm
ℜgr 248 1/µH z0 109 e µm
ℜbc 1 1/µH ba 0.4 f mNs/m
ℜflc 100 b 1/µH ca 583 e N/m
ℜflm 600 b 1/µH
a Estimated from PM measurements [174]
b Value estimated from measurements on actuator prototypes
c Fitted on nonlinear FEM model with typical value for negative stiffness included
d Design parameter
e Derived value
f Estimated from actuator measurements
Spring force F s
Magnet force F m
Residual force
Equilibrium
−0.05
0 50 100 150 200
Air gap z [µm]
0
Figure 5.8: Forces exerted by the membrane suspension
and the PM respectively as a function of
the axial air gap.
Stiffness [N/m]
800
700
600
500
400
300
200
100
Spring
Magnet
Residual
0
−10 −5 0 5 10
Displacement z [µm]
Figure 5.9: The stiffness of the suspension membrane
and PM as a function of the actuated displacement
za around the equilibrium z0.
The relation between currentIa and displacementza is found by solving the force equilibrium
equation in (5.13) for the actuator currentIa, yielding:
Hcmlmℜflmℜ3 ±
Ia(za) =
−2Agaµ0 ˜ ℜ 2 (za)
Nℜflcℜ2
C1Emt 3 s
r 2 m
zs + C2Emtm
r2 z
m
3
s
(5.15)
where zs is used for brevity of the expression. Since (5.12) and thus (5.13) is quadratic in

5.2 The single actuator 103
the current, there are two solutionsIa(za) – as indicated by the ± sign – of which only the
solution with a plus sign is valid. Since a current yields a displacement za around the z0
for which Ia = 0, both positive and negative currents lead to realistic displacements. The
solution forIa(za) with a minus sign could never lead to a positive current because the first
term(Hcmlmℜflmℜ3) is always negative (Hcm is negative) and for realistic displacements
the square root term is always positive. The viable relation forIa(za) is plotted in figure 5.10
for the physical properties in table 5.1. Note that despite the nonlinearity of the equations,
the relation between current and deflection is highly linear within the intended operating
range of−10µm< za < 10µm.
The mechanical stiffnessca of the actuator can be derived by taking the partial derivative of
the sum of forces in (5.13) with respect to the deflectionza. Although this remains a function
of both Ia and za, these quantities are statically coupled through (5.15). The dependence
onIa can therefore be replaced by an implicit constantIaz = Ia(za) that denotes the static
current required to reach the displacement za = z ′ a. The mechanical stiffness ca(za) can
thus be derived as:
ca(Iaz,za) = − ∂
(Fs(za)+Fm(Ia = Iaz,za))
∂za
= − ∂
−C1
∂za
Emt3 m
r2 zs −C2
m
Emtm
r 2 m
Emt
= C1
3 m
r2 Emtm
+3C2
m r2 (za +z0 −h) 2
− ℜ2ℜ3(NIazℜflcℜ2 −Hcmlmℜflmℜ3) 2
µ 2 0A2 ˜ℜ ga
3 .
(za)
z 3 s − (NIazℜflcℜ2 −Hcmlmℜflmℜ3) 2
˜ℜ 2µ0Aga
2 (za)
By substituting the relation between current Iaz and displacement za from (5.15), an expression
for the actuator stiffness in terms of onlyza is found:
ca(za) = − C2Emtm
r 2 mµ0Aga
˜ℜ(za) z3 s + 3C2Emtm
r2 m
z 2 s −
C1Emt3 m
r2 ˜ℜ(za) mµ0Aga
zs+ C1Emt3 m
r2 m
, (5.16)
where zs was resubstituted for brevity of the expression. Note that this expression is a
function of the initial air gap z0, such that it does not fully reflect the effect of parameters
affectingz0. Moreover, the stiffness is a third order polynomial in the deflectionza, whereas
without the PM this was second order. Based on the numerical solution for z0 and the
other properties shown in table 5.1, the actuator stiffness has been plotted in figure 5.9 for
displacements za within the intended operating range. The stiffness decreases for positive
za and increases for negative za and varies approximately 16.5% over the full intended
operating range. At the equilibriumza = 0 andzs = z0 −h, the actuator stiffness is found
asca(0) =583N/m.
Figure 5.10 also shows the dissipated powerPa(za) calculated via Ohm’s law:
Pa(za) = I 2 a (za)Ra,
where Ra is the resistance of the actuator coil that can be expressed in terms of geometry
5

5
104 5 The variable reluctance actuator
Current I a [mA], Power P a [mW]
40
30
20
10
0
−10
−20
−30
Current
Power
−10 −5 0 5 10
Displacement z [µm]
Figure 5.10: The actuator current Ia and corresponding
power dissipation required for a displacement
za.
and material properties as:
Actuator force F a [mN]
8
6
4
2
0
−2
−4
−6
Ra = 2πNrcaρe
,
Aw
Force at z = −10µm
Force at z = 0µm
Force at z = 10µm
−8
−30 −20 −10 0 10 20 30
Current ∆I [mA]
Figure 5.11: The force that can be exerted by the
actuator on the DM facesheet as a function of the
current Ia at three deflections za.
whererca is the average coil radius,ρe the specific resistance of the coil’s material andAw
the cross-sectional area of the coil’s wire. For the∅50µm copper wire used this leads to the
valueRa ≈39.0Ω found in table 5.1.
In figure 5.11 the generated actuator force Fa is plotted. This is the external force required
to keep the actuator at a fixed operating point za = z ′ a
as a function of a supplied current
offset ∆Ia = Ia −Ia(z ′ a). It is calculated by augmenting the static force equilibrium with
an additional termFa(∆Ia,z ′ a ) and solving for it:
Fa(∆Ia,z ′ a) = Fm(Ia(z ′ a)+∆Ia,z ′ a)+Fs(z0 +z ′ a −h). (5.17)
The common relation between force and current in linear systems is through the motor
constant defined in Newtons per Ampère, whereas (5.17) expresses this relation for the
derived nonlinear system. Figure 5.11 shows that the relation between the current offset
∆Ia and forceFa(∆Ia,z ′ a
) is highly linear within the intended operating range. According
to this figure, the generated force due to a change in current is only marginally different
at the intended extreme operating points. In fact, when nonlinearities are neglected, it can
be observed from figure 5.11 that the force per current unit – i.e. the motor constant – is
approximately 0.2N/A.
5.2.4 A dynamic actuator model
The derived equations that describe the electromagnetic part of the actuator will be extended
with the mechanical equations of motion to a (nonlinear) dynamic model. This model is
linearized to obtain Bode plots of the actuator and linear electromechanical properties such
as motor constant and coil inductance.
The terminal voltageVa(t) over the actuator coil can be expressed as a function of timetin

5.2 The single actuator 105
[s] using the flux linkage termλ(Ia,za) corresponding to the actuator coil as [59]:
Va(t) = Ia(t)R+ ∂λ(Ia,za)
∂t
(5.18)
In the expression for the flux linkage in (5.9), the magnetomotive force of the PM is expressed
by that of a fictitious winding asNfIf . The latter can again be replaced by−Hcmlm
of the PM, leading to:
λ(Ia,za) = N
NIaℜ−Hcmlmℜflcℜflm . (5.19)
˜ℜ(za)
Since this expression forλ(Ia,za) is a function of both currentIa and positionza, the partial
derivative in (5.18) w.r.t. time t expands via the chain rule into:
Va = IaRa + ∂λ ∂Ia ∂λ ∂za
+
∂Ia ∂t ∂za ∂t
= IaR+ ∂λ
Ia ˙ +
∂Ia
∂λ
˙za
∂za
(5.20)
where the dependence on t has been omitted for brevity. The first partial derivative represents
the self-inductance voltage term, whereas the second occurs in product with the actuator
velocity ˙za and is called the speed voltage. The latter is common to all electromechanical
energy-conversion systems and is responsible for energy transfer between the mechanical
system and the electrical system.
Substitution of the flux linkageλ(Ia,za) of (5.19) into (5.20) then leads to:
Va = IaRa + N2ℜ ˙
˜ℜ(za)
Ia +
Nℜflc
Agaµ0ℜ2 ˜ ℜ 2 (za) (Hcmlmℜflmℜ3 −NIaℜflcℜ2) ˙za (5.21)
This equation describes the electromagnetic part of the system that generates the force
Fm(za,Ia) in (5.12) on the suspended mass. The equation of motion of this mass-springdamper
system can be expressed as:
mac ¨za +ba˙za = Fm(za,Ia)+Fs(za +z0 −h), (5.22)
where ˙za and ¨za are the first and second derivatives of za to time t respectively, ba is the
mechanical viscous damping andmthe mass of the ferromagnetic moving core. Recall that
the magnetic forceFm(Ia,za) is defined in (5.12) and the spring forceFs(zs) in (5.1). Together
with (5.21) this equation forms a nonlinear dynamic actuator model. When assuming
a state vectorx = [Ia za ˙za] T , the time derivative of this state vector can be expressed as:
⎡ ⎤ ⎡ ˜ℜ(za)(Va−RaIa)
Ia ˙
⎣˙za
⎦ ⎢ N
= ⎣
¨za
2 ℜflc
−
ℜ(za) Aga µ0Nℜ2˜ ℜ(za)ℜ(za) (Hcmlmℜflmℜ3
⎤
−NIaℜflcℜ2) ˙za
⎥
˙za
⎦
1 (−Fm(za,Ia)−ba˙za −Fs(za +z0 −h))
mac
(5.23)
where ˙ Ia was solved from (5.21) and ¨za from (5.22) after substitution of (5.12) and (5.1). All
state derivatives except ˙za are nonlinear equations in terms of the state variables. However,
if the effect of the nonlinearities on the actuator behavior is small, the nonlinear equations
5

5
106 5 The variable reluctance actuator
will only complicate the design of a controller. Therefore, this effect will be investigated
in the next subsection through linearization of the nonlinear equations. This will also provide
insight into (linear) dynamic properties such as inductance, resonance frequency and
damping.
Linearization of the dynamic model
Linearization is a widely used technique that enables the application of linear, frequency
domain tools on nonlinear systems. However, it can only provide useful insight when the
nonlinearities play a negligible role around a certain operation point or state. To verify this
for the nonlinear system in (5.23), linearizations at several displacements za = z ′ a around
the initial air gapz0 will be derived and their frequency response functions plotted.
The coil terminal voltage Va serves as the system input and is assumed to be supplied by a
voltage source. The system output is the displacementza, where no notational difference is
made between the original and linearized system description as this is always clear from the
context. The linearized system with state xl = [Ia za ˙za] T can thus be expressed as [69]:
˙xl(t) = ∂˙x
∂ x T
x=xz ′ ,Va=Ia
a z ′
Ra
a
xl(t)+ ∂˙x
∂ Va
x=xz ′
a
Va(t),
where the operating pointxz ′ a is chosen as[Iaz ′ z
a
′ a 0] T whereIaz ′ = Ia(z
a
′ a) and the current
required for the actuator displacement za = z ′ a as plotted in figure 5.10. In the operating
point used for linearization the velocity ˙za is assumed to be zero. The output of this system
can then be chosen as displacementza, velocity ˙za and/or currentIa. After taking the partial
derivatives, substitutingx = xz ′ a andVa = IaRa and omitting dependence ontfor brevity
this leads to:
⎡
−Ra/La(z ′ a ) 0 −Ka(z ′ a )/La(z ′ a )
Al
⎤
⎡
1/La(z ′ a )
˙xl = ⎣ 0 0 1
Ka(z ′ a )/mac −ca(z ′ a )/mac
⎦xl
+ ⎣ 0 ⎦Va,
(5.24)
−ba/mac
0
whereca(z ′ a ) was defined in (5.16) and the elementsIa, za and ˙za of the state xl now form
small signal variations around the operating point xz ′ a . Further, Ka(z ′ a ) and La(z ′ a ) are
the motor constant and inductance at the operating point za = z ′ a respectively and can be
expressed as:
Ka(z ′ a ) = Nℜflcℜ2(NIaℜflcℜ2 −Hcmlmℜflmℜ3)
˜ℜ µ0Aga
2 (z ′ a )
La(z ′ a) = N2 (ℜ1ℜ2 +ℜflcℜ2 +ℜflmℜm)
˜ℜ(z ′ a )
Bl
,
⎤
(5.25)
Figure 5.17 shows the values of the motor constant and the coil inductance as function of
. Both the motor constant and the inductance decrease as the air
the operation pointza = z ′ a
gapz0+za increases, but for the latter the influence of the operating point is smaller. Bode
plots of the actuator response are plotted for several operating pointsza = z ′ a in figure 5.12.

5.2 The single actuator 107
Magnitude [m/V]
Phase [deg]
10 −4
10 −5
10 −6
0
−90
−180
z a ’ = −10.0 [µm]
z a ’ = −5.0 [µm]
z a ’ = 0.0 [µm]
z a ’ = 5.0 [µm]
z a ’ = 10.0 [µm]
−270
1000 1500 2000 2500 3000
Frequency [Hz]
Figure 5.12: Bode plots of the modeled transfer
function between actuator voltage Va and
displacement za for various operating points
za = z ′ a.
This shows a first resonance frequency of the system around 2.04kHz, which increases as
the air gapz0+z ′ a increases. It should be noted that the ratioRa/La between the electrical
resistance and inductance of the actuator corresponds to a pole located at approximately
2.1kHz, which is very close to the mechanical resonance frequency. Although this pole will
affect the two poles corresponding to the system’s mechanics, this effect is small.
Although the dynamic system equations were mainly derived to gain insight into the dynamic
behavior of the actuator, the DC-gain of the linearized system also provides a direct
relation between the supplied clamp voltage Va and the deflection za. This DC-gain H(0)
can be derived from the state-space model in (5.24) by first rewriting it to transfer function
form as:
H(s) = 0 1 0 (sI3 −Al) −1 Bl,
where the displacement za is chosen as the output and Al and Bl are the system matrices
defined in (5.24). Further,s = jω is the complex Laplace variable andI3 the identity matrix
of size 3×3. Subsequent substitution ofs = jω = 0 then yields the DC-gainH(0) as:
H(0) = Ka(z ′ a )
Raca(z ′ . (5.26)
a)
Observe that this gain depends on the coil resistance, the motor constant and the resulting
actuator stiffness, which will all vary per actuator due to material and manufacturing tolerances.
As a result, the DC-gain is expected to have a relatively large variation from actuator
to actuator.
A simplified case: no leakage flux
The expressions describing the actuator behavior derived so far are complicated by the presence
of the leakage flux around the coil and the PM. Although this leads to a more realistic
model that is better able to describe the measurement results, it makes the physical interpretation
of the expressions more difficult. Therefore, the case will be considered when the
leakage flux is completely absent. This will provide a clearer understanding of the relations
between quantities such as geometric dimensions, number of coil turns and inductance, motor
constant, etc.
The absence of leakage flux corresponds to the limit of all above electromagnetic equations
forℜflm,ℜflc → ∞. For the fluxesφ2(Ia,za) andφ4(Ia,za) through the coil and the PM
5

5
108 5 The variable reluctance actuator
respectively, this leads to:
φ ′ 2(Ia,za) = lim
ℜflm,ℜflc→∞ φ2(Ia,za) = NIa −Hcmlm
,
(za)
µ0Agaℜ ′ 1
φ ′ 4 (Ia,za) = lim
ℜflm,ℜflc→∞ φ4(Ia,za) = NIa −Hcmlm
,
(za)
whereℜ ′ 1 (za) denotes the sum of all remaining reluctances:
µ0Agaℜ ′ 1
ℜ ′ 1(za) = ℜga(za)+ℜr +ℜbc +ℜgr +ℜm.
Note that since there is only a single flux path left, fluxesφ2 andφ4 are equal.
Similarly, for ℜflm,ℜflc → ∞ the expression for the magnetic coenergy W(Ia,za) in
(5.11) reduces to:
W ′ (Ia,za) = lim
ℜflm,ℜflc→∞ W(Ia,za) = (NIa −Hcmlm) 2
2µ0Agaℜ ′ 1 (za)
and that for the magnetic force in (5.12) to:
F ′ m (Ia,za) = lim
2µ0Aga (ℜ ′ 2
1 (za))
ℜflm,ℜflc→∞ Fm(Ia,za) = −(NIa −Hcmlm) 2
= −1
2 µ0Aga (φ ′ 2 (Ia,za)) 2 .
The fluxes, the coenergy and the magnetic force are all proportional to the total magnetomotive
force in the system and inversely proportional to the total, position dependent
reluctance ℜ ′ 1(za). In fact, the fluxes are linearly proportional to the total magnetomotive
force, whereas the coenergy and force are both quadratically proportional to this. The resulting
cross product between the magnetomotive force NIa of the coil and the Hcmlm of
the PM amplifies the effect of the coil on the electromagnetic force, which is an advantage
of using a PM to pre-load the spring.
Comparison of the derived limit equations for flux, magnetic coenergy and electromagnetic
force with the original equations in (5.7d), (5.11) and (5.12) shows that leakage flux leads
to additional weights on the magnetomotive forces of the coil and the PM as well as on their
cross products.
For the static actuator model, the absence of leakage flux leads to a simplified relation between
currentIa and displacementza:
I ′ a (za) = lim
ℜflm→∞
ℜflc→∞
Ia(za) = Hcmlm
N
+ 1
N
−2(ℜ ′ 2
1 (za))
C1
Emt3 m
r2 Emtm
zs +C2
m r2 z
m
3
s ,
wherezs is again used for brevity of notation. The relation between actuator stiffnessca(za)
and displacementza becomes:
c ′ a (za) = lim
ℜflm,ℜflc→∞ ca(za),
= − 2C2Emtm
r2 m µ0Agaℜ ′ z
1
3 s
3C2Emtm
+
r2 z
m
2 s + 2C1Emt3 m
r2 µ0Agaℜ ′ 1
zs + C1Emt3 m
r2 , (5.27)
m

5.2 The single actuator 109
Siglab
excitation
Current
amplifier
actuator
Laservibrometer
velocity
Figure 5.13: Experimental
setup to measure the behavior
of a single actuator using
a Siglab TM system, a current
source and a laser vibrometer.
where the dependence ofℜ ′ 1 on the actuator deflectionza has been omitted for brevity. Observe
that the actuator stiffness remains a third order polynomial in the membrane deflection
zs, where only the first and third order terms depend on the magnetic circuit and are scaled
by the total path reluctanceℜ ′ 1(za).
When the leakage flux is neglected for the equations describing the dynamic actuator be-
havior in (5.23), the expression for ˙
Ia becomes:
I ˙′
a (za) = lim
ℜflm,ℜflc→∞
Ia(za) ˙ = −ℜ′ 1 (za)
N2
RaIa −Va + N (Hcmlm −NIa)
Agaµ0ℜ ′ 1 (za)
˙za .
The expression for the linearized system in (5.24) is unaffected, but the actuator stiffness
ca(z ′ a) is given by c ′ a(z ′ a) in (5.27) and the motor constant Ka(z ′ a) and inductance La(z ′ a)
becomeK ′ a (z′ a ) andL′ a (z′ a ) respectively, where:
K ′ a(z ′ a) = −N (NIa −Hcmlm)
(ℜ ′ 1 (z′ a)) 2 , L ′ a(z ′ a) = N2
ℜ ′ 1 (z′ a) .
Both the motor constant and the inductance are thus inversely proportional to the total reluctance
ℜ ′ 1(z ′ a). Since the motor constant is inversely proportional to the square of this
reluctance, a modest reduction in reluctance will lead to a significant improvement of the
motor constant.
5.2.5 Measurements and validation
Several prototypes of single variable reluctance actuators have been manufactured and measurements
have been performed to determine their (dynamic) behavior. The quasi-static
behavior of the actuator is governed by the force-displacement characteristic as derived in
(5.15), whereas its dynamics will be analyzed in terms of the parameters of the linearized
system in (5.24) and its mechanical resonance frequency. The measurements were performed
using the test setup as depicted in figure 5.13, consisting of a Siglab TM [35] system
and a Polytec laser vibrometer. The Siglab system was used to generate an excitation signal
that was fed to a current amplifier and also measured back at one of the Siglab inputs. The
laser vibrometer was used to measure the velocity ˙za of the moving core and its output – an
analog voltage – was also fed to a Siglab input. The sensitivity of the laser vibrometer was
set to 25mm/s/V with an output range of -10...10V.
To measure Frequency Response Functions (FRFs), a wide band, white noise excitation
signal was used with an Root Mean Square (RMS) level of ≈1.5mA and DC offsets ∆Ia
varying between -15 and 15mA. The FRFs are estimated using the Siglab software with a
Hanning window of 8192 samples and 50 averages without overlap. The results are shown
in figure 5.14. Observe that the resonance frequency of the actuator lies around 2.1kHz and
varies approximately 2Hz per mA current offset.
5

5
110 5 The variable reluctance actuator
Magnitude [m/s/A]
Phase [deg]
10 3
10 2
10 1
10 0
90
0
∆ I a = −14mA
∆ I a = −7mA
∆ I a = 0mA
∆ I a = 7mA
∆ I a = 14mA
−90
1000 1500 2000 2500 3000
Frequency [Hz]
Figure 5.14: Measured frequency response functions
of a single actuator for various current offsets∆Ia.
10 1
10 0
Resonance frequency [kHz]
Actuator stiffness [N/mm]
Motor constant [N/A]
Damping [mNs/m]
10
−15 −10 −5 0 5 10 15
−1
DC current offset ∆I [mA]
a
Figure 5.16: The viscous damping ba, resulting
actuator stiffnessca, motor constantKa and undamped
mechanical resonance frequency fe as a
function of the DC current offset I ′ a as identified
from measurement data.
Magnitude [m/s/A]
Phase [deg]
10 3
10 2
10 1
10 0
90
0
∆ I a = −14mA
∆ I a = −7mA
∆ I a = 0mA
∆ I a = 7mA
∆ I a = 14mA
−90
1000 1500 2000 2500 3000
Frequency [Hz]
Figure 5.15: Bode plots of the modeled transfer
functions of a single actuator for various current
offsets∆Ia.
10 1
10 0
Resonance frequency [kHz]
Actuator stiffness [N/mm]
Motor constant [N/A]
10
−15 −10 −5 0 5 10 15
−1
DC current offset ∆I [mA]
a
Figure 5.17: The modeled actuator stiffness ca,
motor constant Ka and resonance frequency fe
as a function of the DC current offsetI ′ a.
From the measured FRFs it is also possible to estimate the viscous damping, actuator stiffness
and motor constant. This step requires the linearized model expressed in (5.24) to be
adapted from voltage to current input, as used in the test setup. When the current Ia is
prescribed, the first state (Ia) of the system vanishes and the system is determined by the
equations for the acceleration ¨za and the velocity ˙za. This yields the following, second
order state-space system:
˙za 0 1
=
¨za −ca(z ′ a)/mac −ba/mac
za
˙za
+
0
Ka(z ′ a)/mac
Ia(t),

5.2 The single actuator 111
where ˙za is chosen as the system output corresponding to the type of laser vibrometer measurement
used. This system can be rewritten into transfer function form as:
HI(s,z ′ Ka(z
a ) =
′ a )s
macs2 +bs+ca(z ′ a )
(5.28)
and has an undamped mechanical resonance frequency fe that depends on the operating
pointz ′ a :
fe(z ′
1
a ) =
2π
ca(z ′ a )
To obtain estimates for Ka(z ′ a ), ba and ca(z ′ a ), first a parametric identification on the FRF
will be performed using Matlab’sÒÚÖÕ×function. However, to be able to derive the
desired properties from the estimated coefficients, two modifications must be made to the
above transfer function. Firstly, note that the lowest term of the numerator polynomial
is zero. To prevent the need for a constraint in the parametric identification procedure, the
numerator is divided bys. This corresponds to time domain integration that must be applied
to the measured velocity signal prior to parametric identification. The second change is due
to the fact that of the four coefficients only three can be uniquely identified and a fourth
must be given. Since the value of mac is well defined by manufacturing tolerances, this is
further assumed to be known as listed in table 5.1. The transfer function whose coefficients
will be estimated can thus be expressed as:
mac
.
Ka(z ′ a )/mac
HI(s,z ′ a ) =
s2 +(ba/mac)s+ca(z ′ a )/mac
. (5.29)
The Matlab functionÒÚÖÕ×is then used to estimate the three unknown coefficients from
whichKa(z ′ a ),ba andca(z ′ a ) are determined. The obtained values are plotted together with
the damped resonance frequency corresponding to the poles of (5.29) in figure 5.16.
The following observations can be made by comparing the values obtained from the model
in figure 5.17 and from the measurements in figure 5.16:
• the measured resonance frequency is higher than modeled and since the moving mass
is known, this implies that the actuator stiffness must be higher,
• the stiffness and resonance frequency decrease with an increased axial air gap,
• the measured motor constant is higher than modeled,
• the measured viscous damping depends on the position of the core.
To explain the differences found, a sensitivity analysis of the variable reluctance actuator is
performed.
5.2.6 Sensitivity analysis
Figure 5.18 shows the sensitivity of the resonance frequencyfe, actuator stiffnessca, motor
constant Ka and inductance La of the actuator w.r.t. the height h and the stiffness coefficients
C1 and C2. This figure is obtained by evaluating the expressions for fe, Ka, ca and
La derived in the previous subsections while varying a single actuator property and keeping
all others at their nominal values as listed in table 5.1. These values are marked by the thick,
5

5
112 5 The variable reluctance actuator
10 0
10 −1
10 −2
L a [mH]
f e [kHz]
c a [kN/m]
K a [N/A]
z 0 [mm]
180 200 220 240 260 280 300
Height h [µm]
−0.1 0 0.1
Stiffness coëficient C 1 [−]
0.005 0.01 0.015 0.02 0.025
Nonlinear stiffness coëficient C 2 [−]
Figure 5.18: Sensitivity of the resonance frequency fe, resulting stiffness ca, motor constant Ka and
inductance La of the actuator w.r.t. the height h and the stiffness constants C1 and C2.
The thick, dashed vertical line represents the nominal value of the parameter, as listed
in table 5.1.
dashed vertical lines. The results include the effect of the parameters on the initial air gap
z0 on which all expressions implicitly depend. From figure 5.18 the following remarks are
made:
The heighth.
An increase of the height h causes an increase of the initial air gap and a working
point with less stiffness and lower motor constant.
The stiffness coefficientC1.
A change in the linear stiffness coefficientC1 of the membrane suspension will result
in a change in initial air gap z0. The magnetic force depends on the reluctance of
the magnetic circuit and therefore on the position of the moving core. In the equilibrium
position, the spring force is equal to this magnetic force. The spring force is
proportional to the deflection and the linear and nonlinear stiffness of the membrane
suspension. If the linear stiffness coefficient decreases, the initial air gap z0 will decrease
and the contribution of the nonlinear stiffness on the actuator stiffness will
increase. This effect is attenuated by the stiffness due to the magnetic circuit which
also increases for smaller air gaps. However, this effect is linear with the decrease air

5.2 The single actuator 113
gap whereas the effect on the mechanical spring stiffness is quadratic. The net effect
is an increase in actuator stiffness and resonance frequency.
For a smaller air gap the magnetic reluctance drops, causing the magnetic flux, hence
the forceFm and the motor constantKa to increase.
The stiffness coefficientC2.
A decrease in the nonlinear stiffness coefficientC2 of the membrane suspension will
also result in a smaller initial air gap. In analogy with the linear stiffness coefficient
C1 this leads to a larger deflection to maintain force equilibrium, but this deflection
increase is accompanied by a lower spring stiffness and will therefore – in contradiction
to the stiffness coefficientC1 – result in a lower overall stiffness.
Again with the smaller air gap the magnetic reluctance drops and the motor constant
increases.
Figure 5.19 shows the sensitivity of the resonance frequencyfe, actuator stiffnessca, motor
constantKa and inductanceLa w.r.t. the leakage flux reluctancesℜflc andℜflm of the coil
and the PM respectively and the radial air gap reluctanceℜga. The figure was obtained the
same way as figure 5.18 and the thick, dashed vertical lines represent the parameter values
listed in table 5.1. From figure 5.19 the following is observed:
The coil leakage flux reluctanceℜflc.
The leakage flux reluctance of the coil has no significant effect on the initial air gap
z0, actuator stiffnessca or resonance frequencyfe. As the reluctanceℜflc of the coil
leakage flux decreases to the order of the reluctance ℜbc of the coil core, the motor
constant becomes affected. The major part of the flux generated by the coil will then
flow into the leakage flux path.
The coil inductanceLa decreases for increasingℜflc, since this inductance is proportional
toN 2 / ˜ ℜ(za) and ˜ ℜ(za) is proportional toℜflc.
The PM leakage flux reluctanceℜflm.
If the reluctanceℜflm of the PM leakage flux is decreased, the attraction force on the
ferromagnetic core decreases and the initial air gap will be larger. As illustrated by
figure 5.8, a lower equilibrium force will result in a lower stiffness of the membrane
suspension and resonance frequency. The motor constant Ka decreases when ℜflm
decreases since the equilibrium force decreases and the air gap increases. As long as
ℜflm does not significantly affect the total reluctance experienced by the coil, there
is little change in coil inductance.
The radial air gap reluctanceRga.
The reluctance of the radial air gap forms a significant part of the total reluctance
in the flux path through the axial air gap (φ1). This explains its significant effect of
motor constant and initial air gapz0. When the total reluctance decreases, the flux, the
force and the motor constant increase. The decrease of the axial air gap z0 explains
the increase in actuator stiffness and resonance frequency.
Figure 5.20 shows the sensitivity of the resonance frequencyfe, actuator stiffnessca, motor
constantKa and inductanceLa of the actuator w.r.t. the axial air gap areaAga, the coercivity
of the PM Hcm and the PM thickness lm. The figure was obtained the same way as figure
5

5
114 5 The variable reluctance actuator
10 0
10 −1
10 −2
L a [mH]
f e [kHz]
c a [kN/m]
K a [N/A]
z 0 [mm]
10 7
10 8
10 9
Coil leak flux reluctance R [1/H]
flc
10 7
10 8
10 9
10
Magnet leak flux reluctance R [1/H]
flm 7
10 8
10 9
Radial air gap reluctance R [1/H]
ga
Figure 5.19: Sensitivity of the resonance frequency fe, resulting stiffness ca, motor constant Ka and
inductance La of the actuator w.r.t. the leakage flux reluctances of the coil ℜflc and
the PM ℜflm and the radial air gap reluctance ℜga. The thick, dashed vertical line
represents the parameter value listed in table 5.1.
5.18 and the thick, dashed vertical lines represent the parameter values listed in table 5.1.
From figure 5.20 the following is observed:
The axial air gap areaAga.
Observe that the area Aga is present in the expression for the magnetic force Fm in
(5.12) and thus directly influences the initial air gapz0. However,Aga also affects the
actuator properties indirectly via the reluctanceℜga. This reluctance enters quadratically
in (5.12) and becomes the dominant term for small values ofAga. This leads to
an increased magnetic force and a reduction of the nominal air gap width that explains
the minimum in the graph for z0. The initial air gap also affects the motor constant,
leading to a maximum in the graph for Ka that corresponds to the minimum for the
air gap z0. The actuator stiffness decreases as z0 increases for small values of Aga.
For larger values of Aga, the stiffness ca is dominated by the nonlinear mechanical
spring stiffness that decreases as the air gap z0 increases. The decreased negative
magnetic stiffness will not compensate for the mechanical spring stiffness reduction
and an overall decrease in stiffness results.
The magnetic field intensityHcm of the PM.

5.2 The single actuator 115
10 0
10 −1
10 −2
L a [mH]
f e [kHz]
c a [kN/m]
K a [N/A]
z 0 [mm]
0.5 1 1.5
Axial air gap area A ga [mm 2 ]
−650 −600 −550 −500 −450
PM coercive force H cm [kA/m]
250 300 350
PM thickness l m [µm]
Figure 5.20: Sensitivity of the resonance frequency fe, resulting stiffness ca, motor constant Ka and
inductance La of the actuator w.r.t. the axial air gap radius rga, and the coercivity of
the PM Hcm and thickness lm of the PM. The thick, dashed vertical line represents the
parameter value listed in table 5.1.
If the coercive force of the PM is increased the magnetic force increases. This increase
leads to a smaller air gap with a higher actuator stiffness, resonance and motor
constant.
The thicknesslm of the PM.
A thickness increase of the PM has the same effect as a coercive force increase.
Besides sensitivity of the actuator properties fe, ca, Ka and La on the parameters h, C1,
C2, ℜflc, ℜflm, ℜga, Aga, Hcm and lm, it is relevant to know what causes the possible
differences in these parameters and how to minimize them. First of all, the predictability of
the actuator properties is increased by the measurements performed to identify the coercivity
Hcm of the PMs and the stiffness coefficients C1 and C2 of the membrane suspension.
Besides this, the manufacturing tolerances and assembly tolerances play an important role:
Manufacturing tolerances.
All dimensions of the actuator are subject to manufacturing tolerances. Manufacturing
tolerances are typically in the order of tens of µm. These dimensions directly
determine all modeled magnetic reluctances, the mechanical stiffness coefficientsC1
and C2 and the moving mass mac. Besides dimensions, the magnetic permeability
5

5
116 5 The variable reluctance actuator
of the ARMCO can be influenced by stresses caused by the manufacturing process.
This is likely to play a role for the reluctances of the baseplate, the moving core and
the radial air gap. As a result, all actuator properties will be affected and vary from
actuator to actuator.
Assembly tolerances.
In addition to tolerances on the manufactured parts themselves, the design dimensions
are affected by the assembly process. For instance, the thickness of the glue layers between
(1) the PM and the coil insert and (2) between the baseplate and the membrane
suspension and (3) between the moving core and the suspension membrane lead to
a variation of the height h. Another example is the reluctance of the radial air gap,
which is affected by the in-plane alignment of the moving core w.r.t. the pole shoes.
Tolerances on the assembly process used are expected to be typically in the order of
ten µm. In section 5.3 it will be shown how – by design and assembly – the effect of
manufacturing and assembly tolerances on the actuator behavior is minimized.
In the next section, the results from the modeled actuator (figure 5.17) and the measured
actuator (figure 5.16) will be compared and analyzed based on insights of the sensitivity
analysis. newpage
5.2.7 Lessons learned
Actuator stiffness ca and resonance frequencyfe
The actuator stiffness ca is directly coupled to the mechanical resonance frequency
fe via the mass mac. The stiffness values derived from the measurements are higher
than expected from the model. With the use of figures 5.18 and 5.19 it is shown that
the stiffness ca varies significantly for all considered parameters except for the coil
leakage flux reluctance. This is caused mainly by its dependence on the initial air
gap z0, which is affected by all parameters. Note from figure 5.9 that in general the
stiffness increases asz0 decreases. Variation of parameters that lead to an increase in
z0 will therefore lead to a decrease inca.
The same effect is observed in the variation w.r.t. the operating point. In accordance
to the model, the stiffness is found to change with the deflectionza (orIa): it increases
for negativeza and decreases for positiveza.
Motor constantKa
The measured value (≈ 0.28N/A) for the motor constantKa is higher than the original
design value of≈0.2N/A. As can be observed in figures 5.18 and 5.19, the motor
constant shows considerable variation w.r.t. all analyzed parameters except the coil
leakage flux reluctance ℜflc. Most noticeable is the strong dependence of the radial
air gap reluctance ℜga. A higher motor constant can therefore be partially attributed
to a lower reluctance of the radial air gap.
Finally, note from figure 5.20 that there exists a value for the axial air gap area Aga
where the motor constant has a maximum. This may be exploited in future designs to
obtain a higher power efficiency.
InductanceLa
As can be observed in (5.25), the inductance La of the actuator is a function of the

5.3 The actuator module 117
number of turns N in the coil and the reluctances in the magnetic circuit. Consequently,
only parameters that have a significant effect on the total reluctance will
cause a significant change in La. The dominant reluctances are ℜga(za) of the axial
air gap andℜgr of the radial air gap andℜm of the PM.
Dampingba
In the model, the damping is considered a constant, but the measurement results in
figure 5.16 indicate that the damping varies with the deflection of the moving core.
A possible explanation for this effect is that a so-called squeeze film exists between
the PM and the moving core. When the core moves, air is either expelled from or
compressed between the two objects. Viscosity hampers the flow of air, which leads
to both spring and damper behavior that depends on the distance between the two
objects and the relevant time scale (i.e. frequency of motion) [114].
Although differences between the model and measurements exist and the analysis results
indicate that improvement of the motor constant in particular is well possible, the results are
good enough to proceed with design and integration of actuators in grids.
The design and realization of these actuator modules will be introduced in the next section.
Measurement results, including the variation and spread in actuator properties, on all
actuators of seven prototype grids will be presented in section 5.3.1.
5.3 The actuator module
In the transformation of the single actuator design into a grid actuator module, the philosophy
is to design in layers that extend over many actuators and not to make many individual
actuators that need to be placed and aligned individually. This reduces the number of parts
and the complexity of assembly and improves the uniformity of the actuator properties.
A hexagonal actuator layout is chosen since this gives the highest actuator areal density.
The grids are also given a hexagonal shape to accommodate the assembly of large DMs
from many actuator grids. The grid layout consists of a central actuator surrounded by a
number of hexagonal ’rings’ of actuators. This approach results in a total of 7, 19, 37, 61,
91 or 127 actuators for 1, 2, 3, 4, 5 or 6 rings respectively. For the prototype grids realized,
four rings, corresponding to 61 actuators, are chosen. In this choice, several practical issues
were taken into account, such as the size of the baseplate and its corresponding resonance
frequency. A larger grid would require more thickness, or additional support points to avoid
internal resonances. Further, the actuator coils are connected through flex foils. These will
become more difficult to manufacture as the number of actuators per grid increases. When
the line pattern is made on a single layer there is not enough area for more connections.
Finally, the number 61 is close to a division factor of2 6 = 64, which is likely to be used in
digital electronics for drivers and communication. The actuator grid is shown in figures
5.21 and 5.22. Figure 5.23 shows the exploded view of the actuator grid. The main parts
shown in the exploded view are summarized and will be discussed in detail:
The baseplate
The baseplate serves a the flux carrier for the magnetic circuits for 61 actuators and
is made from from ARMCO . The baseplates are cut from bar and their front and
5

5
118 5 The variable reluctance actuator
Figure 5.21: The standard actuator module with
61 actuators seen from the front.
Figure 5.22: The standard actuator module with
61 actuators seen from the back.
back surfaces are made plan parallel. The serrated circumference is made such that
the actuator grids can be placed adjacent with a 0.3mm gap. The holes in the backside
are made by milling, the circumference and pole shoe contours are made with wire
Electrical Discharge Machining (EDM).
The membrane suspension with moving cores
The membrane suspension is made by laser cutting. The same sheet of rolled Havar
material is used for all actuator grids in the same orientation to obtain uniformity for
the spring characteristic. The moving cores are laser cut and made from a 0.3mm
thick ARMCO sheet.
Inserts with the SmCo5 PMs and coils
The inserts complete the magnetic circuit. The inserts are produced on a Computer
Numerical Control (CNC) lathe. After measurement and selection of the PMs described
in [174], the PMs are glued on the inserts with Araldite 2020. The coils,
made of 50µm copper wire with 500 turns, are fabricated separately from the insert
and have pre-leaded ends. The electrical resistance of each coil is measured before
placement. The inner radius of the coil is made slightly larger than the insert core
to avoid damage to the electrical isolation when placed. The bottom of the inserts
contains a hole and a slot to provide an axial feed through for the coil wires.
The flex foil
A flex foil is designed to connect the coil wires to the driver Printed Circuit Board
(PCB). This flex foil design is shown in figure 5.22. Each of the three branches of
the flex foil connects to a double row, 0.3mm pitch, connector. The central hexagonal
part of the flex foil holds∅2.5mm holes through which the coil wires emerge. At the
circumference of the holes, the two wires of that coil are soldered each on a copper
pad. After soldering, a droplet of silicon glue is placed to encapsulate the fragile
coil wires. A strain relief (figure 5.22) is added to avoid damage to the soldered
connections.
The actuator grid supports

5.3 The actuator module 119
M1x6 (3x)
Membrane suspension
Moving cores (61x)
Baseplate
Coils (61x)
SmCo Magnets (61x)
Inserts (61x)
Flex foil
A-frames (3x)
Strain relief and clamps (3x)
Figure 5.23: An exploded view of the actuator grid shown in figures 5.21 and 5.22.
5

5
120 5 The variable reluctance actuator
Figure 5.24: Bode plot of
the measured FRFs s · H(za)
from a single actuator in a
grid, together with the fitted
model and the original analytic
model. The measured FRF of
the single actuator as shown in
figure 5.14 is also plotted for
comparison.
Magnitude [m/s/A]
Phase [deg]
10 3
10 2
10 1
10 0
10 −1
10 −2
180
90
0
−90
−180
10 1
Grid actuator measurement
Fitted model
Original model
Single actuator measurement
10 2
Frequency [Hz]
The actuator grid is connected with three A-frames to its support structure. Each
A-frames is connected with one point to the baseplate and with two points to the
support structure to avoid moments enforced on the baseplate. The A-frames are
connected with M1 bolts. When the actuator grid is placed facedown on the table,
with the membrane suspensions facing down, the bolt heads support the actuator grid
and avoid damage to the suspension systems.
Details on the manufacturing and assembly procedures of the actuator grids can be found in
[174].
5.3.1 Measurement results
A batch of seven grids was realized using the procedure described in [174]. The only difference
in nominal dimensions with respect to the single actuator is an additional 25µm for the
height h. Although this is known to have a negative effect on the motor constant, actuator
stiffness and resonance frequency, more margin is hereby built in for manufacturing errors
and the risk on ferromagnetic cores to snap down on the PMs is avoided. The reduced motor
constant, actuator stiffness and resonance frequency are estimated from figure 5.18 as
Ka=0.17 N/A, ca=550N/m andfe=1980 Hz.
For each grid all actuators are measured with the same experimental setup as shown in figure
5.13. Figure 5.24 shows a typical transfer function with a parametric model fit of one of
the actuators in the grid. As a reference, the figure also shows the nominal transfer function
derived from the analytical model depicted previously in figure 5.15. Note that because
of the velocity measurement of the experimental setup, the response for frequencies below
the first resonance shows a +1 slope and a -1 slope above the resonance. Besides the first
resonance frequency around 2kHz, a second resonance is visible at approximately 5kHz.
This is the tip/tilt mode of the ferromagnetic core in its membrane suspension that can only
be observed when the laser vibrometer is not perfectly aligned to the center of the moving
core.
10 3
10 4

5.3 The actuator module 121
Table 5.2: Average values and standard deviations of the actuator properties measured over all grid
actuators.
Property Ka ca ba fe
Average 0.12N/A 471N/m 0.36mNs/m 1.83kHz
Std.dev. 0.02N/A 48N/m 0.10mNs/m 95Hz
Estimates for the actuator stiffness ca, resonance frequency fe, motor constant Ka and
viscous damping ba are obtained from the measurement data using the same procedure as
described in section 5.2.5. The average and standard deviation values are listed in table
5.2 and more detailed results are shown in figures 5.25, 5.26, 5.27 and 5.28. The values in
the figures are sorted to provide insight into the statistical spread and differences in median
values between grids. The mean actuator stiffness and resonance frequency are 471N/m and
1.83kHz respectively, which is lower than expected. In figure 5.25 it is shown that the spread
in stiffness values within a grid is similar for all grids, but that the mean differs from grid to
grid. This can be caused by manufacturing and assembly variations that directly affect all
actuators in a module, such as baseplate or suspension membrane thickness variations.
Figure 5.27 shows the values of the motor constants, which are lower than expected. The
analytic model predictedKa=0.17N/A, whereas 0.12N/Ais measured. A possible explanation
for the measurement results is that the leakage flux reluctances for the PM and coil are
smaller in the actuator module baseplate than for the single actuator. The few high values
for damping are explained by rests of glue in between the moving core and baseplate.
5.3.2 Power dissipation
To analyze the expected power dissipation of the actuator, only its static response is considered
and assumed to be linear. The validation measurements have shown this to be an
Actuator stiffness c a [N/m]
650
600
550
500
450
400
350
300
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 5.25: The identified actuator stiffnessca,
for mac =3.6mg sorted for each measured actuator
grid separately.
Resonance frequency F e [Hz]
2100
2000
1900
1800
1700
1600
1500
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 5.26: The identified actuator resonance
fe, for mac =3.6mg sorted for each measured
actuator grid separately.
5

5
122 5 The variable reluctance actuator
Motor constant K a [N/A]
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 5.27: The fitted motor constants Ka, for
mac =3.6mg sorted for each measured actuator
grid separately.
Viscous damping b a [mNs/m]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 5.28: The fitted damping ba, for
mac =3.6mg sorted for each measured actuator
grid separately.
accurate approximation at least up to approximately 1600Hz, where effects such as viscous
damping and eddy currents play a negligible role.
The power dissipated by the currentIa through the actuator coil with resistance Ra can be
expressed as:
Pa = I 2 aRa
Fa
=
Ka
2
Ra,
where for the second step the linearized case Fa = KaIa was used. In chapter 2 the
expected RMS actuator force was derived as 1mN, which was based on an actuator stiffness
ca of 500N/m. Since the stiffness of the realized actuator was found to be very close to this
value, the 1mN RMS force remains a valid basis to evaluate power dissipation. For 1mN
RMS actuator force and a measured average motor constant Ka of 0.12N/A this yields an
RMS actuator current Ia of approximately 8mA. Via the coil resistance Ra ≈ 39Ω, this
corresponds to a power dissipation of approximately 3mW per actuator. This means that the
realized actuators meet the specification in chapter 2 that the power dissipation must be in
the order of mW’s.
5.4 Conclusions
The application of electromagnetic actuators for adaptive deformable mirrors has several
advantages. Electromagnetic actuators can be designed with a limited stiffness, such that
failure of an actuator will cause no hard point in the reflective surface and thus a small optical
degradation compared to e.g. stiff piezo-electric elements. Other advantages over the
latter type of actuators at the relatively low cost, low driving voltages and negligible hysteresis
and drift.
Variable reluctance type actuators were primarily chosen because of their high efficiency
and low moving mass. A nonlinear mathematical model of the actuator was derived describing
both its static and dynamic behavior based on equations from the magnetic, mechanic
and electric domains. The model was linearized, leading to expressions for the actuator

5.5 Recommendations 123
transfer function and linear electromechanical properties such as motor constant, coil inductance,
actuator stiffness and resonance frequency.
Single actuator prototypes were realized and transfer functions were estimated from measurement
data, based on white noise current excitation. This was done for various operating
points by adding static current offsets to the excitation signal, which showed that the effect
of the nonlinearities is indeed small. The resonance frequency and the DC-gain of the
transfer functions showed only marginal variation with respect to the operating point. This
means that a control system will be able to use an LTI control law without sacrificing performance.
The measured nominal resonance frequency is higher than modeled: 2.1kHz instead
of 2kHz, corresponding to an actuator stiffness of≈680N/m. The measured motor constant
is also higher than modeled 0.27N/A instead of 0.2N/A.
Due to the satisfactory measurement results, the design for single actuators was applied with
little modification to the design of standard hexagonal modules with 61 actuators. Only the
nominal heighth of the moving core above the PM was increased by 25µm to limit the risk
of the ferromagnetic cores snapping down onto the PM. Based on the results of a sensitivity
analysis, this modification was expected to reduce the motor constant, actuator stiffness and
resonance frequency toKa=0.17N/A,ca=550N/m andfe=1980Hz respectively.
In this sensitivity analysis the effect was determined of variation in a number of model parameters
on several actuator properties (i.e. stiffness, resonance frequency, motor constant
and inductance) as derived from the mathematical model. In particular, this indicated a
strong influence of the radial air gap reluctance on the motor constant and actuator stiffness,
which can explain the deviation between the modeled and measured properties.
Seven actuator module prototypes were made of which all actuators were measured with
the same setup as the single actuator prototypes. All actuators were found to be functional,
indicating that the manufacturing and assembly process is reliable. From transfer function
measurements, the motor constant, actuator stiffness and resonance frequency were identified.
These properties showed slight deviations from the values derived from the model,
but the statistical spread for the properties was small, stressing the reliability of the manufacturing
and assembly process. The mean actuator stiffness and resonance frequency were
471N/m and 1.83kHz respectively, which are very close to their design values of 500N/m
and 1885Hz. The value derived from the model for the motor constant Ka was 0.17N/A,
whereas on average only 0.12N/A was measured. This may be the result of leakage flux
reluctances in the baseplate being lower – and thus the leakage flux being larger – than for
the single actuator.
Despite the motor constants of the actuators realized being lower than expected, the RMS
power dissipation of the actuators is still low during operation and expected to be≈3mW.
5.5 Recommendations
In a redesign, the large influence of the radial air gap reluctance can be used to increase
the motor constant and reduce power dissipation. A factor of two reluctance reduction will
increase the motor constant to 0.37N/A and increase the actuator stiffness to 750N/m. This
can be realized by a smaller gap width or by a larger gap area.
A reduction of the axial air gap area will lead to a further increase in motor constant. With
these changes, an improvement by a factor four is feasible. Since power dissipation is
5

5
124 5 The variable reluctance actuator
inversely proportional to the squared motor constant, a power dissipation reduction by a
factor 16 is achieved. A convenient side effect is the increased electronic damping of the
mechanical resonance frequency. This is illustrated by the Bode plots in figure 5.29, in
which he magnitude peak of the two systems is almost equal, whereas the DC-gain of the
improved system is four times higher. The relative damping of the resonance has increased
approximately 2.6 times, making it less limiting for the controller performance.
Magnitude [m/V]
Phase [deg]
10 −4
10 −5
10 −6
0
−90
−180
−270
10 2
K a = 0.19N/A
K a = 0.76N/A
Frequency [Hz]
Figure 5.29: Bode plot of the voltage to position actuator model with the currently
measured and four times higher motor constant. Note the
increased relative damping of the resonance mode.
10 3

ÔØÖ×Ü
ÐØÖÓÒ×
This chapter describes the design and realization of the actuator driver electronics
and communication system. Pulse Width Modulation (PWM) based
voltage drivers are selected and implemented in three Field Programmable Gate
Arrays (FPGAs) for 61 actuators (one actuator module). A high base frequency
of 61kHz and an additional analog 2 nd order low-pass filter is used to reduce
the actuator position response ripple due to harmonics of the PWM signal to
less than a quarter of the Least Significant Bit (LSB) of the setpoint. The driver
electronics of each actuator module are contained on a single Printed Circuit
Board (PCB), which is placed behind the actuator module to preserve the modular
concept. The FPGAs receive the voltage setpoints via an Low Voltage
Differential Signalling (LVDS) communication link from the control system.
As no commercial LVDS interface board was available for a standard PC, an
ethernet-to-LVDS communications bridge is developed that translates ethernet
packages into LVDS packages and vice versa. A single flat-cable connects up
to 32 driver PCBs to this communications bridge.
To determine the PWM frequency and evaluate the dynamic effects of the electronics,
the actuator model from chapter 5 is extended with models for the
communication and driver electronics. The communication is modeled as a
pure delay and the driver electronics as an ideal voltage source with a linear,
analog 2 nd order low-pass filter. The dynamic model is validated using white
noise identification measurements on the actuator system. The system is evaluated
on control aspects, showing the dependance of achievable bandwidth on
the sampling frequency. Finally, the power dissipation of the FPGAs is evaluated
and found to exceed the power dissipation in the actuator coils. Concepts
are proposed and analyzed to reduce power dissipation in the digital electronics.
Joint work with Roger Hamelinck [PhD]
125

6
126 6 Electronics
6.1 Introduction
In this chapter, the design and realization of the electronics required to generate the currents
through the coils of the actuators is described. The actuators were described in chapter 5
The electronics consist of two parts: a communication system and driver electronics. The
communication system transmits the current value calculated by the control system to the
driver electronics, which generate the actual currents. The main difficulty of the driver electronics
is the required number of channels combined with the desired high power efficiency.
To determine the values of certain design properties of the electronics and evaluate the effects
of the dynamics of the electronics on the Adaptive Optics (AO) system performance,
a first principles model is derived that is combined with the actuator model from chapter 5.
After the realization of the electronics, the behavior of the full actuator system is analyzed
by comparing it to the model.
In the next two sections design concepts for the driver and communication electronics will
be presented, followed by their implementation and realization in section 6.4. The first
principles model is derived in section 6.5 and compared to measurement results in section
6.7. Since the driver electronics and the actuators are located close to the Deformable
Mirrors (DMs) reflective surface, power dissipation is an important design driver. Therefore,
in section 6.8 the power dissipation of the prototype actuator system is analyzed and
compared to the design requirements. Finally, the main conclusions will be formulated.
6.2 Driver electronics
The driver electronics provide currents through the coils of the variable reluctance actuators.
In the actuator, the current converts into forces that deform the mirror facesheet. In this
section the requirements, design concepts and implementation of the driver electronics will
be discussed.
6.2.1 Requirements
In section 2.3 the required actuator positioning resolution is derived as 5nm. From this
value, the worst-case required range and resolution will be derived of the currents and voltages
that must be provided by the driver electronics. Since the resolution requirement is in
terms of displacement, the mechanical stiffness, motor constant and circuit resistance must
be considered. The mechanical stiffness depends on the resulting deformation of the reflective
surface. The minimal stiffness is used to determine the required resolutions and the
maximal stiffness is used to determine the required ranges. Inertia forces and other dynamic
effects are neglected. The stiffness is minimal when all actuators have the same displacement
and the reflective membrane does not deform. The worst case force resolution for the
actuator stiffnessca =583N/m can thus be derived as Fres = 2.9µN. Via a motor constant
Ka of 0.19N/A (figure 5.11) and an actuator coil resistance Ra of 39.0Ω (table 5.1), this
leads to required current and voltage resolutions of 15µA and 0.60mV respectively.
The required current and voltage range follows from the maximum actuator force, motor
constant and coil resistance. An estimate for the actuator force is obtained by summing the
forces required to overcome the internal mechanical stiffness of the actuator and to over-

6.2 Driver electronics 127
come the stiffness of the facesheet:
Fa = caza +cfzia
where the mirror facesheet stiffness cf is ≈6kN/m ([174]), the actuator stroke za = 10µm
and the inter-actuator stroke zia = 0.36µm as derived in chapter 2. This results in a
range of the required force of ±8mN. Via the motor constant and actuator coil resistance
previously used, this leads to the required current and voltage range of±42mA and±1.6V
respectively.
This implies that the dynamic range (the total range divided by the resolution) of the driver
electronics must be at least 2 · 8 · 10 −3 /(2.9 · 10 −6 ) ≈ 5.5 · 10 3 . For a digital driver
system, this would require at least 13 bits of accuracy. This is a minimum and does not
provide any margin for a stiffer facesheet, initial flattening and alignment of the mirror and
variation in motor constant between actuators. To account for this and acknowledge that
the digital implementation in 16 bits is likely to be more efficient, this is the starting point
in the design. The consequences for the driver electronics from this requirement will be
elaborated in section 6.2.2.
The DM actuators are designed to have a high efficiency. In combination with the low
actuator force requirement, this minimizes power dissipation. As a result no active cooling is
needed and vibrations introduced by such systems are avoided. Through natural convection,
heat is convected without adding significantly to the wavefront disturbances that are to be
corrected. By placing the driver electronics near the electromagnetic actuators the number
of wires to the DM can be limited and spacious connectors avoided. With the short length
of the wires, sensitivity to environmental loads (e.g. lead breakage, magnetic fields from
nearby power sources, etc.) is reduced. However, it also means that the power dissipation
of the driver electronics must be in the order of mW’s, similar to the actuators.
Finally, the electronics should be compact in size, have low cost/actuator and preferably
replaceable. Since the actuator grid is made extendable by means of the standard modules
that hold 61 actuators each, the same should hold for the electronics. All drivers for a single
actuator module should therefore be placed on a single Printed Circuit Board (PCB).
6.2.2 Concepts
For the driver electronics concept, two categories are distinguished: current and voltage
sources. A current driver converts the digital setpoint from the control computer into a controlled
current through the electromagnetic actuator. A voltage driver converts this setpoint
into a voltage over the actuator clamps, upon which the circuit resistance determines the
current. The current may therefore vary due to dynamics or time variance in the electronic
circuit. TheLa/Ra time constant is approximately 2.93mH/39.0Ω ≈ 75µs (section 5.2.5),
which is small compared to the intended sampling time of 1ms. Therefore, an applied voltage
will result in a current without significant delay.
The DC-gain of the actuator system including driver electronics depends on the actuator’s
motor constant Ka, stiffness ca and – in the case of a voltage source – the resistance R of
the actuator circuit. All three properties vary from actuator to actuator and vary with temperature,
causing slow gain variations. A current source will compensate for variations in
the resistanceRa, but variations ofKa andca must still be compensated by the AO control
6

6
128 6 Electronics
I1
b1
b15
I15
b16
!b16
actuator
!b16
Figure 6.1: A design based on current sources that can be efficiently
implemented in an ASIC using current mirrors. The
"!" denotes a logicalÒÓØ.
b16
I ref
I out
Load
V out
Figure 6.2: Schematic of a single
current mirror. The reference current
Iref – here determined by a resistor
– is mirrored to the output current
Iout.
system. Therefore, the conceptual advantage of a current driver over a voltage driver is
small, but concrete designs of both driver types will be discussed before a choice is made.
Current mirrors
In practice, a current source regulates its voltage output based on measurement of the actual
current. This feedback can be done with a linear amplifier circuit, but for this application this
approach has several drawbacks. Firstly, the circuit needs two supply voltages to generate
positive and negative currents. Secondly, the required number of components is relatively
high since no standard ASICs are available that compactly house a large number of accurate,
efficient and low power linear amplifiers. Finally, the amplifier obtains its setpoint current
from an analog voltage input that must be generated from the digital value of the control
system by an additional component such as a Digital to Analog Convertor (DAC).
These drawbacks can be circumvented using a design based on current mirrors, which is
schematically represented in figure 6.1. This design holds 15 current mirrors and can be efficiently
implemented in CMOS technology. A single current mirror (figure 6.2) consists of
two parts: in the first part a reference current is generated that is mirrored with a certain ratio
to the second part that includes the load. The reference currents of the current mirrors will
be permanently flowing, whereas the mirrored load currents can be switched according to
the setpoint bits. The physical dimensions of the current mirror’s two transistors determine
the ratios between the reference and load currents. These ratios can be designed to minimize
the permanently flowing reference currents and thus optimize power efficiency. The total
current can be constructed with 15 fixed current mirrors and a sign-switch corresponding
to the Most Significant Bit (MSB) of the setpoint. The sign-switch is obtained using the
full bridge configuration as depicted in figure 6.1, where the four switches are driven by the
MSB of the setpoint and its inverse. This design is relatively simple, has a linear setpoint to
current characteristic and many of these circuits can be implemented in a single Integrated
Circuit (IC). However, it requires an ASIC, which is expensive to design and manufacture.
Its power efficiency is comparable to that of linear amplifiers and dependent on the ratio
between the Root Mean Square (RMS) and Peak To Valley (PTV) currents: the crest factor.
The current sources regulate their output voltage, leading to internal voltage drops and thus

6.2 Driver electronics 129
dissipation. When neglecting internal current paths of the current sources, the total power
consumptionPtot can be expressed in terms of the supply voltageVcc and the desired actuator
currentI asPtot = VccI. The powerPload dissipated in the load with resistanceRload
is equal to Pload = I 2 Rload and can be expressed relative to Ptot as:
Pload
Ptot
= I Rload
Vcc
= I
.
Imax
This means that for low currents, almost all power is dissipated by the current source and
for the maximum currentImax = Vcc/Rload all power is consumed by the load.
For the RMS current of 8mA (section 5.3.2), a load resistance of 39.0Ω and a supply voltage
of 1.6V, this yields a power efficiency of approximately 20%. This low efficiency could be
improved by adapting the supply voltageVcc in accordance with the desired actuator current
by dividing the output range over a number of supply voltages. However, as this leads to an
even more complex design that can only be realized in an ASIC, a Pulse Width Modulation
(PWM) voltage driver with a higher power efficiency is considered as an alternative in the
next section.
Pulse Width Modulation
The absence of a feedback path to regulate a current simplifies the design in comparison
with linear amplifiers although this advantage is limited when compared to the current mirror
design. A regulated voltage source that generates an analog voltage for the actuator from
the digital setpoint value is essentially a DAC. There exist many types of DACs, but the high
accuracy and low power consumption required for this application limit the options. For instance,
the low accuracy of the resistors of the common resistor ladder network DAC limits
the useful accuracy of this type of converter to 8 bits or less.
For high accuracy applications the PWM principle is often used, in which a digital output
is modulated between high and low states to yield a desired average Direct Current (DC)
value of the output voltage. The desired output voltage is translated into a duty cyclerPWM,
which is the time fraction that the digital output is high during a certain time periodTPWM.
This time period forms the base frequencyfPWM = 1/TPWM of the PWM.
The advantage of a PWM based voltage source over the proposed mirror concept is twofold.
Firstly the PWM generators can be implemented in Field Programmable Gate Arrays
(FPGAs), which reduces the number of components and does not require the expensive
and complex design and realization of an ASIC. Moreover, its power efficiency is superior
to the current mirror driver because it has no internal voltage drop that leads to dissipation.
Dissipation is limited to switching losses and indirect losses of the PWM signal generator
and does not significantly depend on the desired output voltage. Finally, PWM design and
implementation is well understood, which limits development risks. A drawback of PWM
is that it outputs a signal with high-frequency components, which causes a corresponding
ripple on the system output. Using the Fourier series expansion shown in appendix C, for a
constant duty cyclerPWM ∈ [0...1] the PWM output voltage ˜ VPWM that modulates between
0 (low) andVcc (high) can be expressed as the following infinite sum of cosines:
˜VPWM(t) = Vcc
rPWM +
∞
n=1
sin(nπrPWM)
nπ
cos(2πnfPWMt)
, (6.1)
6

6
130 6 Electronics
Table 6.1: Properties of industrial communication standards as derived from data sheets of available
driver ICs.
Standard Bandwidth Multi-drop Predefined protocol
USB2 480Mb/s no yes, high overhead
FireWire 800 800Mb/s no yes, high overhead
CAN 1Mb/s yes yes
Gigabit Ethernet 1Gb/s no yes, high overhead
RS-485 (Profibus) 40Mb/s yes no
LVDS 655Mb/s a yes no
a According to LVDS standard as defined in ANSI/TIA/EIA-644-A
whereVcc is the switched supply voltage. The spectrum of this PWM signal thus only contains
power at frequencies2πnfPWM forn = 1,2,...∞. These harmonic frequencies must
be sufficiently attenuated by the dynamics of the driven system, such that the remaining ripple
on the system’s output is within the accuracy margin of a quarter of the Least Significant
Bit (LSB). To achieve this, not only the base frequency fPWM can be suitably chosen, but
also the system’s response to the PWM signal harmonics can be tailored using additional
filters.
As this drawback can thus well be handled, the driver electronics for the DM actuators will
be based on PWM.
6.3 Communication electronics
The communication electronics send the actuator setpoints from the control computer to the
driver electronics. The communication link should have low latency to allow a high control
bandwidth (e.g. low phase lag) and a high reliability and bandwidth to allow a large number
of actuators to be quickly updated. In addition, the power dissipation, flexibility and
costs are relevant. If for example, the number of actuators, the bandwidth or the number of
setpoint bits changes, the communication link and protocol should allow adaptation. Furthermore
a protocol that can be chosen freely and with low overhead is preferred. To limit
development costs, the choice is limited to industrial standards, such as RS-485 (Profibus),
USB2 (Universal Serial Bus), ethernet, LVDS (Low Voltage Differential Signalling), CAN
(Controller Area Network) and FireWire. A few relevant properties of these standards are
listed in table 6.1. When an update rate of 1kHz and 16 bit setpoint values are assumed,
the minimum bandwidth for 5000 actuators is: 1000 · 5000 · 16 = 80Mb/s. With a protocol
overhead of 10% and latency limited to one quarter of the sampling time (250µs), a
minimum bandwidth of (5000·1000·16·1.1/0.250) ≈350Mb/s is obtained. For the prototypes
developed with actuator numbers up to 427 actuators, approximately 30Mb/s would
already suffice, but with future, larger, systems the CAN bus and RS-485 are no option.
Since the driver electronics will be placed on modules and located close to the DM the
power consumption of the transceivers must be as small as possible and for practical reasons
the number of wires leading to the modules should be small. Both arguments suggest

6.4 Implementation and realization 131
Vpwm
Ll Rl
Low-pass filter coil
La
Cl Ra
Ve
Actuator
Rc
VpwmB
Figure 6.3: The analog electronic
circuit, consisting of a
coarse and a fine PWM generator,
an analog low-pass filter
and the actuator.
the use of a multi-drop topology in which one transmitter communicates to many receivers
on the same bus. The modules are given a unique identification code to allow messages
to be passed to specific modules. For such topologies the modules do not require a power
dissipating termination resistor and the number of communication wires is independent of
the number of receiving modules. However, the power efficiency and speed of the communication
link are not only determined by its hardware alone. If the method requires a
specific protocol with a high overhead, both speed and power efficiency are reduced and
flexibility for future upgrades limited. Both arguments favor the development of a custom
protocol. Development costs of such a protocol will be limited, as a master-slave structure
with a small command set will suffice and throughput is more important than guaranteed
transmission.
The Low Voltage Differential Signalling (LVDS) standard was chosen for the serial communication.
In contrast to USB, this allows for a high bandwidth multi-drop topology for
which low-power transceiver ICs are commercially available. Each transceiver dissipates
only 15mW and requires no termination resistor. A custom communication protocol can be
designed that has a small overhead compared to e.g. the USB, FireWire or ethernet protocols.
Two LVDS wire pairs can be used to keep the protocol as simple as possible: one
command line and one return line.
6.4 Implementation and realization
In this section the implementation of the chosen design concepts for the driver and communication
electronics will be discussed.
6.4.1 PWM implementation
For several reasons the PWM voltage drivers will be implemented in an FPGA. Firstly,
because this leads to a compact design with few components because the FPGA can house
many PWM generators. Moreover, no expensive ASIC has to be designed and realized and
it allows modifications to the implementation through a software update.
As derived in section 6.2.1, the driver electronics require a dynamic range of 16 bits.
The PWM driver electronics will be designed such that the ripple magnitude due to the
harmonic component in the PWM signal at the frequencyfPWM is less than a quarter of the
system’s response to the least significant bit for any duty cycle rPWM. Let H(s) denote the
transfer function between the PWM voltage ˜ VPWM and the position za of a single actuator.
6

6
132 6 Electronics
Figure 6.4: H-bridge construction
to allow the PWM
voltage VPWM of the coarse
PWM to be both positive and
negative.
Vcc
a
b
VPWM
Observe from (6.1) that the worst case magnitude of the first harmonic (n = 1) occurs for
rPWM = 0.5 and is equal toVcc/π. The design condition can thus be formulated as:
π |H(2πjfPWM)| < 1Vcc
4216|H(0)|
Vcc
and thus:
|H(2πjfPWM)| < π
218|H(0)|. (6.2)
In section 5.2.4, the frequency response functionH(jω) has been derived for the case that
the PWM output is directly connected to the actuator coil. A Bode plot ofH(jω) is plotted
in figure 6.10, which shows that thefPWM for which (6.2) is satisfied lies above 100kHz.
When implemented in an FPGA, the PWM generator will consist of a 16 bit counter and a
comparator. The counter value is increased by one at every FPGA clock cycle and resets
to zero at the beginning of each PWM time period. The comparator compares the counter
value to a value corresponding to the setpoint. The PWM output is high if the counter is
higher than this value and low otherwise. The counting and thus clock frequency of the
FPGA can be expressed as:
fFPGA = fPWM2 Nb (6.3)
where Nb is the number of bits of the counter. The clock frequency of currently available
FPGAs is limited to approximately 200MHz, which implies that for Nb = 16 the base frequency
fPWM is limited to approximately 3kHz. The dynamic power dissipation of digital
electronics is for most designs linearly correlated with the clock frequency, which is an important
drive to keep the base frequency as low as possible.
To keep fFPGA below 200MHz while implementing 16bit PWM generators, two modifications
are made. Firstly, an analog2 nd order low-pass filter is added to reduce the system
response magnitude at high frequencies and secondly the PWM is split into a fine part consisting
of 5 bits and a course part of 11 bits. For 11 bits, the PWM base frequency can be
increased to approximately 95kHz.
The analog low-pass filter consists of the inductor with inductanceLl and a capacitor with
capacitance Cl (figure 6.3). It is given a bandwidth of 5kHz that is high enough to have
a negligible influence on the behavior of the system up to the mechanical resonance, but
low enough to reduce the required PWM base frequencyfPWM to less than 95kHz. Assume
that above the resonance at 2kHz the magnitude response of the actuator between driving
voltage and position decays a factor 1000 per decade. Further, the response of the analog
low-pass filter decays a factor 100 per decade above 5kHz. The minimum PWM frequency
c
d

6.4 Implementation and realization 133
A
B
Vpwm
0
1
0
1
0
Vcc
−Vcc
〈Vpwm〉 = 0
time
〈Vpwm〉 > 0
Figure 6.5: Comparison of the traditional and BD modulation schemes. The latter is represented by
the black, solid lines and the first by the gray, dashed lines that are shifted slightly to the
top-right for clarity.
fPWM that satisfies (6.2) is then solved from:
3 fPWM
·
2000
A
B
Vpwm
Vcc
0
1
0
1
0
−Vcc
2 fPWM
= 2
5000
18
yielding fPWM ≈ 35kHz. Without the low-pass filter this would be 128kHz, such that this
filter reduces the required PWM base frequency almost a factor 4.
However, this reduced base frequency is only achievable for currently available FPGAs
when the number of bits is less or equal to 12. Therefore, the PWM has been split into
two parts: the 11-bit course PWM provides VPWM whereas the 5-bit fine PWM provides
VPWMB. The latter can re-use the lowest 5 bits of the course PWMs counter and is connected
through an appropriately chosen resistor Rc to one of the actuator clamps (figure 6.3).
Evaluating (6.3) for Nb = 11 bits, the required FPGA clock frequency becomes approximately
125MHz. The number of bits of the PWM is split into unequal parts on purpose, as
the resistance of Rc is in practice inaccurate and causes an output bias that increases with
the magnitude of the fine PWMs highest bit.
To send positive and negative currents through the actuator coil, the PWM must provide
positive and negative voltages. This is achieved with an H-bridge construction as shown in
figure 6.4. This construction has only been applied for the coarse PWM. Due to the limited
range of the fine PWM, the added value of a sign change does not make up for the extra
FPGA pins and PCB connections. The PWM signals control the switchesa,b,c, anddsuch
that current flows either via a and d or via b and c. Care must be taken that both a andbas
well as c and d are never closed at the same time as this forms a short-circuit. By defining
the PWM output ’low’ as the closing of a and d and the PWM output ’high’ as the closing
ofb andc, the effective voltage over the actuator coil can be varied between−Vcc andVcc.
However, in practice the mean actuator voltage will be approximately zero, which for this
approach corresponds to a duty cycle of 50% (figure 6.5). This means that the voltages over
the coils of the actuator and the low-pass filter will continuously vary, resulting in small,
but significant dissipative currents. These can be prevented by the use of a BD modulation
time
6

6
134 6 Electronics
Clock
LVDS
Actuator controller
200 MHz
125 MHz
LVDS
converters
Master
FPGA
125 MHz clock
data-bus
address-bus
control lines
Slave
FPGA 1
Slave
FPGA 2
Figure 6.6: Actuator controller architecture.
H-bridges
.
.
Actuators
1-31
scheme.
For the BD modulation scheme, the a −d and b − c switches are driven by two different,
but related PWM signals A and B, whereas otherwise these would be complementary (i.e.
A=ÒÓØB). For a zero effective voltage, both signals have a duty cycle of 50% and are fully
in-phase (figure 6.5). In this situation neither thea−d path nor theb−c path will ever conduct
and cause dissipation. If a positive voltage is desired then the period of PWM signal A
is increased whereas that of B is decreased and for a negative voltage vice versa. For both
modulation methods, the signals A and B drive the switches according to a = A, b =ÒÓØA,
c = B and d =ÒÓØB. Therefore, switches b and d have always the opposite state of a and c
respectively to prevent short-circuits.
A second advantage of the BD modulation method is that the output voltage swing is only
Vcc, whereas for the traditional modulation this is2Vcc (figure 6.5). Consequently, the magnitudes
of the harmonics in the frequency spectrum of the traditionally modulated signal are
twice as high as suggested in (6.1).
6.4.2 FPGA implementation
In section 6.2.1 it is explained that for modularity of the DM system, the driver electronics
should be made in PCB modules containing 61 driver circuits and connect to a single
actuator module. To implement the 61 PWM generators and the LVDS communication
protocol, three Altera Cyclone II (EP2C8) FPGAs are present on each electronics module.
One master FPGA handles the LVDS protocol and two identical slaves implement 32 PWM
generators each. The functionality is not realized in a single FPGA to limit the risk of the
number of available logic cells or electrical connections being insufficient to implement the
required functionality. Due to the two-level PWM solution discussed in section 6.2.2, each
actuator requires FPGA connections for each of the four H-bridge switches and one for the
fine PWM signal. This results in 5 connections in total per actuator and thus 305 connections
for 61 actuators.
The master FPGA decodes the LVDS signal using 5-times over-sampling (200MHz) and
interprets the commands. If required, information is sent to or requested from the slaves
via a 16-bit parallel data bus. The slaves each have one counter that is increased with the
frequency of an externally supplied 125MHz clock. There the 11-bit counter signal is fed
to 32 comparator circuits that generate the PWM signals. These circuits are divided into
.
32-61
.

6.4 Implementation and realization 135
Figure 6.7: PCB (top and bottom) containing the analog filter electronics for 61 actuators. (left) PCB
(top and bottom) containing the master FPGA, DC-DC convertors and LVDS drivers.
(right)
Figure 6.8: Top-view of the encased LVDS communications
bridge.
Figure 6.9: Seven PCBs connected to the LVDS
bridge via the multi-drop flat-cable. The bridge
is connected to the laptop via ethernet.
four blocks of eight circuits to prevent a large fan-out of connection wires that bring the
counter to the comparators. Such fan-out limits the switching speed and leads to undesired
dissipation.
Nevertheless, as will be discussed at the end of this chapter, the dissipation of the three
FPGAs is dominant over the RMS dissipation in the actuator coils. In section 6.8.1 several
design concepts will be proposed to reduce this.
Figure 6.7 shows the double-sided PCB with 61 drivers and the PCB with the master FPGA,
the DC/DC convertors and the LVDS drivers. The connector board that connects to the
three flex foil flaps on one side and the analog electronics PCB on the other side is shown
in the lower right photo in figure 7.2.
6

6
136 6 Electronics
6.4.3 The ethernet to LVDS bridge
At the time of design, no general purpose PC expansion card was available to provide an
off-the-shelf PC with two LVDS connections and a fully customized communication protocol.
Therefore, a communications bridge was conceived that bridges the 100Mb/s ethernet
connection of a PC with the custom LVDS connection. The LVDS bridge must relay messages
received over the ethernet connection to the LVDS on the other side and vice versa.
The bridge should be reliable and add little latency – i.e. the delay between reception and
transmission of the first bit of a data package – to the delay of the two-step communication
chain. To limit development time, the LVDS bridge is based on an Altera NIOS-II FPGA
development board (figure 6.8). This board is extended with an ethernet PHY that implements
the MAC layer of the ethernet protocol in hardware to limit latency. A second plug-in
PCB contains the LVDS driver ICs. The NIOS FPGA implements a processor that executes
an open source Internet Protocol (IP) stack that has been optimized for latency. As with any
communication type, transmission errors may occur for which detection methods are usually
implemented. However, for real-time application it is more important to limit latency than
to detect or recover rarely occurring errors. Therefore, the User Datagram Protocol (UDP)
protocol has been chosen (appendix E) for the ethernet communication, whose checksums
to detect faulty data have been disabled or are ignored.
6.5 Modeling
The actuator and its electronic circuit are modeled to determine a suitable base frequency
for the PWM signals and to check whether both the actuator and the electronics behave as
designed. Furthermore it allows validation of the full DM system including its reflective
facesheet (chapter 7) and serves as input for a controller synthesis procedure.
Recall the analog electronic circuit depicted in figure 6.3. Let the circuit be driven by the
PWM voltage VPWM. The effect of the fine PWM signal that connects to the system at a
different location – leading to different dynamics – will further be neglected. The actuator
has been modeled in section 5.2.4, leading to the linearized system in (5.24) on page 106.
The2 nd order analog low-pass filter consists of coilLl with internal resistance Rl in series
with capacitor Cl that is connected in parallel with the actuator. From Kirchoff’s laws it
follows that:
VPWM = VLl +VRl +Va, and IRl = ICl +Ia,
whereVRl andVLl denote the potentials overRl andLl andICl andIRl the currents through
Cl andRl respectively. They are defined through the following constitutional equations:
VRl
= RlIRl , VLl = Ll ˙
IRl , ICl = Cl ˙ Va.
The system will be modeled in the state-update form,
˙x = Ax+BVPWM, (6.4)
with state vectorx(t) = [Ia(t) za(t) ˙
za(t) Va(t) IRl (t)]T . The time derivatives of the state
elements can be derived from the constitutional equations together with the two Kirchhoff

6.5 Modeling 137
Table 6.2: Properties of the components of the 2 nd order analog low-pass filter.
equations, leading to:
Parameter Value Unit Parameter Value Unit
Ll 220 µH Cl 4.7 µF
Rl 2.7 Ω Rc 16.2 kΩ
R ′ l 2.4 Ω R ′ a 3 Ω
˙
IRl = (VPWM −RlIRl −Va)/Ll,
˙Va = (IRl −Ia)/Cl.
These two equations can be combined with the previously derived actuator system equation
in (5.24) and expressed in the state update form of (6.4) as:
⎡
˙
⎤
Ia
⎢za
⎢
˙ ⎥
⎢ ¨za ⎥
⎢ ⎥
⎣ ˙Va
⎦
˙ IRl
=
⎡
−Ra/La 0 −Ka(z
⎢
⎣
′ a )/La 1/La 0
0 0 1 0 0
Ka(z ′ a )/mac
⎤⎡
⎤
Ia
⎥⎢
za ⎥
⎥⎢
⎥
−ca/mac −ba/mac 0 0 ⎥⎢
za ⎥⎢
˙ ⎥
−1/Cl 0 0 0 1/Cl ⎦⎣Va
⎦
0 0 0 −1/Ll −Rl/Ll IRl
+
⎡ ⎤
0
⎢ 0 ⎥
⎢ 0 ⎥
⎣ 0
1/Ll
(6.5)
The output signals that will be used for analysis and testing are the actuator displacement
za(t) = [0 1 0 0 0]x(t) and the voltage Va(t) = [0 0 0 1 0]x(t) that can be measured over
actuator coil. Let the transfer function between the PWM voltageVPWM(t) and the actuator
positionza(t) be denotedH(s). For properties of the actuator and the electronics as in table
6.2, figure 6.10 shows the Bode plots of the resulting transfer functions. Figure 6.10 also
shows the Bode plot of the transfer function when only the current-controlled mechanical
system is considered. The static relation Ia = Va/Ra is used to scale the corresponding
transfer function in (5.28) on page 111 and allow comparison with the full mechatronic
system. When omitting the nominal operating pointz ′ a, this yields the transfer function:
Magnitude [m/V]
Phase [deg]
10 −4
10 −6
10 −8
10 −10
0
−90
−180
−270
−360
−450
H mech (s)
H v (s)
H(s)
18 bit suppression
10 3
Hm(s) = HI(s)/Ra =
macRas2 . (6.6)
+baRas+caRa
10 4
Frequency [Hz]
10 5
Ka
Figure 6.10: Bode diagram of three transfer
functions: Hm(s) from (6.6) (only the mechanics),
Hmv(s) from (5.26) (mechanics including
the actuator coil) and H(s) defined in section
6.5 (mechanics with actuator coil and low-pass
filter).
⎦ VPWM.
6

6
138 6 Electronics
Observe in the Bode plot that the electronics provide a small amount of additional damping
of the mechanical resonance, but have negligible influence on the low-frequent actuator
behavior.
The PWM base frequency
In section 6.2.2 it was discussed that an 18-bit attenuation of the PWM ripple is desirable.
To achieve this with an FPGA based implementation, an additional2 nd order low-pass filter
was added to the design.
The effect of this filter is illustrated in figure 6.10, which contains Bode plots of the actuator
system with and without the filter as described by (6.5) and (5.24) respectively. The 18bit
ripple attenuation is shown in the magnitude plot of figure 6.10 as the dash-dotted line.
Observe that the application of the low-pass filter reduces the PWM frequency requirement
from approximately 128kHz to approximately 35kHz.
Although higher than necessary for DMs with Pyrex facesheets, the base frequency fPWM
is set at 61kHz. This is done because in future developments the replacement of the Pyrex
mirror facesheet by beryllium is foreseen, demanding a higher base frequency. The specific
stiffness of beryllium is more than five times higher , which allows for thinner facesheets
and thus less mass per actuator. With the same actuator stiffness this increases the system’s
eigenfrequency and decreases the attenuation of the PWM ripple.
For the foreseen update rate of 1kHz the base frequency of 61kHz provides 61 times oversampling
of the setpoint signal. This means that cross-harmonics in the PWM output voltage
VPWM resulting from non-constant setpoint signals can be neglected.
Serial communication
The serial communication via both ethernet (UDP) and LVDS will introduce a certain delay
τc of the control output. This delay should be as small as possible and its variation (jitter)
should be restricted to a negligibly small fraction of the delay itself. The communication
latency is in the Laplace domain modeled as Hτc(s) = e −τcs . Due to the definition of the
communication protocol (appendix E) and its serial nature, the latency τc will be different
for each 61-actuator module.
6.6 Evaluation of control aspects
In this section the actuator and electronics design will be evaluated from the perspective of
closed-loop control of a single actuator. In section 5.3.1 it is shown that the damping of
the first mechanical resonance frequency of the actuators is low. Further, the resonance frequency
is higher than the sampling frequency of the foreseen control system. The latter is
mainly limited by the Charge Coupled Device (CCD)-based Shack-Hartmann sensor used
for feedback. As derived in chapter 2, the required bandwidth for position control of the
DM is 200Hz. If this is to be achieved for the full DM system in which there is interaction
between actuators, then it should also be achievable with the single actuator considered
here. However, the lowly damped and potentially aliased resonance may limit the achievable
bandwidth. For this reason, damping should be increased in future designs.
The control and sampling aspects of the actuator system will be evaluated using the Aliased

6.6 Evaluation of control aspects 139
r + e u
z y
C Hτc Hzoh H G
−
ideal
sampler
Figure 6.11: Schematic of the control loop for position control of a single actuator. The grey area
contains the continuous time parts of the loop.
Frequency Response Function (AFRF) as proposed in [94, 163]. Recall H(s) to denote the
transfer function between PWM voltageVPWM(t) and actuator positionza(t). The controller
C(za) will be implemented in discrete time with sampling timeTs and send its voltage setpoint
via the communication link with latencyτc to the driver electronics. The PWM based
driver electronics form a Zero Order Hold (ZOH) mechanism that can be expressed in the
Laplace domain as Hzoh(s) = (1 − e −sTs )/s [62]. Finally, the sensor input of the controller
is considered to be CCD-based, similar to the Shack-Hartmann sensor. It integrates
photons over the exposure time τe. When ignoring measurement noise, a continuous time
measurement ˜y(t) of the signalza(t) at timetcan be modeled as [100]:
˜y(t) = 1
τe
t
t−τe
which can be transformed to the Laplace domain as:
za(τ)dτ,
˜y(s) = G(s)za(s), where G(s) = 1−e−sτe
.
sτe
It is assumed that the continuous time signal ˜y(t) is then sampled by an ideal sampler,
yielding the discrete time signal y(k) for k = nTs, n = 0,1,2,.... Schematically, this
yields the control loop depicted in figure 6.11.
Since the loop contains both discrete time and continuous time components, the loop gain
can be written as:
L(s,z) = Ls(s)C(z),
where
Ls(s) = G(s)H(s)Hzoh(s)Hτc(s) = 1−e−sτe
sτe
H(s) 1−e−sTs
e
s
−sτc .
Note thatz denotes the discrete time Laplace variable. Since this transfer function depends
on bothsandz, use will be made of the AFRF, which is a discrete-time system that models
both the continuous time components and the aliasing effects introduced by the ideal sampler.
This means that the continuous time systems are discretized to the sampling frequency
fs = 1/Ts of the controller, causing the frequencies in the signal ˜y(t) above the Nyquist
frequency fN = fs/2 to be mapped to frequencies 0 < f < fN . The result is a linear,
discrete time system from which performance and stability properties can be derived using
common Linear Time-Invariant (LTI) tools [163].
The AFRF L ∗ (z) of the loop gain L(s,z) is the open-loop transfer function between the
6

6
140 6 Electronics
Magnitude [m/V]
Phase [deg]
10 −3
10 −4
10 −5
10 −6
10 −7
0
−180
−360
10 1
10 2
10 3
Frequency [Hz]
f s = 500 [Hz]
f s = 1000 [Hz]
f s = 1500 [Hz]
f s = 2000 [Hz]
f s = 10000 [Hz]
H(s)
Figure 6.12: The AFRF of the single actuator
system for various sampling frequencies when
assuming τc = 120µs.
10 4
10 5
Figure 6.13: Nyquist plots of the loop gain
L ∗ (z) based on the AFRF of the single actuator
system and the controllerC(z) from (6.8) for
various sampling frequencies and τc = 120µs.
Table 6.3: Achievable bandwidths for an integrator type controller for various sampling frequencies.
Sampling frequency [Hz] 500 1000 1500 2000 10000
Bandwidth [Hz] 98 198 159 213 196
Optimalα[V/m] (×1000) 170 170 86 84 15
Optimalγ [V/m] (×1000) -24 -19 -15 -7 -16
sampled output y(k) and the discrete time reference signal r(k). It can be obtained using
thez-transform denoted by the operatorZ as:
L ∗ −ste 1−e
(z) = Z(Ls(s,z))C(z) = Z H(s)
ste
1−e−sTs
e
s
−sτc
C(z).
When assuming the exposure time τe to be equal to one sampling time – i.e. τe = Ts – and
using thatZ(1−e −sTs ) = 1−z −1 , this reduces to:
L ∗ (z) = (1−z −1 )(1−z −1
H(s)
)Z
s2 e
Ts
−sτc
descriptions are listed in [62], but this conversion is also implemented in Matlab’s
C(z). (6.7)
The involvedz-transform can be derived by first transforming the continuous time transfer
function into pole/residue form (partial fraction expansion). This describes the system as a
sum of first order systems, which have an equivalentz-domain description. These equivalent
command [130], which can also account for arbitrary time-delays τc. In figure 6.12 the
AFRFs are plotted for various sampling frequencies when assumingτc = 120µs. Here it is
clear that the effect of aliased dynamics is very small.
To answer the question whether the bandwidth of 200Hz is achievable, let C(z) be a
discrete time proportional plus integrator controller with integrator gainαand proportional

6.7 Testing and validation 141
gainγ:
C(z) =
α
+γ (6.8)
1−z −1
As discussed in chapter 3, an integrator structure is often used for the controller of an AO
system. The proportional gain term is added to increase the phase margin around the bandwidth.
Note that computational delays are here neglected.
Figure 6.13 shows Nyquist plots of the loop gain L ∗ (z) for various sampling frequencies
after tuning the parametersα and γ for highest bandwidth (0dB crossing of the magnitude
of the open-loop gainL ∗ (z)) while considering a modulus margin [62] of 0.5. The achieved
bandwidths are listed in table 6.3. Observe that the bandwidth varies with the sampling
frequency fs and for fs =2000Hz it is with 213Hz the highest. The reason that a higher
sampling frequency does not lead to a higher bandwidth is that the phase delay due to the
low sampling frequency rotates the phase response at the resonance frequency away past
the (-1,0) point. Consequently, the integrator gain α can be further increased without compromising
the modulus margin.
As discussed in chapter 2 the achievable bandwidth depends strongly on the chosen sampling
frequency fs, but it has here been shown to be amplified by the low damping of the
mechanical resonance. This should be taken into account for future design revisions. Nevertheless,
for fs = 2000Hz the desired control bandwidth of 200Hz can be achieved while
respecting a modulus margin of 0.5.
6.7 Testing and validation
The electronics and the actuator grids were tested. The dynamic response of the actuators
was first measured using a controlled current source. These results were shown in section
5.3.1. In this section, first the test results of the communication between a PC and the LVDS
bridge and between the PC and the driver modules are shown (figure 6.9) followed by a full
system test. Here the dynamic response of the actuators is measured. At the end of the
section the power dissipation of the electronics will be discussed.
6.7.1 Communications tests
To measure the latency of the communications bridge, two of its debug lines were connected
to a logic analyzer. The first line is high while a UDP packet is being received and the second
while an LVDS packet is being transmitted. A second computer was used to send burst
packets (appendix E). These are the most relevant in practice and contain 16-bit setpoint updates
for all 61 actuators corresponding to 1024 bits in total. To minimize ethernet protocol
overhead, each UDP burst packet can contain up to eight LVDS burst packets (appendixs D
and E).
Measurements taken by the logic analyzer show that the transmission time of a UDP burst
packet can be expressed as:
τudp ≈ 4.7·10 −6 +10.24·10 −6 Nm
where Nm is the number of actuator modules within the packet. The constant part is due
to ethernet protocol overhead and the approximately 10µs per module corresponds to 1024
6

6
142 6 Electronics
bits at a rate of 100Mb/s.
Further, the measurements show a time delay of approximately 85µs with a variation (jitter)
of less than 10µs after reception of the UDP packet, before transmission of an LVDS
packet. During this time the bridge processes the packet, splits it into LVDS packets and
copies it to the transmit buffer. Since calculation of the UDP checksum takes a significant
time – approximately(10Nm)µs – this checksum is sacrificed for speed and ignored in the
current implementation. Transmission of the 1024 functional bits over the 40Mb/s LVDS
connection with 16-bit data words separated by 18 pause bits, one start-bit and one stop-bit
should takeτlvds ≈ 28.8·10 −6 Nm, which is confirmed by the measurements.
The total communication latencyτc can thus be expressed as:
τc = τudp +85·10 −6 +τlvds = 89.7·10 −6 +39·10 −6 Nm
Since the communication chain consists of two sequential, buffered links, the maximum
update rate is determined by the slowest link, in this case the LVDS line. This rate equals
1/28.8 · 10 −6 /Nm, which for Nm = 1 and Nm = 7 is approximately 35 and 5kHz respectively.
However, since the LVDS bridge may drop incoming packets during its 85µs
processing time, in practice this latency adds directly to the ethernet latency. For the case
that Nm = 1, this makes the ethernet latency dominant and reduces the maximum update
rate to approximately10kHz.
6.7.2 Parasitic resistance measurements
Before the full actuator systems will be tested, first several properties of the electronics
are measured. Deviations of the expected values measured in the next section can then be
properly attributed to either the electronics or the mechanics.
In practice – due to wiring – the resistances Rl and Ra are assumed to increase by R ′ l and
R ′ a respectively. Resistance measurements of the actuator coils show on average the design
value of 39.0Ω, but the average resistance measured over the capacitor Cl is found to be
42Ω, indicating that R ′ a ≈ 3Ω. The resistance R ′ l will be estimated from a few additional
measurements. Firstly, the PWM setpoints are set such that the measured voltage over
the capacitor Cl is VCl =1.001V. After reconnecting the actuators this yielded an average
voltage drop over the capacitor ofV=0.903V. Using the fact that the current throughRl,R ′ l ,
Ra andR ′ a is equal and the sum of the voltage drops is equal to the PWM voltage of 1.001V,
the resistance R ′ l is estimated as R′ l = 2.4Ω. A 1Ω part of the latter can be attributed to
a safety resistor present in the design for short-circuit protection, whereas the rest must be
attributed to wiring and connector resistance.
Although the supply voltage variations will not have an effect for power dissipation – this
will be compensated by a controller – the power dissipation increases linearly with the
= 3 + 2.4 =
resistance of the current path. The total parasitic resistance of R ′ a + R ′ l
5.4Ω will therefore lead to an undesired increase in power dissipation on the driver PCB of
(R ′ a +R′ l )/(Ra +Rl)·100% = 5.4/(39.0+2.7)·100% ≈ 13%.
6.7.3 Actuator system validation
Whereas dynamic measurements were performed on the single actuator prototypes using a
Siglab system, the setup depicted in figure 6.14 will be used for testing and model validation

6.7 Testing and validation 143
xPC
excitation Ethernet LVDS
UDP switch bridge
UDP LVDS
Electronics
module
actuator
Laservibrometer
velocity
position
Figure 6.14: Setup used to perform the actuator response measurements using the custom built electronics.
of the grid actuators. A Matlab TM xPC-target computer is used to generate a white noise
sequence and send it in UDP burst packets (appendix E) over an ethernet connection. The
sequence is logged internally to be used for analysis later. The ethernet connection goes
via a switch to allow both the xPC target and the dedicated electronics to be controlled and
configured by a host computer. Although this doubles the ethernet latency, this is not critical
for the open-loop validation measurements.
The LVDS bridge then converts the packets into LVDS packets to be sent to the electronics
module corresponding to the targeted actuator. Both the position and velocity response of
the actuator are measured using a polytec laser vibrometer. This outputs the measurements
as analog voltages that are fed back to the xPC target using a National Instruments Analog
to Digital Convertor (ADC) card (NI-6025E) that does not contain any anti-aliasing filters.
The measurements are performed for update rates of 1, 3, 5 and 10kHz to be able to evaluate
the effects of sampling and aliasing.
Let the discrete time frequency response function between the PWM voltage output
and the actuator positionza that includes the effects of sampling and digital communication
Magnitude [m/V]
Phase [deg]
10 −4
10 −6
0
−180
−360
−540
−720
10 1
Measurements
Fitted models
Original model
F s = 1kHz
10 2
F s = 3kHz
Frequency [Hz]
F s = 10kHz
Figure 6.15: Bode plot of empirical frequency
response function estimates ˆH ∗ p,Ts (f) together
with the parametric fit H ∗ p,Ts (za, ˆ θ) and the
nominal model H ∗ p,Ts (za,θ0) at 1, 3 and 10kHz
sampling frequencies.
10 3
Magnitude [m/Vs]
Phase [deg]
10 0
10 −1
10 −2
10 −3
10 −4
180
0
−180
−360
−540
−720
10 1
F s = 1kHz F s = 3kHz
10 2
Frequency [Hz]
F s = 10kHz
Figure 6.16: Bode plot of empirical frequency
response function estimates ˆHv,Ts(f) together
with the parametric fit H ∗ v,Ts (za, ˆ θ) and the
nominal model H ∗ v,Ts (za,θ0) at 1, 3 and 10kHz
sampling frequencies.
10 3
6

6
144 6 Electronics
Actuator stiffness c a [N/m]
650
600
550
500
450
400
350
300
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 6.17: Identified actuator stiffness ca
when assuming mac =3.6mg sorted on value
for each measured actuator grid separately. The
value predicted by the original model is 583N/m.
Resonance frequency f e [Hz]
2100
2000
1900
1800
1700
1600
1500
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 6.18: Resonance frequenciesfe when assuming
mac =3.6mg sorted on value for each
measured actuator grid separately.
be denoted H ∗ p,Ts (za,θ,τc), where the subscript Ts indicates the corresponding sampling
time and the vectorθ contains the physical parametersmac,ba,ca,La,Ra,Ka,Ll,Rl and
Cl. Similarly,H ∗ v,Ts (za,θ,τc) denotes the transfer function to the actuator velocityza. ˙ The
effect of the sampling performed by the NI ADC card can be modeled by assuming a zero
order hold on the excitation signal and applying the z-transform similar to the procedure
described in section 6.6, yielding:
H ∗ p,Ts (z,θ,τc)
Tss 1−e
= ZTs
s H(s,θ)e−τcs
= (1−z −1 −τcs
)ZTs H(s)e /s ,
Motor constant K a [N/A]
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 6.19: The motor constants Ka when assuming
mac =3.6mg sorted on value for each
measured actuator grid separately. The value
predicted by the original model is 0.19N/A.
Viscous damping b a [mNs/m]
1
0.8
0.6
0.4
0.2
0
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 6.20: The viscous damping ba when assuming
mac =3.6mg sorted on value for each
measured actuator grid separately.

6.7 Testing and validation 145
Inductance L a [mVs/A]
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 6.21: Identified actuator coil inductance
La when assumingmac =3.6mg sorted on value
for each measured actuator grid separately. The
value used for the original model is 2.93mH.
Latency τ c [µs]
170
165
160
155
Module 1
Module 2
Module 3
Module 4
Module 5
Module 6
Module 7
10 20 30 40 50 60
Actuator number [−]
Figure 6.22: The communication latency τc
sorted on value for each measured actuator grid
separately.
Table 6.4: Average values and standard deviations of the actuator system properties measured over
all grid actuators using the setup depicted in figure 6.14.
Property Ka ca ba fe La
Average 0.11N/A 473N/m 0.30mNs/m 1.83kHz 3.0mH
Std.dev. 0.02N/A 46N/m 0.11mNs/m 91Hz 0.2mH
H ∗ v,Ts (z,θ,τc) = ZTs
1−e Tss
s
sH(s,θ)e −τcs
= (1−z −1 −τcs
)ZTs H(s)e ,
where H(s,θ) denotes the transfer function H(s) based on the physical parameters
in the vector θ. Let the empirical transfer function estimates of H ∗ p,Ts (za,θ,τc) and
H∗ v,Ts (za,θ,τc) be denoted ˆ H∗ p,Ts (f) and ˆ H∗ (f) respectively, where f is the fre-
v,Ts
quency. These estimates together with the corresponding coherence functions Cp,Ts(f)
and Cv,Ts(f) were obtained from 10s of input-output data logged by the xPC target using
Welch’s averaged periodogram method with a block size of 2048 samples with 70% overlap
and a Hanning window. A typical set of estimates is shown in figures 6.15 and 6.16 for
actuator 10 of grid 1.
The model parametersca/mac,Ka/mac,ba/mac,La and the latencyτc will be identified
from the empirical transfer function estimates. The other model parameters Rl, Ll and Cl
are assumed to be accurately known. A single set of parameters is fit against eight measurements
series: four different sampling frequencies times two measurement outputs (position
and velocity). The optimization is performed w.r.t. the cost functionJp +Jv, in which:
Jp =
Ts∈Ts f∈Fm(Ts)
⎛
⎝ Ĉp,Ts(f)
ˆ
H
Hp,Ts(f)
∗ p,Ts (e2πjf )− ˆ ⎞2
Hp,Ts(f) ⎠
,
6

6
146 6 Electronics
Jv =
Ts∈Ts f∈Fm(Ts)
⎛
⎝ Ĉv,Ts(f)
ˆ
H
Hv,Ts(f)
∗ v,Ts (e2πjf )− ˆ ⎞2
Hv,Ts(f) ⎠
.
The set Ts consists of the sampling times corresponding to 1, 3, 5 and 10kHz update rates
and the sets Fm(Ts) contain the frequencies at which the transfer functions were estimated
for the sampling timeTs. The values of the coherence functionsCp,Ts(f) andCv,Ts(f) are
used as weights to include the reliability of the transfer function estimates in the parametric
optimization problem. To remove the bias introduced by the system magnitude response,
the inverse of this response is applied as a second weight. In the optimization the properties
of the electric components that form the analog low-pass filter are taken from table 6.2 and
include the parasitic resistancesR ′ l andR′ a
. The z-transform is implemented using Matlab’s
function, which also accounts for the latencyτc. Since this cost function is non-linear
w.r.t. the parameters to be estimated, the optimization is performed using Matlab’s nonlinear
least squares solverÐ×ÕÒÓÒÐÒ.
Examples of the estimation result for an arbitrary actuator are depicted as the gray dashed
lines in figures 6.15 and 6.16, showing only very slight deviations between the model and
the measurements. Moreover, it should be noted that both figures are based on the same
set of parameter estimates. The average values and standard deviations of the estimates for
Ka, ca, ba and La when assuming mac =3.6mg are listed in table 6.4. These values agree
well with the results obtained using the current source in the previous chapter as listed in
table 5.2 on page 121, which indicates the robustness of the measurement and identification
process. Further, all estimates are plotted in figures 6.19, 6.17, 6.20 and 6.21. For these
plots the values are sorted per actuator module for better insight into their statistical spread
and correspond well to the values estimated using the current source setup in section 5.3.1
depicted in figures 5.27, 5.25 and 5.28 on page 122. Each module has a few actuators with
significantly different properties, but only a single actuator is malfunctioning. Moreover,
the variation between actuators is not linked to the location of the actuator in the grid. This
is illustrated for the resonance frequency and motor constant in the figures in appendix F.
These show the corresponding values for all actuator grids in relation to the location of the
actuator in the module.
Although the resistance in the current path of an actuator affect this system’s DC-gain, it
cannot be separately estimated. The parasitic resistances R ′ a and R′ l
are only practically
measurable for a few actuators per module. Measurements for approximately 20 actuators
provided the average value used for the estimation of the actuator parameters. The several
percent resistance variation affects the parameter estimates, which together with estimation
errors explains the differences with the values obtained from current source measurements
shown in section 5.3.1.
In addition to the results shown in section 5.3.1, the voltage excitation allows to estimate the
actuator inductance La. Observe from figure 6.21 that also the average of the inductance
differs significantly per actuator module, which is most significant for module 5. This module
also has a relatively low average motor constant (figure 6.19). Based on the sensitivity
analysis performed on the actuator design in section 5.2.6, such variation can for instance
be attributed to an increased radial air gap reluctanceℜgr. This reluctance depends strongly
on the radial air gap width, which is e.g. equally affected for all actuators of a module by
the radius of the mill used in the baseplate milling process.

6.7 Testing and validation 147
Displacement [µm]
30
20
10
0
−10
−20
−30
−40
Response measurement
Linear approximation
Linearity error
−3 −2 −1 0 1 2 3
Voltage setpoint [V]
3
2
1
0
−1
−2
−3
−4
Error [µm]
Figure 6.23: Non linearity and hysteresis
measurements performed on a
single actuator excited by a 4Hz sine
excitation voltage with 3.3V amplitude.
The response is plotted against
the excitation for single (thick solid
line) and the deviation from a linear
response (dashed line) is plotted as
the thin solid line on the right vertical
axis.
Finally, the estimated latency varies between approximately 160 and 170µs, of which
89.7 + 39 = 128.7µs can be attributed to the serial communication (section 6.7.1). Another
8µs can be attributed to communication between the master and slave FPGAs and
also 8µs to the implemented PWM update method that yields an average latency of half a
period of the PWM base frequency. The remaining 20µs are likely caused by overhead in
the XPC target computer, in which ethernet communication is performed by a background
process and not strictly real-time.
6.7.4 Nonlinear behavior
So far only the linear system dynamics were considered, whereas it was shown in section
5.2.3 that both the mechanical and magnetic stiffnesses of the actuator are nonlinear functions
of current and deflection. This is particularly true for large deflections, when the
difference with the operating point used for linearization becomes significant. However,
measurement results show that no significant (i.e. measurable) hysteresis is present (figure
6.23).
Measurements were performed to quantify both effects for the single actuator system.
The previously described setup depicted in figure 6.14 was used to excite the system with
a low frequent sine signal and measure its deflection response. Differential measurement
capabilities of the laser vibrometer were used to limit drift due to e.g. air motion and reduce
external disturbances such as floor vibrations. The excitation frequency is chosen at 4Hz
such that only the system’s static behavior (stiffness) plays a role and not its resonances.
An amplitude of 3.3V corresponding to the maximum available input voltage is used. The
sampling frequency of the measurements is chosen as 10kHz to minimize effects of aliasing
and the results depicted in figure 6.23 have been compensated for the discussed latencies
that yield a spurious hysteresis loop. The figure shows that hysteresis is negligible and of
similar order of magnitude as the drift of the laser vibrometer. A linear function is fit to the
response and shown as the dashed line. The difference with the actual response is plot as
the dash-dotted line against the right y-axis. Although for large deflections the nonlinear
actuator stiffness becomes visible, for the intended±10µm deflection, the actuator linearity
error is less than 5%.
6

6
148 6 Electronics
6.8 Power dissipation
As stressed in chapter 2, power dissipation forms a design driver for the DM system. In this
section, the power dissipation measured will be discussed.
For the analysis only the static response of the system is considered, which is assumed to
be linear w.r.t. the PWM voltage setpoint. The validation measurements have shown this to
be an accurate approximation at least up to approximately 1600Hz. Non-static dissipative
effects such as viscous damping and eddy currents play a negligible role in this frequency
range. The remaining power dissipation of the single actuator system consists of several
parts. Firstly, there is the power dissipated as a direct consequence to the actuator current.
This flows through the actuator coil with a resistanceRa+R ′ a and the coil of the analog lowpass
filter with resistanceRl +R ′ l . For the static case, the corresponding power dissipation
can be expressed as:
Pa = I 2 a(Ra +R ′ a +Rl +R ′ 2 Fa
l) = (Ra +R
Ka
′ a +Rl +R ′ l),
where for the second step the system was assumed to be linear such thatFa = KaIa.
For the expected RMS actuator force of 1mN, a measured average motor constant Ka ≈
0.12N/A and Ra + R ′ a + Rl + R ′ l ≈ 39 + 3 + 2.7 + 2.2 ≈ 46.9Ω, this corresponds to
3.2mW per actuator. For the design values in tables 5.1 and 6.2 this power dissipation
would be 1.4mW, which means that the actual dissipation will be approximately 2.3 times
higher than expected.
Besides direct dissipation of the electronics, there is also indirect dissipation. This consists
of the power dissipated by the FPGAs to generate the PWM signals and handle the communication
and dissipation of the Field Effect Transistor (FET) switches of the H-bridges,
LVDS drivers and voltage converters. These contributions have been quantified by measuring
the supply current to a single electronics module for various configurations using a
Fluke digital multi-meter.
The static power dissipation of the three FPGAs is provided by the manufacturer as approximately
40mW. The summed power dissipated by the master FPGA, the voltage converter
and the LVDS driver has been obtained by measuring the supply current with only the master
print connected. The power dissipated by the slave FPGAs that generate the PWM signals
has been obtained by measuring the supply current with the resulting PWM outputs disabled.
Losses in the analog part of the electronics and due to the switching of the H-bridge
were obtained by measuring the supply current for various actuator setpoints, but without
the actuators being connected. This prevents DC currents from flowing and allows the measuring
of parasitic effects only. Further, the difference in supply current to the case that
the actuators are connected can be attributed to actuator currents and resulting dissipation.
Finally, for all measurements the DC-DC convertor was assumed to have an efficiency of
85%, leading to the results plotted in figure 6.24.
For an output voltage of 0V the dissipation consists only of the mentioned indirect losses,
whereas for non-zero voltages the dissipation is proportional to the square of the voltage
setpoint divided by the total resistance. The results in the figure confirm this resistance to
be around 40Ω. The RMS voltage setpoint expected in practice is derived from the expected
RMS actuator force of 1mN derived in chapter 2 by division by the motor constant
≈ 46.9Ω, yield-
Ka ≈ 0.12N/A and multiplication by the total resistanceRa+R ′ a +Rl+R ′ l

6.8 Power dissipation 149
Power per actuator [mW]
35
30
25
20
15
10
5
DCDC−convertor
Actuators
Analog electronics
Slave initialization
Slaves
Master
0
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
PWM voltage V [V]
pwm
Figure 6.24: Measured
power dissipation of the
single actuator system including
driver and communication
electronics. Results
are split into the contributions
of several components.
ing 0.39V. Observe in figure 6.24 that for this voltage setpoint value the power dissipated
in the FPGAs dominates the power dissipated in the actuator to generate force. A reduction
of the total power consumption will thus be most effectively achieved by a reduction
of the power consumption of the FPGAs. The FPGA implementation has been analyzed in
detail by William de Bruijn, who also proposed design modifications to improve the power
efficiency. His work has been documented in detail in [42], but only the main findings and
design proposals will be summarized.
6.8.1 Optimizing the FPGA power efficiency
The driver electronics developed for the DM system use three FPGAs to implement 61
PWM signal generators and the LVDS communication with a control computer. The power
dissipated by an FPGA can be divided into static and dynamic dissipation. The first is
dissipated regardless of the program loaded or configuration, but depends highly on the
specific IC. For each of the Altera ICs this dissipation is approximately 40mW. The dynamic
dissipation depends on the program loaded and can be approximated as:
Pdyn = Psc(fclk)+αCV 2
cc fclk, (6.9)
where Psc is the short-circuit power dissipation that is linearly proportional to the clock
frequency fclk. Further, α denotes the switching activity (the average number of 0 to 1
transitions per clock cycle), Vcc the supply voltage and C the total capacitance. The latter
is a measure for the amount of hardware (transistors, interconnection wires, etc.) in use.
The static power dissipation and the supply voltage Vcc are determined by the choice for
a particular FPGAs. Both are likely to decrease for future models due to technological
progress. A reduction of the dynamic power dissipation can be achieved by reducing the
capacitance C and the total number of state switches per second expressed by the product
αfclk. Such a reduction requires more insight into the distribution of the dissipation over
6

6
150 6 Electronics
Slaves
Master
Clocks
RAM
LVDS receiver
PWM counter
Comparators
Other
0 10 20 30 40 50 60 70 80 90 100
Fraction of total power dissipation [%]
Figure 6.25: The power usage of the master and slave FPGAs subdivided into several functional
categories.
specific parts of the current FPGA program.
To determine the contribution of code parts, the existing FPGA programs were first ported to
a Xylinx University Program (XUP) development board. This development board contains
a single Xylinx chip instead of the three Altera’s, but measurements confirmed that the
dynamic FPGA power dissipation is of the same order of magnitude: approximately 550mW
instead of 673mW. The latter is derived from figure 6.24 by summing the contributions of the
master, the slaves and the initialization thereof atVPWM = 0. These measurements were used
as a benchmark to compare with the output of the XPower simulation tool [195] by Xylinx
that uses the post-place and routing net-lists of the ModelSim PE simulator [131]. Since the
simulation results corresponded well to the measurements [42], the more detailed results
of the XPower tool were trusted to provide an accurate subdivision of the total dissipation.
This is shown graphically in figure 6.25 for the master and slave FPGAs. Observe that the
dissipation of the master FPGA can be mainly attributed to a RAM module and that the
dissipation of both FPGA types is for over 50% attributable to clock signals. This means
that power dissipation is reduced by:
1. Removing the Random Access Memory (RAM) module.
2. Reducing the clock frequencyfclk.
3. Reducing the amount of clocked hardware.
The RAM module is currently used to buffer the incoming LVDS messages, which is not
necessary. According to (6.9) the dynamic dissipation is linearly proportional to the clock
frequency fclk. A reduction is not possible for all functionality of the FPGA, since this is
linked to the PWM base frequency (section 6.2.2), but for some parts it is. The dissipation
due to clock signals can be significantly reduced using an asynchronous design in which
logic cells are synchronized using local handshakes instead of global clock signals. Such
an asynchronous design can be efficiently written in a parallel language such as Tangram
or HASTE [93], but the achievable reduction in dissipation depends highly on the software
tools used for mapping this code onto a specific FPGA. For maximum effect the design
should be mapped to an ASIC instead of an FPGA. Nevertheless, three new designs were
proposed that have been conceptually analyzed and evaluated using both simulations and

6.8 Power dissipation 151
implementations on the XUP development board. This has lead to the following observations:
• The master functionality can be efficiently expressed in an asynchronous language.
After implementation without RAM module, a reduction in the power dissipation of
the master FPGA of 40% was measured.
• Although asynchronous designs have a high potential for power reduction, this cannot
be fully exploited by implementation on FPGAs, because these devices are inherently
synchronous and must in fact simulate asynchronous programs.
• The slave functionality was most efficiently implemented using a recursive PWM
driver designs. This design has no central counter, but uses a counter for each channel.
A power reduction is achieved via a recursive counter design in which the bits are
clocked at their rate of change (lower bits have a higher clock frequency).
• A combination of a recursive counter implementation for each PWM driver and the
asynchronous master lead to a reduction in overall power dissipation of approximately
29%. This reduction is significantly larger when the design is mapped to an ASIC.
6.8.2 Cooling
The improvements proposed in section 6.8.1 are not yet implemented. The power dissipation
in the FPGAs will therefor still exceed the power dissipation of the coils by far. To
avoid this heat to be transferred to the ambient air with detrimental air flow in the path of
light as a result, a possibility for active cooling is added. This cooling system consists of an
aluminium fin which is placed between the master and two slave FPGAs. The aluminium
fan is connected with a the aluminium block, which holds the cooling channels. The cooling
liquid cools the block and thereby the aluminium fan. In case of, for example, an electronic
failure, the PCB together with the aluminium fin can be disconnected from the cooling
system without disconnection of the fluid system. The flow is regulated by a commercial
Central Processing Unit (CPU) cooling system. The aluminium block is suspended in a thin
plate to allow dimensional change of the PCB as well as tolerances on the PCB size. In
figures 6.27 and 6.26 the active cooling system for the FPGAs is shown.
Figure 6.26: The cooling system seen from the
back.
Figure 6.27: The cooling system seen from the
front.
6

6
152 6 Electronics
6.9 Conclusions
The electronics for the prototype DMs consist of two parts: the communication electronics
that supply the setpoints as computed by the control system and the driver electronics that
generate the corresponding actuator voltages. The requirements for both part are derived.
Both current and voltage drivers are considered. Current mirrors were not implemented because
of their complexity to produce them and the lower efficiency, especially with the large
dynamic range required. Since theLa/Ra time of the actuator is short: 75µs, the advantage
of current control over voltage control is limited. The motor constantKa, stiffnessca and –
in the case of a voltage source – the resistance R of the actuator circuit, will vary from actuator
to actuator and vary with temperature, causing slow gain variations. A current source
will compensate for variations in the resistance R, but variations of Ka and ca must still
be compensated by the AO control system. PWM based voltage drivers are chosen because
of their high efficiency and capability to be implemented in large numbers with only a few
electronic components.
A LVDS based serial communication bus was chosen for its low power consumption
(15mW/transceiver), high bandwidth (up to 655Mb/s) and consequently low latency, low
communication overhead and extensive possibilities for customization. The driver electronics
for 61 actuators are located on a single, multi-layer PCB and consist of FPGAs to
generate the PWM signals, FETs for the H-bridge switches and a coil/capacitor pairs that
form 2 nd order low-pass filters. The FPGAs that generate the PWM signals also control
two LVDS communication connections – one up- and one downlink – to receive setpoint
updates and to report status information. A 16-wire flat-cable connects up to 32 electronics
modules to a custom designed communications bridge, which translates ethernet packages
into LVDS packages and vice versa. The ethernet side of the communications bridge is
connected to the control computer at a speed of 100Mbit/s and uses the UDP protocol to
minimize overhead and latency.
The actuator model from chapter 5 was extended with models for the communication and
driver electronics. The communication is modeled as a pure delay and the driver electronics
as a voltage source with an analog 2 nd order low-pass filter. The model is used
to select a suitable PWM base frequency for which the position response from the voltage
ripple due to higher harmonics of the PWM signal is less than a quarter of the LSB
of the setpoint. This frequency should be higher than 40kHz for the DMs with Pyrex
facesheets, but is set at 61kHz to be suited for the replacement of these facesheets by
beryllium. The actuator model including its communication and electronics was validated
by measurements. The measurements include communication tests, static and dynamic
response measurements and power dissipation measurements. It is shown that the
communication latency is well represented by τc = 89.7 · 10 −6 + 39 · 10 −6 Nm, where
Nm is the number of the actuator grid. With the actuator response measurements, actuator
properties as stiffness (ca =473±46N/m), motor constant (Ka =0.11±0.02N/A),
damping (ba =0.30±0.11mNs/m), inductance(La =3.0±0.2mH) and resonance frequency
(fe =1.83±91Hz) are verified. These properties showed some variation between actuators,
but this could not be attributed to the location of the actuator in the grid (appendix F).
The time domain response of an actuator to a 4Hz sine voltage was used to determine hysteresis
and semi-static nonlinear response of the actuator. This showed the first to be negligible
and the second to remain below 5% for the intended±10µm stroke.

6.9 Conclusions 153
Finally, power dissipation was measured. Unintended resistances in the paths between the
voltage source and the actuator, combined with the lower motor constant showed to lead
to 2.3 times higher power consumptions of the actuators: 3.2mW instead of 1.4mW. Measurements
also showed that in the expected operating range, the total power dissipation is
dominated by indirect losses in the FPGAs. An alternative FPGA implementation is investigated.
A reduction of 40% in the master FPGA and 29% in the slave FPGAs is thereby
achieved.
6

ÔØÖ×ÚÒ
ËÝ×ØÑÑÓÐÒÒ
ÖØÖÞØÓÒ
The developed actuator modules (chapter 5) and electronics (chapter 6) are
integrated with the reflective facesheet [174] to form a complete Deformable
Mirror (DM) system. The static and dynamic system behavior is modeled and
compared to measurement results. The reflective deformable facesheet, which
couples all actuators, is modeled with a biharmonic plate equation and an analytic
solution for the surface shape under a regular actuator grid is found.
The model is used to derive the actuator influence functions. The static model
is extended with lumped masses to include the dynamic behavior. From the
model, the transfer functions, impulse response functions and mode shapes are
derived. The verification of the static behavior of the DM system is done using
an interferometer setup. The dynamic system identification is performed using
white noise excitation on the actuators and displacement and velocity measurement
of the mirror facesheet with a laser vibrometer. With these measurements
the model modal analysis is compared with the measurements.
sections 7.2, 7.3 and 7.4 are joint work with Roger Hamelinck
155

7
156 7 System modeling and characterization
7.1 Introduction
In this chapter the developed actuator modules (chapter 5) and electronics (chapter 6) will be
combined with a reflective facesheet (for design considerations, see [174]) into a complete
prototype Deformable Mirror (DM) system. First, the integration of these parts is described,
after which the behavior of the DM system will be analyzed. An analytical model for the
reflective facesheet is derived that – combined with the DC-gain of the single actuator system
provides a static model for the DM system. This model describes the actuator influence
functions that will be compared with measurements on the DM system. Further, measurement
results are presented that show the initial flatness of the DM and its ability to form
Zernike mode shapes. Finally, the Direct Current (DC) model is used to determine the expected
average power dissipation of the DM when correcting Kolmogorov type wavefront
disturbances.
An analytical dynamic model for the system is then derived based on the available model
for a single actuator system from chapter 6. From this model the expected resonance frequencies
and modal shapes are derived that will be compared to measurement results on
the DM system. Finally, the dynamic behavior of the DM system will be evaluated w.r.t.
discrete time control aspects.
7.2 DM integration
A single actuator grid with 61 actuators is integrated with a 100µ thick ∅50mm Pyrex
facesheet and a single Printed Circuit Board (PCB) to form the first prototype. In a second,
larger prototype, 7 actuator modules are placed on a reference base and connected to a
single,∅150mm Pyrex facesheet.
7.2.1 Integration of the 61 actuator mirror
In figure 7.1 the integration the first DM prototype is shown at different stages. The actuator
struts are first connected to the back of the mirror facesheet and then connected to
the actuator grid. With the struts attached, only the out-of-plane DOFs of the facesheet are
constrained. The in-plane DOFs are still free and will be constrained by the three folded
leafsprings shown in figure 7.1. The folded leafsprings are placed in their aluminium mount
and glued to the backside of the facesheet. Figure 7.2 shows the DM with the folded leafsprings
in place and figure 7.3 shows the 61 prototype DM including its electronics and
protective cover.
7.2.2 Integration of the 427 actuator mirror
The second DM prototype with 427 actuators is assembled similar to the single actuator
grid DM. First the 7 actuator grids are placed on a reference base (figures 7.4 and 7.5). The
corrugated edges of the actuator grids are separated by 0.3mm. The base is made from a
40mm thick aluminium block, perforated with 7 large holes (∅30mm) and a pattern of small
holes to accommodate the M2-bolts to attach the A-frames that connect to the actuator grids.
These bolts are mounted from the back. The base itself is placed vertically and supported
by three larger A-frames.

7.2 DM integration 157
The PCBs with the driver electronics are placed in one box (figure 7.6). The PCBs are
mounted similar to figures 6.27 and 6.26. Via the slits in the front plate, the flex foils
connect to the connector boards and PCBs. Figure 7.7 shows the electronic box connected
to the actuator grids.
The electronic box is decoupled before the mirror facesheet is connected to the actuator
grids by means of the actuator struts and small droplets of glue. First the connection
struts are glued to the backside of the mirror facesheet, during which the mirror facesheet
is supported by a porous air bearing. After curing of the glue, the mirror facesheet with
the struts is glued to the actuator modules. Details on the procedure can be found in [174].
Finally, the folded leafsprings needed to constrain the in-plane DOFs are placed (figure
7.8) and the flex foils are connected to the electronics (figure 7.9). The mirror is now fully
assembled, except for the protective cover ring. Unfortunately, while placing the cover ring
the mirror got damaged before any measurements could be obtained from the completed
DM system. The results presented in the sequel of this thesis originate from the single
actuator module prototype shown in figure 7.3.
Figure 7.1: The DM prototype with 61 actuators shown during final assembly. The upper left figure
shows 61 struts attaching the mirror facesheet to the actuator module. The module is
connected to the (black) base with three A-frames. The flex foil is fed through a central
hole in the base. On the right, the folded leafsprings that constrain the facesheet’s inplane
DOFs are shown prior to assembly. In the lower left, one of the folded leafsprings
is located a little below the mirror facesheet, before it is translated to make the glued
connection with the facesheet.
7

7
158 7 System modeling and characterization
7.3 Static system validation
In this section, the static behavior of the DM is modeled, providing a description for
the actuator influence functions. This involves modeling of the mirror facesheet and
combining this with the static model of the actuator system from chapter 6. The actual
influence functions of the DM system are measured using a Wyko interferometer to which
the modeled influence functions are compared. This was done for all 61 actuators in
the ∅50mm DM. Further, the influence matrix derived from the measurements is used
to form the mirror facesheet into the first 28 Zernike mode-shapes including the piston
term that represents the best flattened mirror [86, 87]. The measured shapes are compared
to the perfect Zernike modes and to the least square fit based on the DC model derived.
Figure 7.2: The 61 actuator DM. The protective cover is not shown, to see the inner parts. The
connector board, described in section 6.4.2 is shown in the lower right photo.

7.3 Static system validation 159
7.3.1 Modeling
Figure 7.3: The 61 actuator DM including
its electronics.
First a model for the reflective facesheet is derived, leading to an expression for the influence
function matrix. Finally, this matrix is used in an algorithm to calculate the actuator
commands that provide a facesheet that best approximates a certain Zernike shape in a least
squares or absolute-error sense.
Facesheet modeling
The mirror facesheet is modeled as a circular plate with free edges, subjected to point forces.
Although the facesheet has a large diameter to thickness ratio, the facesheet is still a plate
with significant bending stiffness, particularly on the spatial scale of the actuator pitch. In
contrast to a true membrane, there is no pre-tension from which it derives its stiffness and
resonance frequency. Since the connection struts are only 100µm thick – which is small
in comparison to the pitch – the forces exerted on the reflective facesheet are considered
to be point-forces. The in-plane stiffness is provided at the circumference by three folded
Figure 7.4: Seven actuator modules placed on
the reference base. The actuator grids are separated
by 0.3mm.
Figure 7.5: The backside of the reference base
with the 7 actuator modules mounted. The flex
foils are visible through the larger holes.
7

7
160 7 System modeling and characterization
Figure 7.6: The 7 PCBs with driver electronics assembled in the electronics box. The cooling, similar
to figure 6.27 is visible. Via the slits in the front plate, the flex foils connect to the
connector boards and PCBs.
leaf springs at 120 ◦ intervals. The out-of-plane stiffness that these springs contribute is
negligible in comparison to the facesheet and can therefore be neglected . The edge of the
facesheet is thus considered to be free.
Let r (i) and ρ (j) for i = 1...Nr and j = 1...Na be complex values corresponding to
coordinates in the complex plane. The deflection zf(r (i)) at coordinate r (i) of a circular,
Figure 7.7: The DM ready for testing. Each actuator is tested individually (section 6.7.3). After
testing the facesheet is assembled.

7.3 Static system validation 161
Figure 7.8: Detail of one of the three folded leafsprings
that constrain the in-plane DOFs.
Figure 7.9: The assembled DM with 427 actuators.
Figure 7.10: The broken DM.
thin plate of Hookean material, with radiusrf and free edge conditions due to a point force
F (j) located at ρ (j) can be derived from the biharmonic plate equation [177] in terms of the
Laplacian operator∇ 2 and the plate’s flexural rigidityDf as:
where ∇ 2 (r) =
δ 2
δRe 2 (r) +
∇ 4 zf(r) = F (j)r2 f
,
Df
δ 2
δIm 2 (r) , and Df =
Eft 3 f
12(1−ν 2 f ).
Further, Re(r) and Im(r) denote the real and imaginary parts of r respectively, Ef is the
plate material’s Young’s modulus, νf its Poisson ratio and tf its thickness. The deflection
Z(r (i)) can be expressed analytically in terms ofF (j) as [122]:
Z(r (i)) = F (j)r2 f
W(r (i),ρ (j))+wp +wxRe(r (i))+wyIm(r (i)),
16πDf
7

7
162 7 System modeling and characterization
Table 7.1: Dimensions and material properties of the reflective facesheets of the 61 and 427 actuator
DM prototypes.
Parameter Value Unit
rf for the 61 actuator DM 25.4 mm
rf for the 427 actuator DM 76.2 mm
Ef 64 GPa
ρf 2230 kg/m 3
tf 100 µm
νf 0.2 -
wherewp,wx andwy denote the rigid body motions in the out-of-plane direction and around
thex- andy axes respectively. The functionW(r (i),ρ (j)) is defined as:
W(r (i),ρ (j)) = ̺ (i,j)̺ ∗ (i,j)
+
ln(̺ (i,j))+ln(̺ ∗ (i,j) )+1–νf
ln(1–r(i)ρ
3+νf
∗ (j) )+ln(1–r∗ (i) ρ (j))
(1–νf) 2
(1+νf)(3+νf) r (i)r ∗ (i) ρ (j)ρ ∗ (j) +
8(1+νf)
(1–νf)(3+νf)
+k(r (i)ρ ∗ (j) )+(1–r∗ (i) ρ (j))ln(1–r ∗ (i) ρ (j))+k(r ∗ (i) ρ (j))
where the superscript ∗ denotes the complex conjugate and
̺ (i,j) = r (i) −ρ (j) and k(x) =
x
0
(1–r (i)ρ ∗ (j) )ln(1–r (i)ρ ∗ (j) )
,
ln(1−ς)
dς = −dilog(1−x).
ς
This analytic expression allows the spatial grids to be discretized without loss of accuracy.
The values of the geometric and material parameters for the 61 and 427 actuator DM prototypes
can be found in table 7.1.
Influence function modeling
The shape of the influence functions depend on the stiffness of the facesheet and actuators
and the lay-out of the actuator grid. Linearity is assumed to allow linear superposition of
multiple point forces. For convenience, matrix-vector notation is used, where matrices are
set in a bold typeface.
Let z f,(i) = Z(r (i)), z a,(i) = Z(ρ (i)) and F ρ,(j) = F (j) be elements of the vectors zf , za
and Fρ respectively. Similarly, the coordinatesr (i) and ρ (j) form the i th and j th elements
of the vectors r and ρ and Ω rρ,(i,j) = w(r (i),ρ (j)) the elements of the matrix Ωrρ. The
facesheet deflectionzf can then be expressed as:
zf = ΩrρFρ +Urwpxy, (7.1)
where Ur = [1 Nr×1 Re(r) Im(r)] and wpxy = [wp wx wy] T . The reflective facesheet is
supported by actuators with effective mechanical stiffnesses that exert forces denoted by the

7.3 Static system validation 163
vectorFa. Since these stiffnesses can be considered linear (section 6.7.4 on page 147), the
following force equilibrium must be satisfied at the actuator locations in the vectorρ:
Fa −Caza −Fρ = 0, (7.2)
where Ca is a diagonal matrix whose i th diagonal element is the stiffness ca of actuator
i and it is assumed that the facesheet deflection at the actuator locations is equal to the
actuator deflectionza.
Since the rigid body modes are not constrained by the free edge condition of the plate, the
moments due to the net plate forces Fρ around the x and y axes and the net force in the
out-of-plane direction should be zero. This leads to the extra condition U T ρ Fρ = 0, where
Uρ is defined similar to Ur as Uρ = [1 Nr×1 Re(ρ) Im(ρ)]. When (7.1) is evaluated only
on the actuator grid – i.e. r = ρ – it can be expressed as za = ΩρρFρ +Uρwpxy, where
Ωρρ = Ωrρ| r=ρ . Together with the rigid body constraint this can be written in matrix form
as:
za
=
0
which can be inverted to:
Fρ
wpxy
=
Ωρρ Uρ
UT
Fρ
,
ρ 0 wpxy
Ωρρ Uρ
U T ρ 0
−1
za
=
0
Km
Kz
za, (7.3)
in which the matrices Km and Kz are implicitly defined. Substitution of this result for Fρ
into the force equilibrium in (7.2) then yields:
Fa −Caza −Kmza = 0
and thus the facesheet deflectionza at the actuator positions is related to the actuator forces
Fa as
za = (Km +Ca) −1 Fa, (7.4)
The static forceFa of a certain actuator due to a supplied Pulse Width Modulation (PWM)
voltageVPWM can be expressed as the quotient of the motor constantKa and the total electric
resistance: Fa = Ka/(Ra +R ′ a +Rl +R ′ l )VPWM. This can be written in vector notation
for all actuators as:
Fa = Ka(Ra +R ′ a +Rl +R ′ l )−1 VPWM, (7.5)
where the ith diagonal elements of the diagonal matrices Ka, Ra, R ′ a, Rl and R ′ l are the
values of the corresponding (regularly typefaced) symbols for all actuators i = 1...Na.
Further, the vector VPWM stacks the PWM voltages VPWM of all actuators. Substitution of
(7.5) into (7.4) then leads to:
VPWM, (7.6)
za = (Km +Ca) −1 Ka(Ra +R ′ a +Rl +R ′ l) −1
Bρ
where Bρ is the influence matrix that links PWM voltages to facesheet deflection at the
actuator locations. The plate deflections due to point forces at positionsρ can also be evaluated
over the arbitrary grid with complex coordinate vector r. The results from (7.3) can
7

7
164 7 System modeling and characterization
LVDS
Ethernet
LVDS
bridge
Driver
electronics
UDP
DM
Ethernet
switch
setpoint
UDP
PC
Wyko 400
interferometer
Intelliwave TM
software
Figure 7.11: The measurement setup which is used to measure the influence functions of the DM.
be substituted into the plate equation in (7.1) together with (7.4), yielding:
zf = (ΩrρKm +UrKz)Fa
= (ΩrρKm +UrKz)Bρ VPWM, (7.7)
Bf
whereBf is the influence matrix that links PWM voltages to deflections at an arbitrary grid
of points on the facesheet.
7.3.2 Measurements and results
This section describes the setup and procedure used to measure the mirrors influence functions
and low order Zernike modes.
Interferometric measurement setup
The verification of the static behavior of the DM system is done using an interferometer
setup. A Wyko 400 interferometer available at TNO Science and Industry measures the
surface shape of the DM. Intelliwave TM software is used to perform the reconstruction of
the actual wavefront from the measured fringe patterns. Figure 7.11 shows the schematic
of the measurement setup, where a PC sends desired setpoint commands VPWM via the
ethernet/Low Voltage Differential Signalling (LVDS) communication link to the DM. All
shapes and measurements in the coming sections are considered w.r.t. an arbitrary grid as
determined by the interferometer’s Charge Coupled Device (CCD) camera. Since the interferometer
cannot observe the piston mode corresponding to a non-zero average deflection, it
is assumed that all measurementsˆz are piston-free. When considering the zero-mean, white
measurement noisen, this allows the measurementsˆzf ∈ R Nw to be expressed as:
ˆzf = Pzf +Pn, (7.8)
where the rank deficient matrix P = I −pp T projects out the piston term denoted by the
vectorp whose elements are all equal to1/ √ Nw s.t. p T p = 1.

7.3 Static system validation 165
Assuming the static response of the DM to be linear, let the shape zf of the DM facesheet
be expressed as:
zf = Bf,wVPWM +zf,0,
wherez f,(0) is the initial unactuated shape of the DM facesheet and the matrix Bf,w is the
influence matrix Bf w.r.t. the measurement grid of the Wyko interferometer. Substitution
of this expression for zf into (7.8) then yields the measurement corresponding to a certain
actuator commandVPWM as:
ˆzf = PBf,wVPWM +Pzf,0 +Pn (7.9)
This measurement equation will be used in the following two subsections to estimate the
influence function matrixBf,w and fit the DM facesheet to a desired set of shapes.
Influence function measurements
As described in section 7.3.1, the influence functions are the static responses of the DM to
actuator commands. Analytically they are expressed over the actuator grid by the matrix
Bρ in (7.6) and over an arbitrary grid asBf in (7.7). In this section the method is described
that is used to measure the influence functions of the DM prototypes. The most obvious
method to determine the influence functions is to individually poke each actuator, measure
the response and then compute the influence function. Multiple measurements must be used
per actuator to reduce the measurement noise and at least two different command values
are required to determine the influence function as a linear relation between command and
deflection. More command values and measurements can be used to distinguish any nonlinear
behavior. Each Wyko measurement takes approximately 8 seconds including data
processing. When several commands are used for each actuator and the number of actuators
is large (e.g. 427) the measurements would take several hours to complete. And even then,
each influence function has to be estimated from only a few measurements, still leading to a
high sensitivity to measurement noise. Better and more efficient methods can be used. For
instance, in [92, 109] columns of scaled Hadamard matrices [22] are used as the actuator
command vectors. This setpoint choice will minimize the mean standard deviation of the
estimation error of the influence matrix due to measurement noise and thus requires fewer
measurements.
All elements of a Hadamard matrixQn ∈ Rn×n are either 1 or -1 and the matrix is orthogonal
s.t. QnQT n = nI. Although it is yet unknown whether Hadamard matrices exist for all
n ∈ N + , algorithms are available for specific dimensions. When an algorithm is unavailable
for n = Na, it is argued in [109] that virtual actuators can be added that do not influence
the DM shape, but allow the use of a larger Hadamard matrix of size Nav > Na that does
exist at the cost of additional measurements.
Accordingly, for the 61 actuator DM prototype a 64 × 64 Hadamard matrix is used that
is generated by Matlab’sÑÖfunction. For the 427 actuator DM, the 428 × 428
Hadamard matrix derived in [111] can be used. In the procedure described below, the influence
matrix ˆ Br is estimated that includes influence functions of both the real and virtual
actuators. After estimation, the columns corresponding to the virtual actuators are ignored.
By individual scaling of the rows of the Hadamard command matrix it is possible to compensate
for the lower stiffness at the edges of the DM that would lead to a larger deflection
7

7
166 7 System modeling and characterization
Figure 7.12: The 61 influence functions of the DM prototype shown in figure 7.3. Each influence
function is downsized and placed on the location of the corresponding actuator in the
grid.
than at the center. This is done for optimal use of the measurement range of the interferometer
and avoid large deflections that would exceed the measurement range.
Let measurements be expressed as in (7.9) using actuator setpoints VPWM taken as the
columns of the matrixV that consist of the vectorsv (i) s.t.
V = [v (0) v (1) ... v (Nav)] ∈ R Nav×Nav+1 .

7.3 Static system validation 167
Herev (0) = 0 is a zero voltage command vector that is included to allow direct estimation of
the unactuated shapez (0). Further,v (i) = Λq (i) fori = 1...Nav are the command vectors
based on the Hadamard matrixQNav, whereQNav = [q (1)...q (Nav)]. The diagonal matrix
Λ scales the rows of the Hadamard matrix QNav and provides the mentioned individual
setpoint gains for all actuators. This matrix will be derived from the DM system model.
For the influence functions identification, (7.9) becomes:
ˆz f,(i) = Pz f,(0) + ˜ Bf,wv (i) +Pn (i),
where ˜ Bf,w = PBf,w. Before stating the estimation problem, let all measurements be
expressed in matrix form as:
ˆZf = ˜ Bf,wV+Pz f,(0)1 T + Ñ =
Bf,w ˜ 0 ΛQNav
Pzf,(0) 1 1
X
T
+
Υ
Ñ,
where
ˆ Zf = [ˆz f,(0) ˆz1...ˆzNav],
Ñ = P[n (0) n (1) ... n (Nav)],
and all elements of the column vectors1 and0are equal to 1 and 0 respectively.
The matrixXof unknowns is estimated as:
ˆX = argmin Tr ˆZf −XΥ ˆZ T
f −Υ
X T X T
,
= ˆ
T
ZfΥ ΥΥ T −1
,
= ˆ
−
Zf
1
Nav 1TQT NavΛ−1
1
,
0
1
Nav QT Nav Λ−1
where the second step follows from a completion of squares argument and the last follows
from the orthogonality of the Hadamard matrixQNav. The sought estimate of the influence
matrix forms the first Na columns of ˆ X and that of the unactuated shape Pz f,(0) the last
column.
Although the Hadamard matrix approach has minimal sensitivity to measurement noise, it
assumes linearity of the DM system and does not lead to an overdetermined set of equations
from which to estimate the influence matrix. As a result, neither the quality of the estimate
nor the linearity of the DM can be verified using criteria such as the Variance Accounted
For (VAF). Such information can be obtained by extending the set of measurements, e.g.
by repeating the measurements using scaled versions of the actuator setpoints.
Finally, it should be noted that the same procedure can be used to determine the influence
functions from measurements of a Shack-Hartmann sensor.
Influence function results
The above described procedure is used on the 61 actuator DM prototype, where the command
vector scaling matrixΛwas obtained as:
γwdiag(Bρ)
Λ =
−1
0
,
0 I
7

7
168 7 System modeling and characterization
Here, the bottom-right identity matrix corresponds to the three virtual actuators and is of
size 3 × 3. The scalar γw determines the range of the facesheet deflections and thus the
measurement range of the interferometer. A value γw = 2µm has been used. The matrix
Bρ was computed for this DM from (7.6) by substitution of the relevant parameter values in
tables 5.1, 6.2 and 7.1. Further, diag(·) denotes the diagonal operator that sets all elements
of the matrix between brackets to zero except for the diagonal entries.
The estimated influence functions are shown in figures 7.12, 7.13 and 7.14. Figure 7.12
shows the 61 influence functions downsized and placed on the location of the corresponding
actuator. The facesheet deflection due to a unit voltage increases for actuators near the edge
of the facesheet, which is the result of the decreased stiffness due to the facesheet boundary
and the smaller number of surrounding actuators. As can be expected from a hexagonal
actuator layout, a 60 ◦ symmetry is observed.
Figures 7.13 and 7.14 show the cross sections over two indicated axes of the measured and
modeled influence functions. From the right figure the actuator coupling η can be well
observed as the ratio between the the deflection value at a radius of 6mm (a single actuator
spacing) from the maximum and the maximum deflection value. For the central actuator
this leads to η ≈ 0.52, whereas for the edge actuators this reduces to η ≈ 0.3 due to the
reduced facesheet stiffness at the edge. Although the influence function measured for the
center actuator matches almost perfectly to the one derived from the model, the errors vary
per actuator. This is attributed to the measured variation between actuators in properties that
determine its DC-gain, i.e. motor constantKa, stiffnessca and the total electrical resistance
Ra +R ′ a +Rl +R ′ l .
Zernike mode measurements and results
With superposition of the influence functions, the DM facesheet can be fit to a desired shape.
This shape may be entirely flat, in which case the actuators must compensate for the initial
unflatness of the DM. Here it is only possible to correct for spatial frequencies up to the
Displacement [µm]
2.5
2
1.5
1
0.5
0
Measurements
Model
−20 −10 0 10 20
Radius [mm]
Figure 7.13: The cross-section of the five influence
functions of figure 7.12 along the x-axis.
The thick lines represent the functions as derived
with the model.
Displacement [µm]
2.5
2
1.5
1
0.5
0
Measurements
Model
−20 −10 0 10 20
Radius [mm]
Figure 7.14: The cross-section of nine influence
functions in figure 7.12 along the axis rotated 30degree
counter-clockwise. The thick lines represent
the functions as derived with the model.

7.3 Static system validation 169
Nyquist frequency, which is determined by the actuator spacing. Higher order deformations,
e.g. caused by the shrinkage of the glue that connects the actuator struts to the reflective
facesheet, or initial waviness in the polished facesheet, cannot be corrected.
The flat surface shape corresponds to the first Zernike mode: piston. With the influence
function superposition also higher order Zernike modes are fit. The shape errors can be
minimized for both the Peak To Valley (PTV) and RMS norms, corresponding to theℓ1 and
ℓ2 norms respectively.
Let a shape measurement be denoted by the vector ˆzf as defined in (7.9) and subject to
white measurement noise. Let the command vector ˆ Vℓ for which the difference between
the measured facesheet shape ˆzf and the desired shape zd is minimized w.r.t. an arbitrary
normℓ:
ˆVℓ = arg min ˆzf −zd
V
ℓ
PWM
= arg min PBfVPWM +Pzf,(0) +Pn−zd .
ℓ
V PWM
In practice the PWM voltage is limited to Vmax = 3.3V , leading to the constrained optimization
problem:
ˆVℓ = arg min PBfVPWM +Pzf,(0) +Pn−zd
V ℓ
PWM
subject to −Vmax ≤ VPWM ≤ Vmax.
The effects of the measurement noise n can be reduced by taking ˆz f,(0) as the average of
several measurements. Since the actual influence matrixBf is not known, the productPBf
shall be replaced by the piston removed estimate ˆ Bf .
For the ℓ1 norm (i.e. minimization of the PTV value), the optimization problem becomes a
linear programming problem:
ˆVℓ1 = arg min γ subject to
VPWM ,γ
−γ ≤ ( ˆ BfVPWM +Pz f,(0) +Pn−zd) ≤ γ and −Vmax ≤ VPWM ≤ Vmax.
For the ℓ2 norm, the optimization problem becomes a quadratic programming problem:
ˆVℓ2 = arg min
VPWM ˆ
BfVPWM +Pzf,(0) +Pn−zd
ℓ2
subject to −Vmax ≤ VPWM ≤ Vmax.
However, due to e.g measurement noisenand nonlinear behavior of the DM prototypes, the
DM model in (7.9) with PBf replaced by ˆ Bf will not be correct. The estimated command
vectors ˆ Vℓ1 and ˆ Vℓ2 will thus neither minimize PBfVPWM + Pz f,(0) − zdℓ for the ℓ1
norm, nor for the ℓ2 norm. Therefore, an iterative process with iteration indexmis used to
derive the vector ˆ V (m)
ℓ :
ˆV (m+1)
ℓ
= ˆ V (m)
ℓ
+ arg min
∆V PWM
ˆ Bf∆VPWM +ˆz (m)
f −zd
,
ℓ
7

7
170 7 System modeling and characterization
Figure 7.15: The first 28 Zernike modes made with the DM from figure 7.3. The inset shows the
RMS fitting errors w.r.t. the desired shape for the model and the measurements.

7.3 Static system validation 171
Figure 7.16: The assembled dummy mirror
placed vertically in front of the Wyko 400 interferometer.
Flattening force [mN]
1.5
1
0.5
0
−0.5
−1
−1.5
50 100 150 200 250 300 350 400
Actuator number [−]
Figure 7.17: Forces required to obtain the flattened
shape of the 427 struts glue experiment in
figure 7.18. Actuators are sorted on value for insight
into the statistical spread.
where the minimization is subject to−Vmax − ˆ V (m)
ℓ ≤ ∆VPWM ≤ Vmax − ˆ V (m)
ℓ .
Each new setpoint yields the measurement:
ˆz (m)
f = PBf ˆ V (m)
ℓ +Pz f,(0) +Pn (m) .
Here again the effect of measurement noise can be reduced by taking ˆz (m)
f as the average
of several measurements.
The above procedure has been applied for the 61 actuator DM prototype, using an
∅44mm circular aperture area that corresponds the inscribed circle of the hexagonal actuator
grid. The results for the first 28 Zernike modes with an RMS amplitude of 400nm
are shown in figure 7.15. Since the DM has only a limited number of regularly spaced actuators,
there will be a fitting error. The inset in figure 7.15 shows the RMS errors w.r.t.
a perfect Zernike mode together with the numerically evaluated fitting error of the derived
static model.
Since the latter is not subject to measurement noise, nonlinearities or initial unflatness,
this forms the error contribution only due the limited number of actuators. From the piston,
tip and tilt mode it becomes clear that the higher order unflatness is ≈25nm RMS. This
unflattness can not be compensated by the (limited) number of actuators. For the higher
order Zernike modes this unflattness can result in better or in worse fitting.
Flattening of the∅150mm dummy mirror
Since the 427 actuator DM prototype was broken during the last assembly step, it was not
possible to measure its initial unflatness, static or dynamic behavior. Also the effect of
combining multiple actuator modules into a DM system with a single continuous facesheet
could not be analyzed. However, a dummy mirror (figure 7.16) was assembled from an
identical facesheet and 427 struts glued using the same procedure as used for the broken
7

7
172 7 System modeling and characterization
Figure 7.18: Measured shape in µm of the 427 struts glue experiment (left). Shape in in µm after
correction using hypothetical actuator influence functions (right). The unsupported edge
area of the facesheet is not shown as this provides no insight into the achievable flatness.
DM prototype. The shape of this mirror, measured with the Wyko 400 interferometer is
depicted in figure 7.18 on the left and shows a PTV unflatness of approximately 4.5µm.
Since the results of the above influence function measurements for the 61 actuator DM
indicate that the influence functions are well predicted by the model, it is inferred that the
same is true for the 427 actuator DM. With the influence matrix Bf evaluated with the
parameters from tables 5.1, 6.2 and 7.1, the dummy mirror is flattened fictitiously. The
flattening procedure, described in the previous section, is here not applicable, since the
facesheet is here supported by 427 struts and a rigid reference plate instead of soft actuators.
The shape of interest is the initial unflatness of the facesheet, when it would have been glued
to soft actuators instead of to a stiff base. The soft actuators will deflect due to stresses in the
facesheet, whereas the stiff base does not. Therefore, the flattening must thus be performed
after ’replacing’ the stiff base with soft actuators.
This is analyzed as follows. Let the vector ˆzf,b contain the deflections measured at the
strut positions andˆzf,a the deflections after placing the facesheet on the soft actuators. The
force Fa that the actuators must exert on the facesheet to keep the shape ˆzf,b is obtained
using (7.3) as Fa = Kmˆzf,a. This force is generated by deflection of the actuators from
their measured ˆzf,b positions via the effective actuator stiffness as Fa = Ca(ˆzf,b − ˆzf,a).
Combination of both equations yields:
ˆzf,a = (Km +Ca) −1
T
Caˆzf,b. (7.10)
Observe that T is the influence matrix that links force to deflection and Ca is a diagonal
matrix with the actuator stiffnesses. Since application of the first part to a vector has a lowpass
filter effect and the latter part is only a linear scaling, the transformation from ˆzf,b to

7.3 Static system validation 173
ˆzf,a is a smoothing operation. The strength of this smoothing effect depends on the widths
of the influence functions and thus on the facesheet and actuator stiffnesses.
At this point the fictitious DM with initial unflatnessˆzf,a is flattened w.r.t. an RMS criterion
by applying a set of suitable fictitious actuator forcesFf a. The forcesFf a that minimize the
unflatness w.r.t. the weighting function ˆzf,a − TFa2 F are found using a completion of
squares argument as:
F f a = (TT T) −1 T T ˆzf,a = Caˆzf,b,
where the second step uses the definition ofTin (7.10). The resulting forcesF f a are plotted
in figure 7.17. Note that these forces are independent of the influence functions and just
overcome the actuator spring forces corresponding to the deflectionˆzf,b. Figure 7.18 shows
the hypothetical unflatness that remains when fictitious, soft actuators exert the force F f a .
For this dummy mirror the residual RMS unflatness is approximately 20nm, requiring an
RMS actuator force of 0.4mN. With the average values of the actuator properties Ka, Ra,
Rl as measured in the previous chapter, the latter corresponds to an RMS actuator voltage
of 0.17V which is approximately 5% of the available maximum voltage.
7.3.3 Power dissipation
Based on the measured influence functions and flattening commands, the expected average
power dissipation to correct atmospheric wavefront distortions, is estimated. For realistic
results, a trade-off is made between power dissipation and performance in terms of fitting
error. The RMS power Pa, dissipated by the actuators, can be expressed in terms of the
actuator voltage setpoint and the total resistanceRa+R ′ a +Rl+R ′ l estimated in sections
6.7.2 and 6.7.3 as:
Pa = (Ra +R ′ a +Rl +R ′ l )−1 V 2 PWM (t) , (7.11)
where (·) 2 denotes the element-wise square and t the time. The actuator voltages consists
of the static voltagesVf required for the initial flattening of the DM and the dynamic voltages
Vd(t) required to correct atmospheric wavefront disturbances. The value V 2 PWM (t)
in (7.11) can be expressed accordingly as (Vd(t)+Vf) 2 . This reduces to V 2 d (t) +V 2 f
when the atmospheric wavefront distortion and thusVd is a zero-mean signal that is therefore
not correlated with the constant signal Vf . Application of these simplifications to
(7.11) yields:
Pa = (Ra +R ′ a +Rl +R ′ l )−1V 2 d (t) +V 2 f . (7.12)
An estimate for the vector Vf with flattening voltages is obtained from the measurements
described above. To quantify V2 d (t) , consider the atmospheric wavefront disturbance
to have a Von Karmann spatial spectrum (section 2.1.2) with covariance matrix
Cφ = φ(t)φ T
(t) . Here the vectorφ(t) denotes the wavefront distortion over a fine grid
over the telescope aperture in radians. The matrix Cφ is numerically approximated using
the approach described in [95] with the modification that the Kolmogorov structure function
has been replaced by the Von Karmann structure function [98] corresponding to the power
spectrum given in (2.4) on page 28:
D vk
5
3
1/3 2
7/3
r r r r
φ (r) = 6.88 1−1.485 +5.383 −6.381 .
r0 L0 L0 L0
7

7
174 7 System modeling and characterization
The resulting covariance matrix is an approximation, since the continuous spatial integrals
are replaced by numerical sums over a discrete grid of points within the telescope aperture.
The incoming wavefront is corrected by reflection on the DM. Let the DM shape zf(t) be
expressed by (7.6), based on the estimated influence matrix ˆ Bf . Although inertial forces
are neglected, the static model accurately describes the DM facesheet deflections, since its
first resonance frequency lies far above the control bandwidth. A fine grid is used to model
the wavefront distortions and the DM influence functions. Hereby a realistic estimate of
the fitting error and power dissipation is made. When considering the open-loop controlled
case, let the actuator voltages be chosen as the minimizing argument of a quadratic cost
function that weights both fitting error and control effort. Let the fitting error, in meters, be
expressed as:
efit(t) = λ
2π φ(t)− ˆ BfVPWM(t)
where λ is the wavelength of the incoming light. The optimal actuator command vector
Vd(t) is chosen as:
efit(t) Vd(t) = arg min
VPWM (t)
T efit(t) 2
+γ
F
V T d (t)2
F
= λ
2π ˆ B +
f φ(t), (7.13)
where ˆ B +
f = (ˆ B T f ˆ Bf +γ 2 I) ˆ B T f is a regularized pseudo inverse of ˆ Bf andγ is a weighting
factor for the control effort. For this command vector the fitting errorefit(t) becomes:
f )
ˆB
−
f
φ(t) (7.14)
efit = (I− ˆ Bf ˆ B +
and the command signal covariance matrixCVd can be expressed as:
CVd = Vd(t)V T d(t) =
λ 2 ˆB +
2π f Cφ
T ˆB +
f ,
where the second step follows after substitution of (7.13). The diagonal elements of the
covariance matrix CVd now form the vector V2
d . Similarly, using (7.14) the fitting error
covariance matrixCfit can be expressed as:
Cfit = efit(t)e T fit (t) =
λ 2 ˆB −
2π f Cφ
from which the RMS fitting errorσfit,d can be derived as:
σfit,d =
Tr(Cfit)
,
nf
T ˆB −
f ,
where nf is the number of grid points used. To provide insight in the power dissipation,
based on measurements on a 61 actuator prototype, several parameters must be scaled.
Firstly, in chapter 2 the number of actuators for an 8 meter telescope is 5000, which implies
that for the same actuator density, the 61 actuator DM prototype should be used on a 0.9m
telescope. Secondly, a continuous facesheet type DM can only prescribe the slope of the

7.3 Static system validation 175
Power [W]
10 −2
10 −3
10 −4
10 −5
10 −8
10 −6
10 −7
RMS fitting error [nm]
Illuminated area
Edge
Uncorrected error
10 −6
Figure 7.19: Relation between the fitting error
σfit,d and the average power dissipation per actuator
based on the measured influence matrix of
the 61 actuator DM prototype. Power dissipation
is differentiated between actuators in- and outside
the ∅32mm illuminated area (λ=550nm).
Power [W]
10 −2
10 −3
10 −4
10 −5
10 −8
10 −6
10 −7
RMS fitting error [m]
Illuminated area
Edge
Uncorrected error
10 −6
Figure 7.20: Relation between the fitting error
σfit,d and the average power dissipation per actuator
based on the modeled influence matrix of
a 427 actuator DM. Power dissipation is differentiated
between actuators in- and outside the
∅102mm illuminated area (λ=550nm).
Table 7.2: The expected average power dissipation in mW per actuator based on the influence function
measurements for a 0.9m telescope using a ∅32mm illuminated area on the DM and a
Von Karmann spectrum with a Fried parameter r0=0.16m (λ=550nm) and an outer scale
L0=100m.
Atmospheric Flattening Total
turbulence
Edge 1.5 23.8 25.3
Illuminated area 1.4 5.5 6.9
Full area average 1.4 14.5 15.9
facesheet at the aperture edge if there is at least one ring of actuators outside the illuminated
aperture. Therefore, an illuminated diameter of 32mm will be considered over which
the Von Karmann spectrum is to be corrected. Actuators outside this area are only used
for flattening and for prescribing the boundary conditions. By varying the control effort
weighting factor γ and assuming a wavelengthλ = 550nm, evaluation of (7.12) yields the
relation between fitting error and power dissipation shown in figure 7.19. When a fitting
error of 35nm is taken, which is 2.5% above the best achievable value, the corresponding
average power dissipation of the actuators is listed in table 7.2. A distinction is made
between actuators in the illuminated area and those at the edges. Also the two causes for the
dissipation – i.e. the correction of wavefront errors due to atmospheric turbulence and the
correction of the initial DM unflatness – are stated separately. The dissipation for actuators
in the illuminated area required for atmospheric wavefront correction is approximately
1.4mW. Although this is less than the 3.2mW estimated in section 6.8, it depends on the
chosen regularization factor γ. Further, the dissipation required for flattening is larger than
7

7
176 7 System modeling and characterization
Table 7.3: The expected average power dissipation in mW per actuator based on the influence function
measurements for a 2.3m telescope using a ∅102mm illuminated area on the DM and a
Von Karmann spectrum with a Fried parameter r0=0.16m (λ=550nm) and an outer scale
L0=100m.
Atmospheric Flattening Total
turbulence
Edge 0.2 0.4 0.6
Illuminated area 0.6 0.2 0.8
Full area average 0.4 0.2 0.6
the functional dissipation, due to significant initial unflatness of the DM, especially at the
edge of the DM.
For the ∅150mm dummy DM a similar table is derived based on an illuminated area
of ∅102mm. This corresponds to the largest diameter, which is fully filled with actuators,
as can be observed in figure 7.23. With the same actuator density as the 5000 actuator DM
proposed for an 8m class telescope, this DM is suitable for a 2.3m wavefront. The forces as
derived in section 7.3.2 are assumed to be indicative of the forces required to compensate
the initial unflatness of the ∅150mm facesheet. Figure 7.20 shows the trade-off between
performance and effort for this DM and table 7.3 shows the estimated power dissipation for
the same fitting error σfit,d of 35nm as used for the 61 actuator DM. The table shows that
both the power dissipated to correct the initial unflatness as well as the power dissipated to
correct the atmospheric wavefront distortions with a Von Karman spectrum is smaller than
for the 61 actuator DM. However, this only holds true for the regularization factor chosen.
7.4 Dynamic system validation
In section 7.3 the static behavior of the DM system was validated. In this section the dynamic
behavior will be added to the DM facesheet model and combined with the actuator
models. The resonance frequencies, mode shapes, modal damping and transfer functions
from this model will then be compared to a black-box model, identified from measurement
data, using the PO-Multivariable Output-Error State-sPace (MOESP) subspace identification
algorithm.
7.4.1 Dynamic modeling
There is no known analytic solution available for the biharmonic plate equation for a circular
plate including inertia and viscous damping terms subjected to multiple point forces. The
dynamic behavior can be modeled using a Finite Element Model (FEM) approach, but here
it will be done by combining the derived models for the actuators and the facesheet and
extending this with lumped masses and dampers.
Let the force equilibrium in (7.2) be extended accordingly to:
Fa −Fρ −Caza −Ba˙za −Maf¨za = 0, (7.15)

7.4 Dynamic system validation 177
where each (i,i) element of the diagonal matrices Ba and Maf are the viscous actuator
dampingba and the sum of the lumped facesheet mass and the moving actuator mass at coordinateρi
respectively. Note that for simplicity it is assumed that the actuator and lumped
mass/damper locations coincide, but this can be generalized by using an arbitrary grid and
appropriately attributing mass, stiffness and damping values. The mass distribution chosen
leads to adequate approximations of the mode shapes with spatial frequencies significantly
below the Nyquist frequency of the actuator grid, which corresponds to the lower eigenfrequencies.
These are the most relevant for the achievable correction quality, since they pose
the tightest limit on the achievable control bandwidth. Moreover, the stiffness matrix used
corresponds to the solution of the biharmonic plate equation under the assumption of pure
bending. For high spatial frequencies, shear forces become dominant and this assumption
loses its validity.
When the result in (7.3) for Fρ is substituted into (7.15) and transformed to the Laplace
domain this yields the dynamic system:
Caf +Bas+Mafs 2 za = Fa, (7.16)
where Caf = Km + Ca. The undamped mechanical eigenfrequencies fe,(i) and mode
shapesx (i) fori = 1...Na can be obtained by solving the generalized eigenvalue problem:
Caf −λ (i)Maf x(i) = 0,
wheref e,(i) = λ (i)/2/π.
This procedure has been performed for two different cases. Firstly for a DM with regularly
placed actuators with 6mm pitch in a hexagonal pattern. Equal lumped masses were added
at all actuator grid points and all actuator and facesheet properties were used as given in
tables 5.1, 6.2 and 7.1. The first 100 resonance frequencies for this model are plotted as
circles in figure 7.21 and the first twelve resonance modes are shown in figure 7.22. From
figure 7.21 it is clear that the first resonance modes occur in a small frequency band and
from figure 7.22 it is clear that the modal shapes correspond very well to the Zernike polynomials
[136]. Traditionally, these polynomials are used to describe both the aberrations in
the optical domain as well as the dynamic modes of the wavefront corrector in the mechanical
domain.
However, the resonance frequencies and corresponding modal shapes are influenced by the
edge conditions of the reflective facesheet. When considering the 427 actuator DM prototype
that consists of seven hexagonal actuator modules, observe that the gaps between
the hexagons at the outside have no actuators that support the mirror facesheet. In the dynamic
model in (7.16) leads to ’zero’ elements on the diagonals of the matrices Ca and
Ba and to lower values of the corresponding diagonal elements in Maf due to the absence
of moving actuator masses. The lack of support stiffness in the edge areas leads to lower
resonance frequencies with local mode shapes. To properly attribute lumped mass fractions
of the facesheet to grid points in edge areas, its mass is distributed based on the Voronoi
diagram of the grid points (figure 7.23). The Voronoi diagram is the dual of the Delaunay
grid triangulation [130, 159] that is frequently used to draw a surface defined at arbitrary
grid of points. The Voronoi diagram creates a polygon area around each grid point in which
all points are closest to that particular grid point. For most grid points this yields a closed
polygon, but for edge points this is open towards the edge. The polygon area determines the
7

7
178 7 System modeling and characterization
Frequency [Hz]
1800
1600
1400
1200
1000
800
600
400
200
427 actuators in seven grids of 61
559 actuators in a full hexagonal array
20 40 60 80 100
Mode number [−]
Figure 7.21: The lowest 100 undamped mechanical
resonance frequencies.
Figure 7.22: The lowest 12 undamped mechanical
resonance modes corresponding to the frequencies
plotted as circles in figure 7.21.
lumped fraction of the facesheet mass and can for closed polygons be computed using Matlab’sÔÓÐÝÖfunction
[130]. The open edge polygons are first extended with two points
on the circular facesheet edge by which the polygon is closed. The total area is then the sum
of the area of the artificially closed polygon and the area between these two points and the
circle, which follows from the distance between the two additional edge points. Since all
areas can be calculated analytically, no approximations are made and the summed area for
all grid points exactly equals πr2 f . For the 427 actuator DM prototype the result is plotted
in figure 7.23.
The lowest 100 eigenfrequencies for this case are plotted in figure 7.21 as the solid dots.
The first resonance frequencies are lower than in the homogenously supported case and
the corresponding modal shapes are local bending modes of the unsupported edge areas.
Clearly, the actuator layout of future DMs should be chosen such that the facesheet edges
are uniformly supported.
Figure 7.23: Voronoi diagram for the hexagonally
arranged grid points marked with a small
dot. The actuator grid of the 427 actuator DM
prototype is marked with ”o”.

7.4 Dynamic system validation 179
Magnitude (abs)
Phase (deg)
10 −5
10 −10
180
0
−180
−360
−540
10 1
Central actuator
To first neighbor
To second neighbor
To third neighbor
10 2
Frequency [Hz]
Figure 7.24: Bode plots of the modeled transfer
functions between the PWM voltage of the central
actuator and the position of itself and three
neighbors.
10 3
Displacement [µm]
2.5
2
1.5
1
0.5
0
Central actuator
To first neighbor
To second neighbor
To third neighbor
−0.5
0 10 20 30 40 50
Time [ms]
Figure 7.25: Step response of the modeled transfer
functions between the PWM voltage of the
central actuator and the position of itself and
three neighbors.
Although the mechanics largely determine the behavior of the DM in terms of resonance
frequencies, the electronics influence the damping, response time, gain, etc. Therefore, the
state-space description for the single actuator behavior in (6.4) on page 136 is extended to
describe the behavior of multiple actuators and combined with the mechanical equations of
motion in (7.15). The scalar statesIRl ,Ia,VCl ,za and ˙za become the vectorsIRl ,Ia,VCl ,
za and ˙za whose ith elements contain the corresponding states of the ith actuator. Similarly,
the scalar parametersKa, Ra, Rl, La, Ll andCl for each actuatoribecome the(i,i)
diagonal elements of the square diagonal matricesKa,Ra,Rl,La,Ll andCl respectively.
Further, instead of the mechanical equation of motion for the single, uncoupled actuator in
(5.24), the dynamic equation in (7.16) will be used, leading to:
⎡ ⎤
˙Ia
⎢ ˙zf ⎥
⎢ ⎥
⎢¨zf
⎥
⎢ ⎥
⎣ ˙Va
⎦
˙IRl
=
⎡
−L
⎢
⎣
−1
a Ra 0 −L−1 a Ka L−1 a 0
0 0 I 0 0
M −1 −1 −1
afKa−M afCaf−MafBa 0 0
−C −1
l 0 0 0 C −1
l
0 0 0 −L −1
l −L−1
l Rl
⎤⎡
⎤
Ia
⎥⎢
zf ⎥
⎥⎢
⎥
⎥⎢
˙zf ⎥
⎥⎢
⎥
⎦⎣Va⎦
IRl
+
⎡
0
⎢ 0
⎢ 0
⎣ 0
L −1
⎤
⎥
⎦
l
Afm
Bfm
VPWM
(7.17)
The transfer matrixH(s) between the PWM voltageVPWM and the facesheet deflectionzf
can be expressed accordingly as:
H(s) = 0 I 0 0 0 (sI−Afm) −1 B T fm.
The DC-gain of this transfer matrix is equal to (7.6) and can be derived by evaluatingH(s)
fors = 0:
H(0) = − 0 I 0 0 0 A −1
fmBTfm = C−1
afKt(Ra +Rl) −1 = Bρ, (7.18)
7

7
180 7 System modeling and characterization
whereAfm andBfm are defined in (7.17) and the final expression follows from the definition
ofBρ in (7.6).
The model in (7.17) has been generated numerically for a∅150mm reflective facesheet that
is regularly supported by 6mm spaced actuators over its entire area. The actuator, electronics
and facesheet properties used are given in tables 5.1, 6.2 and 7.1. In figure 7.24 Bode
plots are shown of the entries of the transfer matrix corresponding to the PWM voltage of the
central actuator and the position of itself and three neighbors at 6, 12 and 18mm distance. It
shows that the static DC response to neighboring actuator positions indeed decays rapidly
with the spatial distance, but that the global shapes of the lightly damped, lowest dynamic
modes (figure 7.22) lead to a strong coupling between the actuators at high frequencies. To
illustrate this low damping, figure 7.25 shows the step response of the same actuators due
to a step input at the PWM voltage of the central actuator. In practice the damping will be
higher due to the presence of air above the facesheet, intrinsic damping of the facesheet and
strut materials, deformation of the glue between the struts and the facesheet, etc. To quantify
this effect, the relative damping of the resonant modes of this model will be compared
to the damping derived from a modal analysis in section 7.4.3. The consequences of these
observations for control performance will be discussed in section 7.5.
7.4.2 System identification
Modal (or structural) analysis is frequently performed using the Eigensystem Realization
Algorithms (ERAs) [15, 60] and its variants [116]. The ERAs are a subspace based identification
methods that estimate a state space by taking the Singular Value Decomposition
(SVD) of a Hankel matrix of the system’s impulse response function. This impulse response
function is either measured directly from impulse excitation or estimated from more generic
input-output data. For open-loop measurement data of Multi-Input Multi-Output (MIMO)
systems the MOESP algorithm [183] and its variants [186] are very suitable. For closedloop
identifications the Predictor Based Subspace IDentification (PBSID) identification algorithm
[29, 31] can be used, which has been applied for the identification of a DM with
60 actuators and 104 sensors of the MAD (Multi-conjugate Adaptive-optics Demonstrator)
system in [30]. However, the DM can here be identified in open-loop, making the
added complexity of the PBSID algorithm superfluous. Besides subspace based identification
algorithms that use state-space parameterizations, other algorithms can be used for
MIMO system identifications with other parameterizations. For instance, in [170] a MIMO
Transfer Function (TF) parametrization is shown to be very efficient in both the number of
parameters and the required computational effort, when compared to subspace algorithms.
The identification algorithm used for modal analysis of the DM prototype is chosen based
on several requirements. It must be able to deal with 61 simultaneous inputs and at least
as many outputs. Since the sampling frequency of the setup used is limited to 10kHz, the
dynamics of the electronics that become dominant above 5kHz will not be well observable
from the measurement data. Although the DM facesheet dynamics are of infinite order,
the low frequent resonance modes can be adequately described using a limited number of
lumped masses. This means that the system order required for identification is at least twice
the number of lumped masses, which in this case is in the order of hundreds. It also means
that the model parametrization used must allow the large number of poles and zeros to be
independent to properly describe the numerous resonance modes of the DM facesheet. The

7.4 Dynamic system validation 181
poles will be used after the identification step to compute the resonance frequencies, their
relative damping and the modal shapes.
Since for identification of a state-space model a high state dimension and large numbers of
in- and outputs are required, the identification algorithm used must be efficient w.r.t. both
memory and computation steps. Moreover, the method must be suitable for the available
measurement setup, which is the same as used previously to identify the behavior of single
actuators and depicted in figure 6.14. In this setup, the deflection of the DM facesheet can
be measured only at a single point at a time with a Polytec laservibrometer. The obtained
measurement data is thus expected to be significantly corrupted both by measurement and
process noise, since the measurements are performed in a noisy environment without vibration
isolation facilities. The identification algorithm must be robust for these types of noise.
On the other hand, since all quantization is performed in the control PC in an open-loop
setting there is no quantization noise.
Unbiasedness to noise can be achieved using instrumental variable techniques [119, 170,
186] that e.g. exploit the fact that the excitation signal is uncorrelated with measurement
or process noise but correlated with the future system output. Instrumental variables can be
used in both the MOESP algorithm – e.g. PI-MOESP uses past inputs and PO-MOESP uses
also past outputs as instrumental variables – and the TF method in [170], which uses future
inputs as instrumental variables.
The method in [170] assumes a model in a canonical MIMO TF form of which it is argued
by the authors that – in general – it has a smaller number of unknown coefficients than
state-space parameterizations. The parametrization is based on a denominator polynomial
with scalar coefficientsak and a numerator polynomial with matrix coefficientsBk:
n
aky(t−k) =
k=0
n
Bku(t−k)+
k=1
n
akv(t−k),
wherea0 = 1 andv(t) ∈ N(0,Cv) is white measurement noise. However, when the number
of in- and outputs as well as the degree of any irreducible matrix fraction description
of the system become large, the number of parameters to be identified for this parametrization
becomes larger than that of a corresponding state-space parametrization. Since this is
the case for the system to be identified, the PO-MOESP algorithm will be used for system
identification.
Identification using the MOESP algorithm
The MOESP algorithm is a subspace identification algorithm that uses a QR decomposition
[77] of the input-output data matrix to compress the data and thus improve the computational
efficiency. The algorithm and its variants are found in literature [184–186] and Matlab implementations
are readily available. Since the identification is subject to both measurement
and process noise, the PO-MOESP algorithm will be used, which uses past inputs and outputs
as instrumental variables to provide unbiased estimates w.r.t. measurement and process
noise. Since in literature this variant of the algorithm is generally intended when referring
to MOESP, the prefix PO- will here also be neglected. For the MOESP algorithm, the data
k=0
7

7
182 7 System modeling and characterization
generating system is described in innovation form as:
x(t+1) = Ax(t)+BVPWM(t)+Ke(t)
ˆzf(t) = Cx(t)+DVPWM(t)+e(t)
where e(t) ∈ N(0,Ce) expresses the effect of both process and measurement noise. The
Kalman gainKis not of interest here and will not be included in the estimation procedure.
Furthermore, the system has at least one sample delay, hence the direct feed-through term
D is assumed to be zero.
The MOESP algorithm estimates a minimal realization of a system by determining the
column space of an estimate of the system’s extended observability matrix. To be able to
determine this column space, the rank of this matrix must be equal to the system order n,
which for an arbitrary observable Linear Time-Invariant (LTI) system requires it to have at
least n block-rows. Consequently, the number s of block-rows of the data Hankel matrix
used in the MOESP algorithm should also be larger or equal to the chosen system order n.
Due to the large number of inputs and outputs together with a large number of recorded
samples, for the DM system prototypes this would lead to huge memory requirements
beyond the specifications of the available computers. To relax this requirement, note that
for systems with multiple outputsl the rank of the observability matrix may become equal
to the system order n for less than n block-rows with a minimum set by l ·s > n. This is
not unlikely for the system at hand, since the number of outputs is large compared to the
system order when considering lumped masses to be located at actuator positions only. For
the DM system the minimum possible number s of block rows to be used for the MOESP
algorithm is then around 5: the number of block-states in (7.17). However, since in reality
the reflective facesheet forms an infinite order system that was modeled using a finite
number of lumped masses, choosing a suitable numbers of block-rows is not trivial.
After identification, the quality of the obtained system is evaluated by applying the
estimated system realization to a validation data sequence and computing the VAF, defined
as: ⎡
⎤
VAF =
⎢
⎣1−
(ˆzf(t)−˜zf(t)) T (ˆzf(t)−˜zm(t))
ˆz T f (t)ˆzf(t)
N
t0
N
t0
⎥
⎦·100%,
where ˜zf(t) is obtained by simulating the identified model for the known excitation signal
VPWM(t) and ˆzf(t) is the vector of facesheet deflections measured by the laser vibrometer.
The VAF represents the fraction of the signal variance that is accounted for by the model
and should be close to 100%.
7.4.3 Modal analysis
The modal analysis of the identified system begins with a modal decomposition of its Amatrix.
This decomposition is such thatΛÂ = MÂ, where the diagonal matrixΛcontains
the (complex) eigenvalues of and the columns ofMcontain the (complex) eigenvectors.
The matrix M forms a state transform matrix that diagonalizes the system and yields a

7.4 Dynamic system validation 183
state-space description with statex(t) whose state transition matrix is equal toΛ:
˜x(t+1) = Λ˜x(t)+M −1 Bu(t),
zf(t) = CM˜x(t),
where the influence of process and measurement noise is neglected. For the discrete time
system, each eigenvalue λ (i) on the diagonal of Λ is related to the resonance frequency
ω n,(i) and relative dampingζ (i) of modei as [62]:
λ (i) = e Ts
−ζ (i)ωn,(i)±jωn,(i) 1−ζ2
(i) = |λ (i)|·∠λ (i),
whereTs is the sampling frequency and
|λ| = e −Tsζ (i)ω n,(i), ∠λ = e ±Tsjω n,(i)
1−ζ 2
(i) .
Inversely, the resonance frequenciesω n,(i) and damping ratio’sζ (i) can be computed as:
ω n,(i) = ln∠λ−ln|λ|
Ts
ln|λ|
and ζ (i) = − . (7.19)
ln|λ|−ln∠λ
Further, the vectors of the matrix CM form the facesheet shapes corresponding to the system’s
eigenfrequencies. Since complex poles occur in conjugate pairs (a,b) with the same
modal frequency ωn,a = ωn,b, the matrix’s complex part and columns corresponding to
complex conjugate eigenvalues will for the modal analysis be ignored.
It should be noted that the estimated system matrices are corrupted by process noise – e.g.
due to external vibrations – and measurement noise in the laser vibrometer and the data acquisition
card. Moreover, they are subject to aliasing effects and an ill-chosen system order.
This leads to errors in the derived eigenvalues and eigenvectors, some of which may be nonstructural
extraneous [60]. Various criteria have been developed to separate the structural
and extraneous modes, e.g. using the concept of modal amplitude coherence (MAC) [60] or
modal dispersion analysis (DA) [57]. Besides these criteria that select modes considering
the estimated system matrices, statistical techniques (e.g. bootstrap or Monte-Carlo) can
be used to estimate confidence intervals for the desired modal parameters [102]. However,
for the MIMO system at hand the computations involved to derive reliable statistics are
extremely time consuming. Confidence in the identified system behavior will therefore be
derived in a qualitative manner from the application of the PO-MOESP algorithm with variations
in the number of block rows s and the obtained VAF values and consistency of the
identified system modes. Also the DC-gain of the identified realizations should match the
influence matrix previously identified in section 7.3.2 at the laser-vibrometer measurement
locations.
Results
For the 61 actuator DM prototype, the facesheet response is measured on the 79 points
shown in figure 7.27. These are the 61 actuator locations and 18 points on the facesheet
along its edge. The signal acquisition and DM setpoint update rate is chosen as high as
possible to minimize the effects of sampling and aliasing. The 10kHz sampling frequency
7

7
184 7 System modeling and characterization
Table 7.4: VAF values obtained for the PO-MOESP algorithm on the laser-vibrometer measurement
data of the 61 actuator DM prototype for various values of the number of block-rows s.
s [-] 8 9 10 11 12 13 14 15 16 17
n [-] 105 108 110 112 109 108 108 110 113 113
VAF [%] 93.3 93.7 93.9 94.0 94.1 94.1 94.2 94.4 94.4 94.4
used, is close to the upper limit of the serial communication chain but still significantly lower
than the PWM actuator voltage base frequency. The laser vibrometer is pointed at each
grid location for 10 seconds, producing 100.000 measurements. A zero-mean, bandlimited,
white noise sequence VPWM(t) ∈ N(0,σ 2 e I) is generated with σe = 0.13V and t =
0...10s and applied on each location. Except for small variations in initial conditions and
timing (i.e. jitter), the obtained data is equal to the data that would have been obtained when
the response of all points is measured simultaneously.
Before applying the MOESP system identification algorithm, the suitability of using a small
number of block rows is tested by evaluating the model in (7.17) for properties of the system
to be identified. The properties of all 61 actuators are taken equal to the averages of the
identified values (table 6.4) and lumped masses are assumed to be located at all points in
the measurement grid (figure 7.27). According to (7.17) this yields a state-space system of
ordern = 2·79+3·61= 341, which consists of twice the number of lumped masses (2·79)
and three times the number of actuators (3·61). The system is subsequently discretized to a
10kHz sampling frequency assuming a Zero Order Hold (ZOH) input using Matlab’s
function. The rank of the observability matrix of the resulting system reaches the system
order n for six block-rows with a condition number of 10 8 , which indicates the suitability
of small numberssof block-rows for the MOESP algorithm.
However, the available efficient implementations of the MOESP algorithm do not allow to
chooses smaller than the system ordernsuch that a customized implementation is required.
To reduce memory requirements, batch-wise computation of the economy size R-factor of
the data Hankel matrix is used together with the efficient algorithm described in [185] for
estimating the matrixBfrom the already computed R-factor and corresponding SVD.
The obtained measurement data set is split into an identification set of 85.000 samples and
a validation set of 15.000. The MOESP system identification method is applied to the first
part of this input-output data and the VAF value is computed after simulation on the second
part. The number of block rows s is varied between 8 and 17 and the model order n is
chosen from the singular values of the data Hankel matrices as the break-point of the initial
negative slope visible in figure 7.26, which shows the singular values for two instances of
s. The singular values are normalized to the largest one to clarify that the breakpoint of
the initial slope hardly changes with the number s of block-rows used. The VAF-values
obtained for the PO-MOESP algorithm with various values of s are listed in table 7.4 and
are generally around 95%. Despite the presence of process and measurement noise, the
derived models are consistent with the measurement data.
Figures 7.28 and 7.29 show the first 12 resonance frequencies and the corresponding
modal shapes derived from the analytic model in (7.17) and the black-box model identified
with the MOESP algorithm. For the modal analysis of the analytic model, the average
measured actuator properties listed in table 6.4 are used. When all actuators have equal

7.4 Dynamic system validation 185
properties, the modal shapes show a high degree of symmetry (figure 7.28). In practice the
actuator properties vary, leading to the asymmetric mode shapes in figure 7.29. The lowest
resonance mode of the system lies at ∼725Hz and corresponds to a motion of the lowerleft
edge area of the facesheet. This frequency is lower than expected. Since the edges are
supported by a few actuators, the low resonance may be caused by a lower stiffness ca of
Singular value [−]
10 0
10 −1
10 −2
10 −3
s = 8
s = 17
50 100 150 200 250 300 350 400
Singular value index [−]
Figure 7.26: Singular values of the data Hankel
matrix in the PO-MOESP algorithm normalized
to the largest one. The drop around the 100 th
singular value forms an indication of the system
order.
Figure 7.28: The first 12 modal shapes derived
from the analytic model in (7.17) using the average
actuator properties listed in table 6.4 and
the facesheet properties listed in table 7.1.
66
19
67
18
20
14
65
15
64
10
68 17
16 12
13
8
9
4
5 63
36
69
46
11
6
7
2
3
26
31
62
37
45 1 21 32
70 51 44 41 27 38 79
50 43 22 33
56
71
55
49
48
42
23
28
34
39
78
61
72 60
54
53
47
24
29 40
35 77
59
73
58
74
52
57
30
76
25
75
Figure 7.27: The grid of points at which the
response of the 61 actuator DM prototype was
measured. Points 1...61 correspond to the actuator
locations.
Figure 7.29: The first 12 modal shapes derived
with the MOESP system identification method for
s = 17.
7

7
186 7 System modeling and characterization
only a few actuators. The relative damping ζ has been computed for the identified system
modes using (7.19) and plotted against the eigenfrequency for the models identified using
MOESP for various values ofsin figure 7.30. Since for most eigenvalues of the system the
corresponding eigenfrequencies, relative damping and modal shapes are independent of the
value of s, these estimates are considered to be reliable. However, note from the figure that
this is not the case for all eigenvalues and these results have therefore not been included in
figure 7.29.
Figure 7.30 also shows the relative damping for the analytical model based on the average
actuator properties listed in table 6.4. The relative damping for this model is significantly
lower than for the identified black-box models, which means that the damping observed
in the DM system is not entirely due to the actuators. This suggests the presence of other
dissipative processes such as intrinsic material damping in the facesheet material, glued
connections or damping due to the movement of air above the facesheet. The latter is a very
likely explanation, since the facesheet vibrations due to the noise excitations used were
clearly audible.
A Bode plot of the identified modelfor s = 17 is shown in figure 7.31 for the transfer
functions from the actuator voltage at actuator 2 to the displacement at its first, second and
third neighboring actuators 41, 42 and 47. This shows the high peaks in the magnitude
response as a result of the low relative damping. However, where the magnitude of the
peaks does not decrease with frequency in the Bode plot of the analytical model in figure
7.24, they do in the model as identified, which suggests the presence of additional dissipative
processes at high frequencies.
Further, the influence functions have been derived from the model identified by computing
the DC gain matrix of the system according to (7.18) as Ĉ(I − Â)−1ˆ B whose columns
contain the influence functions. The influence function of actuator 2 is shown in figure 7.32
together with the influence functions of the same actuator derived from the analytic model
Relative damping [−]
10 0
10 −1
10 −2
MOESP (s=8)
MOESP (s=12)
MOESP (s=17)
Analytical model
10
500 1000 1500
−3
Frequency [Hz]
Figure 7.30: Relative damping ζ corresponding
to the eigenfrequencies of the models identified
with the MOESP algorithm for various values of
the number of block-rows s and of the analytical
model from (7.17).
Magnitude [m/V]
Phase [deg]
10 −6
10 −8
0
−180
−360
10 1
Poked actuator
To first neighbor
To second neighbor
To third neighbor
10 2
Frequency [Hz]
Figure 7.31: Bode plot of the model identified
with the MOESP algorithm for s = 17 between
the command voltage at actuator 2 and its first,
second and third neighboring actuators 41, 42
and 47 respectively.
10 3

7.4 Dynamic system validation 187
and the Wyko measurements of section 7.3.2. The shape and magnitude match qualitatively,
but some quantitative error can be observed.
This is partly due to the poor alignment accuracy of the laser vibrometer spot.
Finally, the step response functions derived from the model from the central actuator voltage
setpoint to the displacement of four points on the reflective surface are shown in figure 7.33.
The four points are the location of the actuator itself and that of three neighbors with 1, 2
and 3 actuator spacings distance. In figure 7.34, the corresponding step response derived
from the analytical model is shown for the same actuators. A comparison with figure 7.33
confirms that in the analytical model of the 61 actuator DM damping is underestimated. A
comparison with figure 7.25 on page 179 is also of interest as the settling time of the 427
actuator DM model is significantly shorter than that of the 61 actuator DM. Although this
suggests that the scale of the system has an influence, this difference is caused by differences
in the system properties used to generate the figures. For instance, for figure 7.25 a damping
constant ba = 0.4mNs/m and motor constant ka = 0.19N/A was used, which are higher
than the ba = 0.30mNs/m andKa = 0.11N/A used for figure 7.34. Nevertheless, to better
understand the effect of the number of actuators on the DM system damping, consider the
lowest resonance mode – which has a global modal shape – to form a mass-spring-damper
system. For such a system the relative damping coefficient ζ is related to the damping
coefficientb, massmand resonance frequencyωn as2ζωn = b/m, hence:
ζ = b
.
2mωn
As discussed in [174], the first facesheet resonance frequency (ωn in the above equation)
is scale independent because each increase in facesheet mass is matched by a proportional
increase in stiffness. On the other hand, this means that the modal massmis proportional to
the number of actuators Na. When considering only viscous actuator damping, the modal
dampingb is also proportional to Na such that ζ should not depend on the number of actuators.
Figure 7.32: The influence function of actuator 2 (figure 7.27) as derived from the analytical model
(left), from the Wyko measurements of section 7.3.2 (middle) and from the model identified
using MOESP (right).
7

7
188 7 System modeling and characterization
Displacement [µm]
1.2
1
0.8
0.6
0.4
0.2
0
Central actuator
First neighbor
Second neighbor
Third neighbor
−0.2
0 10 20 30 40 50
Time [ms]
Figure 7.33: Step response of the central actuator
in the model identified with the MOESP algorithm
for s = 17. The response is shown of four
locations: the central actuator and three actuators
at 1, 2 and 3 actuator spacings distance.
7.5 Discrete time control
Displacement [µm]
2
1.5
1
0.5
0
Central actuator
First neighbor
Second neighbor
Third neighbor
−0.5
0 10 20 30 40 50
Time [ms]
Figure 7.34: Step response of the central actuator
of the analytical model in (7.17), using the
average actuator parameters listed in table 6.4.
The response is shown of four locations: the central
actuator and three actuators at 1, 2 and 3
actuator spacings distance.
To investigate the relevance of the high frequent dynamic behavior of the DM for controller
design, the discrete time behavior of the system is of interest. The discrete time black-box
model identified in the previous section can be used for this purpose, but this is a discrete
time model with a 10kHz sampling frequency. Although the data acquisition devices used
to obtain the measurements did not include anti-aliasing filters, the aliasing effects caused
by sampling will have affected the behavior measured below the 5kHz Nyquist frequency.
However, the ZOH filter on the system input has a limited bandwidth and the damping
of DM system resonances increases for high-frequent resonances (figure 7.31), suggesting
that aliasing effects are likely limited. Under this assumption, the identified model can be
resampled to other sampling frequencies without significant error.
It has been assumed throughout this thesis that when the DM system is used in closed-loop
control for the purpose of Adaptive Optics (AO), the sensor used for feedback is a CCDbased
WaveFront Sensor (WFS). As outlined in the introductory chapter, such a sensor
integrates photons over time and thus introduces additional temporal dynamics. In section
6.6 these dynamics were modeled in the continuous time domain and it was shown how
they affect the discrete time system behavior. Discrete time system models for arbitrary
sampling frequencies are here derived by first transforming the identified model back to
continuous time under the assumption of a ZOH input signal. Subsequently, thez-transform
in (6.7) on page 140 is used to obtain a model for the system behavior for a desired sampling
frequency, where the CCD integration time is assumed to be equal to the sampling time
and communication delays are neglected. Note that the z-transform procedure described
in section 6.6 can also be used to obtain the corresponding discrete time behavior for the
analytical, continuous time model in (7.17). However, here only the resampling procedure
for the identified model will be considered. The impulse responses of so obtained models for

7.5 Discrete time control 189
sampling frequencies of 500Hz and 1, 2 and 10kHz are plotted in figure 7.35 for the poked
actuator 2 and its first, second and third neighboring actuators 41, 42 and 47. The figure
shows that the settling behavior significantly improves for a reduced sampling frequency.
In fact, for the design value of the sampling frequency of 1kHz the figure suggests that
the DM system may be well modeled using a low order Finite Impulse Response (FIR)
description. Such a model structure has been previously described in AO literature for
different correctors and successfully used for controller synthesis [100]. In fact, most AO
literature considers the DM to be a static gain without any temporal dynamics other than
Response [µm]
Response [µm]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
f s = 10000Hz
−0.8
0 5 10 15 20 25
Time [ms]
f s = 1000Hz
−1.2
0 5 10 15 20 25
Time [ms]
Response [µm]
Response [µm]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
0 5 10 15 20 25
Time [ms]
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
f s = 2000Hz
f s = 500Hz
Poked actuator
To first neighbor
To second neighbor
To third neighbor
−1.2
0 5 10 15 20 25
Time [ms]
Figure 7.35: The impulse response function of the MOESP identified model for s = 17 resampled
to different sampling frequencies including CCD-type temporal behavior. The poked
actuator is actuator 2 and the response is also plotted for actuators 41, 42 and 47.
7

7
190 7 System modeling and characterization
delays [191].
Now consider the traditional controller structure for AO as described in section 3.2.2. When
disregarding the spatial dynamics introduced by the WFS, this consists of a discrete time
integrator with tunable parameters α and β and the inverse of the static system response
matrixH(0) as in (7.18):
C(z) = α(I−βz −1 I) −1 H −1 (0).
Since the controller contains the inverse of the static response H(0), it diagonalizes the
loop gain for low frequencies. Up to the first resonance mode, the magnitude response of
the discretized plant is highly constant such that cross-coupling can be neglected. For small
values of β, the parameter α is thus equal to the bandwidth defined as the 0dB crossing
of the loop gain. Using this controller for the identified model resampled to 1kHz and
augmented with the temporal dynamics introduced by a CCD-based WFS, a bandwidth
of approximately 60Hz can be achieved. This is significantly less than the design goal of
200Hz (section 2.2.3) and is mainly due to the low damping of the system. The bandwidth
can be increased as suggested in chapter 6 by adding a proportional term to reduce the
phase lag around the bandwidth or by using an optimal controller synthesis approach in
which dynamic models for both the DM system and the atmospheric disturbances to be
suppressed are considered.
Unfortunately, due to lack of time, no results of the closed loop performance of the designed
AO system on either a breadboard setup or an actual telescope can be presented here, leaving
the final system verification for future research.
7.5.1 A note on distributed control
Although throughout this thesis the feasibility of a modularly distributed control system was
investigated, the realized 61 actuator DM prototype has too few actuators to demonstrate the
feasibility and benefit of such a controller structure.
7.6 Conclusions
The assembly of two DMs prototypes is shown: a ∅50mm DM with 61 actuators and a
∅150mm DM with 427 actuators. In the first prototype a single actuator grid is used,
whereas for the second prototype modularity is shown by the assembly of seven identical
grids on a common base. The seven actuator grids, with accompanying dedicated driver
boards, are attached to a single, continuous, facesheet.
The actuator model derived in the chapter 6 is extended with a linear model of the continuous
facesheet, based on an analytic solution of the biharmonic plate equation and point
forces. Lumped masses are added to obtain the dynamic behavior. Both static and dynamic
performance is validated on the∅50mm DM using measurements. Scaled Hadamard matrices
for the actuator voltage command vectors are used to measure the 61 influence functions
in front of a Wyko interferometer. This approach minimizes the variance estimation errors
of the influence matrix due to measurement noise.
The measured actuator coupling for the central actuator is 52%, which is close to the value

7.6 Conclusions 191
obtained from the static DM system model. Nevertheless, variation in accuracy was observed
between actuators, which is attributed mainly to unknown variation in the mechanical
stiffness, motor constant and electrical resistance of the actuators.
The measured shape and amplitude of the influence functions agree with the prediction with
the static model (figures 7.13 and 7.14). This includes the increased static gain (m/V) of
the actuators at the edge of the DM due to the reduced facesheet stiffness from the mirror’s
free-edge boundary condition. The variation in the static gain is observed and attributed to
variation in motor constant, actuator stiffness and electrical resistance and driver circuits.
The influence matrix derived from the measurements is used to shape the mirror facesheet
into the first 28 Zernike modes, which includes the piston term that represents the best flat
mirror. The interferometrically measured shapes are compared to the perfect Zernike modes
and to the Zernike modes as made with the limited number of regularly spaced actuators in
the actuator grid. The total RMS error is ≈25nm for all modes, whereas the inevitable fitting
error varies between 0 and 23nm depending on the mode.
The power dissipation in each actuator of the ∅50mm mirror to correct the Von Karman
turbulence spectrum (D/r0 = 5.4, L0 = 100m) is estimated. Actuators outside the illuminated
area are distinguished from those inside this area. Furthermore, the estimated power
dissipation is split into turbulence correction and mirror flattening. For the turbulence correction,
1.5mW for the outer and 1.4mW for the inner actuators, is dissipated. For static
flattening these values are 23.8mW and 5.5mW respectively.
The 427 actuator DM prototype was broken during placement of its protective cover, therefor
it was not possible to measure its unflatness, static or dynamic behavior. A dummy
mirror (figure 7.16) with similar facesheet specifications, which was used to test the assembly
procedure of the 427 connection struts, was interferometrically measured instead. This
mirror showed≈4.5µm PTV unflatness. This mirror was fictitiously flattened, with the use
of the modeled influence matrix and requires 0.17V RMS actuator voltage, which corresponds
to 5% of the available output voltage range and a power dissipation of 0.2mW per
actuator.
The predicted dynamic behavior of the DM is validated by measurements. A laser vibrometer
is used to measure the displacement of the mirror facesheet, while the actuators are
driven by zero-mean, bandlimited, white noise voltage sequence. Using the MOESP system
identification algorithm, high-order black-box models are identified with VAF values
around 95%. This identified model is compared with the derived analytical model. The
latter uses the average actuator properties as measured in chapter 6. The first resonance frequency
identified is 725Hz, and lower than the 974Hz expected from the analytical model.
This is attributed to the variations in actuator properties, such as actuator stiffness.
The relative modal damping of the model identified is an order of magnitude higher than the
damping in the analytical model, where only actuator damping is considered. The difference
is attributed to the presence air damping and damping in the glue used for the connection
struts. To evaluate the behavior of the realized DM system for discrete-time closed-loop
control, the identified black-box model is resampled to the foreseen AO system sampling
frequency of 1kHz, while considering the temporal dynamics of a CCD-based WFS. When
neglecting spatial dynamics of the WFS and considering a traditional integrator-type control
law a bandwidth of approximately 60Hz can be achieved. This is lower than the design
goal of 200Hz and is the result of the low damping of the system’s resonances. To achieve
a higher bandwidth, either a more complex controller structure should be considered or a
7

7
192 7 System modeling and characterization
more suitable sampling frequency should be chosen for which the system dynamics become
the most desirable. Nevertheless, for future DM designs damping of the system resonances
should be considered an explicit design requirement.

ÔØÖØ
ÓÒÐÙ×ÓÒ×ÒÖÓÑÑÒØÓÒ×
In this chapter the conclusions over all chapters will be summarized and recommendations
for future research will be given.
193

8
194 8 Conclusions and recommendations
8.1 Conclusions
Over the last half a century Adaptive Optics (AO) has provided a solution for the wavefront
distortions introduced by the earth’s atmosphere that limit the resolution of ground based
optical telescopes. To increase this resolution, the size of telescopes must increase together
with the correction quality of their AO systems. This thesis presents the design, realization
and testing of a new Deformable Mirror (DM) system concept. The main design drivers of
this concept are low cost, low power dissipation, low hysteresis and drift and high linearity.
The developed system consists of the DM including the required driver electronics and
control system, but excludes the WaveFront Sensor (WFS) that is assumed to be of the
Shack-Hartmann type. This thesis considers the specific application of the DM system for
an 8m class telescope and discusses extendibility of its design towards larger telescopes.
Detailed design requirements for the DM system are derived based on general assumptions
on the properties of atmospheric wavefront disturbances modeled by Kolmogorov
statistics and the Taylor hypothesis (frozen flow) for representative conditions (r0 = 0.16m
for λ = 550nm and fG = 25Hz). The derivation is driven by a desired optical quality in
terms of a Strehl ratio of 0.85, which is linked to a certain residual wavefront variance. This
residual error is budgeted over the main sources of error that are identified as the fitting error
of the DM due to the limited number of actuators and the temporal error due to the limited
control bandwidth. A trade-off based on foreseen challenges on both fields of competence
leads to a control bandwidth of 200Hz with 5000 actuators having at least 5.6µm stroke and
0.36µm inter-actuator stroke.
The foreseen design consists of a single reflective, deformable facesheet supported via struts
by hexagonally arranged actuators with a pitch of 6mm on a∅500mm DM. The hexagonal
actuator layout is chosen since this gives the highest actuator areal density. The actuators
have a low mechanical stiffness that is only high enough to provide a first mechanical
resonance frequency of approximately 1kHz. This choice not only reduces the required
complexity of a control system and keeps power dissipation to a minimum, but also limits
the detrimental effect of a failed actuator on the DM shape. The actuators are chosen of
the magnetic reluctance type because of their high efficiency, low driving voltage and low
moving mass. Moveover, this type of actuators shows little hysteresis or drift. They are built
as modules of 61 on a stiff base-plate that also serves as a carrier for magnetic flux. These
modules are designed in layers that extend over many actuators to reduce the number of
parts and the complexity of assembly and improve the uniformity of the actuator properties.
Multiple actuator modules are supported by a stiff backing structure. Extendibility of the
design concept towards larger telescopes requiring more Degrees Of Freedoms (DOFs) is
achieved through modularity of the design in terms of mechanics and electronics.
To allow for modularity of the control system hardware, a distributed controller design
is required. A spatially distributed architecture is proposed in which each DM actuator is
matched to a controller node that can only communicate with a small number of nearby
nodes and receives only a local subset of the available WFS measurements. To allow this
architecture to be extendible towards DMs with more DOFs, both the amount of information
exchanged per sample with neighboring nodes as well as the size of the local subset of
the WFS measurement may not vary with the number of DOF.
The computational cost of existing control methods is investigated together with their suit-

8.1 Conclusions 195
ability for implementation on the modularly distributed hardware architecture. This shows
that even for the most efficient methods available the wavefront reconstruction step required
for Shack-Hartmann type WFSs involves a number of computations that is more than linearly
proportional to the number of DOF of the WFS. This renders the wavefront reconstruction
step to be the most critical part in the design of a distributed controller for which
new algorithms need to be developed. Several concepts for such algorithms are explored,
starting from a standard Steepest Descent (SD) solver whose computations are shown to
have a distributed structure. Simulations showed that by extending this solver with an Least
Mean Squares (LMS) based wavefront prediction mechanism, the required number of computations
does not increase with the number of DOFs of the WFS. The distributed dynamic
wavefront reconstructor is also generalized towards a network of output-interconnected controllers
with an Auto-Regressive Moving Average (ARMA) structure. By assuming this
parametrization it is implicitly assumed that the structure of the disturbance generating system
in innovations form is also a network of output interconnected ARMA systems. Performance
loss is thus expected if this is not the case in practice.
Using a two-stage algorithm the unknown controller parameters are identified. Two methods
are presented to enforce stability of the resulting closed-loop. However, both identification
steps must be solved off-line and the first is a centralized operation that involves the measurement
data of all sensors.
Results presented on measurement data obtained from an optical breadboard and on synthetic
data show the performance to depend on the chosen communication radius, but even
for very small radii it is found to exceed that of a random walk baseline strategy. However,
the performance shows a decays w.r.t. the number of DOF of the WFS, which is in contrast
with the SD/LMS based reconstructor and is neither the case for the structured Finite
Impulse Response (FIR) approximation obtained as an intermediate result at the first identification
step. Future research is required to analyze such scaling properties in more detail,
preferably using wavefront disturbance measurements from an actual large telescope.
The second part of this thesis discusses the electromagnetic actuators, the electronics
and the system modeling and validation. The actuators consist of a closed magnetic circuit
in which a Permanent Magnet (PM) provides static magnetic force on a ferromagnetic core
that is suspended in a membrane. This attraction force is influenced by a current through a
coil, which is situated around the PM to provide movement of the core. With the direction
of the current the attractive force of the PM is either increased or decreased, allowing
movement in both directions. The actuators are free from mechanical hysteresis, friction
and play and therefore have a high positioning resolution with a high reproducibility. The
stiffness of the actuator is determined by both the membrane suspension and the magnetic
circuit. There exists a large design freedom for both.
To drive a desired current through the actuator coils the computed command value of the
control system must be communicated to driver electronics. In chapter 6 these two parts of
the required electronics are discussed. Since the inductance-over-resistance time-constant
of the actuator is short (75µs), voltage control is chosen over current control. The motor
constant, stiffness and resistance of the actuator circuit will vary from actuator to actuator
and vary with temperature to cause slow gain variations. Current control would compensate
variations in the resistance, but still leave variations of the motor constant and stiffness for
the AO control system. Pulse Width Modulation (PWM) based voltage drivers are chosen
because of their high efficiency and suitability to be implemented in large numbers with
8

8
196 8 Conclusions and recommendations
only a few electronic components. A drawback of this choice is that suppression of the
PWM ripple requires a second order low-pass filter per actuator. The driver electronics for
61 actuators are located on a single, multi-layer Printed Circuit Board (PCB) and consist
of Field Programmable Gate Arrays (FPGAs) to generate the PWM signals, Field Effect
Transistors (FETs) for the H-bridge switches and coil/capacitor pairs for the 2 nd order
low-pass filters. A serial communication system is chosen based on the Low Voltage Differential
Signalling (LVDS) standard for its low power consumption (15mW/transceiver), high
bandwidth (up to 655Mb/s) and consequently low latency, low communication overhead
and extensive possibilities for customization. A flat-cable connects up to 32 electronics
modules to a custom designed communications bridge, which translates ethernet packages
into LVDS packages and vice versa. The ethernet side of the communications bridge is
connected to the control computer at a speed of 100Mbit/s and uses the User Datagram
Protocol (UDP) protocol to minimize overhead and latency.
Two DMs prototypes were successfully assembled: a ∅50mm DM with 61 actuators
and a ∅150mm DM with 427 actuators. In the first prototype a single actuator grid is used,
whereas for the second prototype modularity is shown by the assembly of seven identical
grids on a common base. All actuators from the seven grids are attached to a single,
continuous, facesheet.
A nonlinear mathematical model of the actuator is derived that describes both its static and
dynamic behavior based on equations from the magnetic, mechanic and electric domains.
This model is linearized to obtain expressions for general actuator properties such as motor
constant, inductance, stiffness and resonance frequency. Frequency response function
measurements are performed on each actuator using a general purpose current source and
a laser vibrometer, showing all actuators to be functional. From these measurements the
motor constant, actuator stiffness and resonance frequency are identified. On average, these
properties deviate slightly from the modeled values, but their statistical spread is small,
stressing the reproducibility of the manufacturing and assembly process. Moreover, the
average actuator stiffness and resonance frequency of 471N/m and 1.83kHz respectively
are close to their design values of 500N/m and 1885Hz. The fact that the measured average
motor constant of 0.12N/A is lower than the modeled value of 0.17N/A, can be partly
attributed to leakage fluxes.
The frequency response measurements are repeated using the custom built communication
and driver electronics. The expected change in behavior is modeled by extending the
actuator model with a pure delay for the communication electronics and a voltage source
with an analog 2 nd order low-pass filter for the driver electronics. Measurement show the
communication latency to be well represented by τc = 89.7·10 −6 +39·10 −6 Nm, where
Nm is the number of the actuator grid. From the frequency response measurements the
actuator properties are again identified, yielding and average a stiffness of 473±46N/m,
a motor constant of 0.11±0.02N/A, a damping of 0.30±0.11mNs/m, an inductance of
3.0±0.2mHand a resonance frequency of 1.83±91Hz. These properties show some
variation between actuators, but this cannot be attributed to the location of the actuator in
the grid.
The time domain response of an actuator to a 4Hz sine voltage over the full stroke shows
hysteresis to be negligible and static nonlinearities in the response of the actuator to remain
below 5% for the intended±10µm stroke. Measurements also showed that in the expected

8.1 Conclusions 197
operating range, the total power dissipation is dominated by indirect losses in the FPGAs
to generate the PWM signals. Solutions are sought in alternative FPGA implementations,
yielding a reduction of 40% in the master FPGA and 29% in the slave FPGAs.
The actuator model is extended with a linear model of the continuous facesheet, based
on an analytic solution of the biharmonic plate equation in presence of point forces. The
static performance is validated on the ∅50mm DM using interferometric measurements.
Scaled Hadamard matrices for the actuator voltage command vectors are used to measure
the 61 influence functions in front of a Wyko interferometer. This approach minimizes
the variance of estimation errors in the influence matrix due to measurement noise. The
measured shape and amplitude of the influence functions agrees with the prediction of the
static, linear model. This includes the increased static gain (m/V) of the actuators at the
edge of the DM due to the reduced facesheet stiffness from the mirror’s free-edge boundary
condition. The variation in the static gain is observed and attributed to variation in motor
constant, actuator stiffness and electrical resistance and driver circuits. The measured
actuator coupling of the central actuators of 52% is close to the modeled value.
The influence matrix derived from the measurements is used to shape the mirror facesheet
into the first 28 Zernike modes, which includes the piston term that represents the best
flat mirror. The interferometrically measured shapes are compared to the perfect Zernike
modes and to the best-fit Zernike modes created by the static DM model. The total Root
Mean Square (RMS) error is ≈25nm for all modes, whereas the inevitable fitting error
varies between 0 and 23nm depending on the mode.
The static DM model is extended with lumped masses for the facesheet to obtain insight
in the dynamic behavior of the DM system. This behavior is validated by measurements
in which a laser vibrometer measures the displacement of each of a number of points on
the mirror facesheet, while the actuators are driven by a zero-mean, bandlimited, white
noise voltage sequence. Using the Multivariable Output-Error State-sPace (MOESP)
system identification algorithm, high-order black-box models are identified with Variance
Accounted For (VAF) values around 95%. The so obtained models are compared with the
analytical model in which the measured average actuator properties as substituted. The first
identified resonance frequency of 725Hz is lower than the 974Hz expected from the model,
which is attributed to the variations in actuator properties, such as actuator stiffness. The
relative modal damping of the identified model is an order of magnitude higher than the
damping in the analytical model, where only the identified actuator damping is considered.
The difference is attributed to the presence of air damping and damping in the glue used for
the connection struts.
The power dissipation in each actuator of the∅50mm mirror to correct a Von Karmann
turbulence spectrum with D/r0 = 5.4 and an outer scale of 100m is estimated. Approximately
1.5mW is dissipated for the purpose of turbulence correction by actuators near the
edge and 1.4mW by the inner actuators. These values are 23.8mW and 5.5mW respectively
to perform static flattening of the DM.
8

8
198 8 Conclusions and recommendations
8.2 Recommendations
In this thesis the feasibility of a modularly distributed controller architecture is investigated.
The main focus lies with the controller synthesis and how to implement it, but as the WFS
is excluded from this research project the efficient transmission of the measurements to the
distributed controller nodes has received no attention and requires further research.
A distributed adaptive reconstruction and prediction algorithm is presented that requires
further research into several of its properties. Firstly the proposed cost function does
not weight the unseen modes of the WFS such that the update law allows them to grow
arbitrarily large. Incorrect values for the unseen ’waffle’ mode of the Fried geometry
will compromise performance and incorrect values for the ’piston’ mode will require an
overly large stroke of the corrector. Weighting of these modes in the distributed setting
is not possible without affecting the overall AO performance. A strategy (e.g. through
weighting) must therefore be sought that constrains the unseen modes with the least effect
on performance.
Further, the stability and performance of the algorithm has to be determined on actual
measurement data from a large telescope and also realistic DM models have to be incorporated
into the algorithm, which currently assumes this to be the identity operator. When
the influence of the DM actuators is local, the latter may be approached by estimating
the coefficients of a second distributed filter by minimization of a quadratic cost function.
Similar to the separation principle in centralized control design, a distributed controller is
then obtained that consists of two parts: a reconstructor/predictor and a regulator.
Finally, the effect of sampling frequency on the performance trade-off has not been
considered. Sampling frequency largely determines the Signal to Noise Ratio (SNR)
of the WFS measurements. It is conceivable that in a distributed setting the sampling
frequency that gives optimal performance in terms of correction quality is different than in
the centralized setting, which makes it an interesting tuning parameter that deserves future
attention.
An entirely different approach to efficient control may be to investigate the relation between
the number of sensors and the required number of states for a atmospheric disturbance
model identified from measurement data to maintain a sufficient level of accuracy. It is not
unconceivable that the accuracy of models identified using black box system identification
techniques (e.g. subspace identification) does not (significantly) deteriorate as the number
of sensors in the AO increases while keeping the model order fixed. This still allows the
number of model coefficients and the corresponding computational load to increase linearly
with the number of sensors.
To further improve the properties of the actuator considered in chapter 5 it is recommended
to lower the reluctance of the radial air gap. This can be realized by a smaller gap
width or by a larger gap area. A sensitivity analysis of the actuator model showed this to
strongly affect the motor constant and actuator stiffness. A factor two reduction of this reluctance
will increase the motor constant to 0.37N/A and increase the actuator stiffness to
750N/m. When neglecting the effects of actuator stiffness, the factor four increase of the
motor constant leads to a factor 16 reduction in power dissipation. A higher motor constant
also leads to increased electronic damping of the mechanical resonance frequencies of the

8.2 Recommendations 199
DM system such that they become less limiting for the closed-loop performance without
adding complexity to the actuator or electronics design.
8

200
ÔÔÒ×

ÔÔÒÜ
ËÀÖØÑÒÒ×ÔÓØÔÓ×ØÓÒÒ
The size and alignment of the Hartmann array relative to the actuator grid is very
important for the quality of measurements from a Shack Hartmann sensor (SHS) for use in
closed loop control. The sensor should be well able to observe the Deformable Mirror (DM)
shape. In fact, as noted by [65, 66], the Hartmann grid affects both the controllability as
well as the observability of the system and thus its closed loop performance.
For a distributed control setting it is important that all local controllers have access to
measurements containing relevant information. When optical, spatial aliasing filters are
used, the spatial resolution of the sensor does not need to exceed that of the DM actuators.
This allows the sensor spots to form a regular pattern together with the DM actuator grid
and only leaves the position of the spots relative to the actuators to be chosen arbitrarily. It
will here be shown that this alignment makes a big difference for the observability of the
DM shape.
Let the deformationds as measured by a SHS in terms of spatial gradients of the mirror
surface be expressed in terms of the actuator command vectoruas:
ds = Bu,
where B is the DM influence matrix and temporal dynamics are neglected. The conditioning
of this matrix B determines the observability and controllability of the DM system and
therefore the achievable suppression of the wavefront disturbance [65]. However, since the
number of slope measurements is often larger than the number of actuators the matrix B
is often rectangular. This yields zero-valued singular values, corresponding to modes in
sensor space that cannot be the result of actuator actions and will therefore be neglected.
To determine the best spot location, the condition number of the matrix B has been evaluated
for various locations of the sensor spots relative to the actuator positions. The matrix
was calculated using equation (7.7) on page 164 with the DM parameters from table 7.1
using a hexagonally arranged actuator grid with 547 actuators over a circular aperture. The
command-to-slope influence function matrix was obtained from four command-to-phase
matrices based on slightly displaced (1/60 th actuator spacing) instanced of the sensor grid.
The two grids displaced in x direction provided the slopes in x direction, whereas the two
displaced in y direction provided the slopes in y direction. To reduce the effect of DM
edges, the outer three rings of actuators and sensor spots were modeled, but not considered
in the computation of the condition number.
The relation between the sensor-to-actuator grid alignment and the condition number of B
is plotted in figure A.1. This shows that the best (lowest) condition numbers are obtained
201

202 Appendices
Figure A.1: Condition number of the matrix B
as a function of the spot displacement in X and Y
directions w.r.t. the actuator grid (boxes).
Condition number [−]
10 4
10 3
10 2
10 1
500 1000 1500 2000
Actuator stiffness [N/m]
Figure A.2: The condition number of the matrix
B versus the actuator stiffness ca for a spot displacement
in y direction of 2.6mm.
when the sensor spots not not lie on the actuator grid lines. The best location shows a
∼70 times better conditioning than the worst location, when the sensor spots are exactly
aligned with the actuators. In that case, the influence of an actuator is not observed at
all by the corresponding sensor spot, since the spatial gradients remain practically zero.
The influence of an actuator is only observed at the location of its neighbors, where the
gradients are already much smaller.
Besides the grid alignment, the mechanical stiffness of the DM actuators influences
the condition number. This significantly affects the width and amplitude of the influence
functions and thus the magnitude of the slope measurements to a unit actuator command.
This relation has been evaluated for a number of different actuator stiffness values and is
shown in figure A.2. The condition number first improves for higher actuator stiffnesses,
but a minimum is obtained around 1650N/m after which it increases again.

ÔÔÒÜ
ÇÒÙ×ÒÐÓÐÔ××ÚØÝØÓÒÓÖ
ÐÓÐ×ØÐØÝ
The concept passivity can be used in the context of stability of interconnected systems
[8]. Passivity can be posed as a condition to guarantee stability and it can be shown that the
interconnection of two passive systems is again passive [133]. Various definitions of passivity
exist that can be further divided in several sub-classes [110, 133]. A passive system is a
special type of dissipative system. Let passivity here be defined using quadratic Lyapunov
functions on the system state, which is in literature referred to as internal passivity [110].
To show stability and interconnection stability of passive systems, consider two discrete
time Single-Input Single-Output (SISO) systems S1(z) and S2(z) that are given in state-
space form as:
Si(z) :
xi(t+1) = Aixi(t)+Biui(t)
yi(t) = Cixi(t).
(B.1)
Let these systems be called strictly passive if there exist Lyapunov functions Vi(xi(t)) =
x T i (t)Pixi(t) and matrices Pi ≻ 0 such that for the supply function si(yi(t),ui(t)) =
2yi(t)ui(t) the following holds for any inputui(t) ∈ R at any time instantt:
Vi(xi(t+1))−Vi(xi(t)) < si(yi(t),ui(t)). (B.2)
Asymptotic stability requires that for the autonomous system (ui(t) = 0) and all t that
Vi(xi(t+1))−Vi(xi(t)) < 0, which is implied by the above definition of strict passivity.
However, note that this is a sufficient condition for stability – not a necessary condition –
since the Lyapunov functions are here restricted to quadratic functions.
Now consider the negative feedback interconnection of the systemsS1(z) andS2(z) (figure
B.1). To show that the negative feedback interconnection is also strictly passive (and thus
stable), consider the candidate Lyapunov functionV(x1(t),x2(t)) = V1(x1(t))+V2(x2(t))
and supply function s(y1(t),u(t)) = 2y1(t)u1(t). By summation of the passivity conditions
in (B.2) for both systems and substitution ofu1(t) = u(t)−y2(t) and the definitions
ofV(x1(t),x2(t)) ands(y1(t),u(t)), the following is obtained:
V(x1(t+1),x2(t+1))−V(x1(t),x2(t)) < s(y1(t),u(t)).
This means that the interconnected system is again strictly passive. Note that a similar proof
can be given for the fact that a series connection of two passive systems is again passive.
This result can be used to derive conditions for stability of the interconnected network of
203

204 Appendices
Figure B.1: Feedback interconnection
of the local controllers
Ri(z) and Rm(z)
from chapter 4.
ˆφm
sl
ˆφk
-1
+
ûm
ˆφmi
Rm
Rim(z)
Rmi(z)
controllers Ri(z) in chapter 4. To properly state these conditions, first use linearity of the
local controllers to express ˆ φi(t) as the sum of filtered inputs:
ˆφi(t) =
Rim(z) ˆ φm(t)+
m∈Ci
m∈Mi
ˆφim
Ri
ûi
R (s)
im (z)sm(t).
The systems R (s)
im (z) and Rim(z) thus describe the contributions from sm(t) and ˆ φm(t)
onto the output ˆ φi(t) respectively. A sufficient condition for stability of the interconnected
network is that allRim(z) fori = 1...Nn andm ∈ Ci are passive and all interconnections
are negative feedback interconnections. This is illustrated in figure B.1, showing the output
feedback interconnection of two local controllers.
However, observe that the local controller parts Rim(z) are implicitly defined in (4.4)
of chapter 4 and have no direct feed-through terms. This implies that the feedback interconnected
subsystems have a full sample delay that – as will now be shown – makes passivity
unfeasible. This can be observed by making the passivity condition in (B.2) explicit for any
subsystem by substituting the system equations from (B.1), yielding:
(Aixi +Biui) T Pi(Aixi +Biui)−x T i Pixi < 2x T i CT i yi,
where for brevity the dependence on time t has been omitted. This can be rewritten into a
matrix form:
T
T
xi Ai PiAi −Pi AT i PiBi
T
T
xi xi 0 Ci xi
<
.
Ci 0
yi
B T i PiAi B T i PiBi
Since this inequality must hold for all input and state trajectoriesyi(t) ∈ R andxi(t) ∈ Rn ,
the following matrix inequality must hold:
T Ai PiAi −Pi AT i PiBi −C T
i ≺ 0.
yi
B T i PiAi −Ci B T i PiBi
The two Schur complements for this inequality are:
T Ai PiAi −Pi ≺ 0,
BT i PiBi − BT i PiAi
T −1
T −Ci Ai PiAi −Pi Ai PiBi −C T
i ≺ 0,
yi
+
sj
ˆφq
ˆφi
yi

Appendix B On using local passivity to enforce global stability 205
B T i PiBi ≺ 0,
A T i PiAi −Pi − A T i PiBi −C T i
T −1
T Bi PiBi Bi PiAi −Ci ≺ 0.
The second complement is not feasible by the fact that Pi ≻ 0 and thus B T i PiBi must
be positive (semi-)definite. To see that the first complement is not feasible, first note that
A T i PiAi −Pi can be negative definite. However, in that case the matricesB T i PiBi and the
product−X T (A T i PiAi−Pi) −1 X for any matrixX will be positive semi-definite. As their
sum will also be positive definite, this is in contradiction with the second part of the first
Schur complement. Therefore, this concept of passivity cannot be used to enforce stability
of the interconnected network of discrete time controllersRi(z) in chapter 4.

ÔÔÒÜ
ÓÙÖÖ×Ö×ÓÈÙÐ×ÏØ
ÅÓÙÐØÓÒ ÈÏÅ ×ÒÐ
A PWM signal is a periodic signal with period TPWM that switches between a high
value and a low value depending on a duty cyclerPWM ∈ {0,1}. For the BD modulation
principle used in section 6.2.2, the high value is equal to the clamp voltageVcc and the low
value equal to zero. Let the PWM signaly(t) be defined as a function of time t as:
Vcc, for −TPWMrPWM/2+kTPWM ≤ t ≤ TPWMrPWM/2+kTPWM,
y(t) =
0, otherwise,
(C.1)
wherek ∈ Z.
According to Fourier theory, any periodic signal can be written as a sum of sines and cosines.
Since this signal is symmetric int = 0, it can be expressed in cosines only as:
y(t) = a0 +
∞
ancos(2πnfPWMt), (C.2)
n=1
where fPWM = 1/TPWM. The Fourier coefficients an for n ≥ 1 can be determined by
integrating the product ofy(t) with a single cosine over one full period. Let this integralIn
be defined as:
In =
=
TPWM/2
−TPWM/2
TPWM/2
−TPMW/2
y(t)cos(2πfPWMnt)dt,
a0 +
∞
amcos(2πmfPWMt) cos(2πfPWMnt)dt,
m=1
where the second step follows from substitution of (C.2). This can be further simplified
to In = anTPWM/2 for n ≥ 1 and to I0 = a0TPWM for n = 0 using the following
goniometric identities:
TPWM/2
−TPWM/2
⎧
⎪⎨ 0 forn = m,
cos(2πfmt)cos(2πfnt) = TPWM forn = m = 0,
⎪⎩
TPWM/2 forn = m ≥ 1.
206

Appendix C Fourier series of a PWM signal 207
Conversely, the coefficientsan forn ≥ 1 can be calculated as:
an = 2In
TPWM
=
=
2
TPWM/2
TPWM
−TPWM/2
2
τ/2
TPWM
−τ/2
= 2Vccsin(2πfPWMnt)
2πfPWMnTPWM
= Vcc
= Vcc
y(t)cos(2πfPWMnt)dt,
Vcccos(2πfPWMnt)dt, (C.3)
τ/2
,
−τ/2
sin(πfPWMnτ)
,
πn
sin(πnrPWM)
. (C.4)
πn
where τ = TPWMrPWM . In (C.3), the signal y(t) from (C.1) is substituted and the remaining
steps use TPWMfPWM = 1 and other common algebra identities to simplify the
result.
The derived result is not valid forn = 0, buta0 can be obtained directly asa0 = I0/TPWM,
which reduces to:
TPWM/2
1
a0 =
TPWM
−TPWM/2
y(t)dt = Vcct
τ/2 = VccrPWM.
TPWM −τ/2
Summarizing, the PWM signaly(t) as defined in (C.1) can be expressed as an infinite series
of cosines by substituting thisa0 andan from (C.4) into (C.2), yielding:
∞
sin(πnrPWM)
y(t) = Vcc rPWM + cos(2πnfPWMt) .
πn
n=1

ÔÔÒÜ
ÌÄÎËÔÖÓØÓÓÐ
The electronics modules communicate with the communication bridge over an Low
Voltage Differential Signalling (LVDS) connection. This communication method uses a
current source to transmit information instead of a voltage source, which makes it much
less sensitive to the cable length or resistance. The sign of the current defines the binary
high and low values.
The LVDS connection does not use a clock signal to synchronize the communication. Each
message is preceded by 18 pause bits on which a message pointer can be synchronized and
one start and one stop bit on which the 16-bit data words can be synchronized. A start bit
is high and the pause and stop bits are low. No parity bits are used.
Each message consists of four parts: 18 pause bits, a header, the data and a checksum. The
header consists of one 16-bit word of which the lower eight bits are formed by the module
identifier and the upper eight by the message identifier:
byte index number of bits description
0 8 module identifier
1 8 message identifier
2 depends on message type command data
? 16 checksum
The sum of all data words – including the checksum itself – equals zero, which makes
the checksum the 2’s complement of the sum of the preceding words. The data structure
depends on the message identifier, which can be one of the following.
Burst write (1)
A burst-write message contains setpoints for all 61 actuators on the specified module. The
message identifier for this message is 1 and the command data field is defined as:
byte index number of bits description
0 16 setpoint for actuator 1
2 16 setpoint for actuator 2
.
.
.
120 16 setpoint for actuator 61
208

Appendix D The LVDS protocol 209
Register write (2)
A register write message can be used to modify a specific register on one of the modules.
The message identifier for this message is 2 and the command data field is defined as:
Register read (3)
byte index number of bits description
0 16 register number
2 16 new register value
A register read message can be used to read back a specific register on one of the modules.
The message identifier for this message is 3 and the command data field consists only of a
16-bit register number.
The module will respond with a message with the format of a register write and the module
and message identifiers set to the values used in the requesting register read message.
Registers
Each module contains the following registers:
index description
0x00. . . 0x1e PWM setpoints for actuators 1. . . 31
0x1f not used
0x20. . . 0x21 Enable PWM A bits for actuators 1. . . 31
0x22. . . 0x23 Enable PWM B bits for actuators 1. . . 31
0x24. . . 0x25 Enable PWM C bits for actuators 1. . . 31
0x26. . . 0x27 Coil integrity control bits for actuators 1. . . 31 (read-only)
0x28 Global settings register for actuators 1. . . 31
0x29. . . 0x3f not used
0x40. . . 0x5d PWM setpoints for actuators 32. . . 61
0x5e. . . 0x5f not used
0x60. . . 0x61 Enable PWM A bits for actuators 32. . . 61
0x62. . . 0x63 Enable PWM B bits for actuators 32. . . 61
0x64. . . 0x65 Enable PWM C bits for actuators 32. . . 61
0x66. . . 0x67 Coil integrity control bits for actuators 32. . . 61 (read-only)
0x68 Global settings register for actuators 32. . . 61
0x69. . . 0x7f not used
0x80. . . 0xff reserved
Global settings registers
Each slave Field Programmable Gate Array (FPGA) has a global, 16-bit settings register
with which its global behavior can be controlled. Its bits have the following meaning:

210 Appendices
Bit index Description
0. . . 3 Dead-time in 8ns steps (default = ’0111’)
4. . . 6 Not used
7 Set global PWM B to high (used for coil integrity check)
8 Global PWM C enable
9 Global PWM B enable
10 Global PWM A enable
11 Testmode (used for coil integrity check)
12:15 Revision number (= ’0001’)
Coil integrity check
The FPGA slaves can be put into test-mode to perform the so-called coil integrity check.
This can be used to determine whether the actuator coil conducts electricity or not.
First, the fine PWM C signal that is directly provided by the FPGA is set to high (figure 6.4)
and the course PWM signals are disabled. If the actuator coil does not conduct (e.g. because
of a broken wire), this will charge capacitorCl (figure 6.3) and build up a capacitor voltage.
Otherwise, the actuator coil will prevent this build-up and the voltage will quickly drop to
zero when PWM C is disabled. This behavior can be checked by reversing the directionality
of the FPGA pin corresponding to the PWM C signal and using it to measure the capacitor
voltage. If the voltage is zero, the actuator coil is fine and if it’s high, it does not conduct
electricity.
These operations are controlled via specific bits of the global settings registers. The coil
integrity check procedure should be as follows:
1. Enable bits 7 and 11 of the global settings registers. This enables the test mode for
which the course PWM signals are disabled and the fine PWM signal is permanently
high.
2. Wait at least four times the time constantRaCl of the system to allow charging of the
capacitor.
3. Set bits 7 of the global settings registers to zero, causing the FPGA pins corresponding
to PWM’s C to go into tri-state and allowing them to measure the capacitor voltages.
4. Disable the test-mode by setting bits 11 of the global settings registers to zero. The
results of the voltage measurements can now be found in the coil integrity control registers.
These should be zero for conducting actuator coils and one for malfunctioning
ones.

ÔÔÒÜ
ÌÍÈÔÖÓØÓÓÐ
User Datagram Protocol (UDP) is a very lean communication protocol, since no feedback
is given whether the message is properly received. Each message is a stand-alone
message and not part of a stream such as a Transmission Control Protocol (TCP)/Internet
Protocol (IP) message. Besides the required IP header, the message contains a small UDP
header that contains the source and destination ports, the message length and a checksum:
byte index number of bits description
0 16 source port
2 16 destination port
4 16 message length
6 16 checksum
8 ?? UDP data
The checksum value is ignored by the LVDS communications bridge to improve latency.
The data of the UDP message comes after the checksum field and has a substructure that
consists of the following parts: a module identifier (8 bits), a message identifier (8 bits) and
the command data:
byte index number of bits description
0 8 module identifier
1 8 message identifier
2 ?? command data
The module identifier byte specifies the module for which the message is intended and the
message identifier specifies the type of message that follows. The command data specification
depends on the type of message used. The message type can be one of the following.
Burst write
A burst-write message contains setpoints for all 61 actuators on the specified module. The
message identifier for this message is 1 and the command data field is defined as:
211

212 Appendices
byte index number of bits description
0 16 setpoint for actuator 1
2 16 setpoint for actuator 2
.
.
.
.
.
.
120 16 setpoint for actuator 61
122 16 Reserved
124 16 Sequence number (for diagnostics)
Together with the module and message identifier bytes, a burst write message is thus exactly
128 bytes in length. It is possible to include up to eight of such burst messages in one single
UDP message. Since the UDP-to-LVDS communication bridge has a processing time of
85µs per message, this significantly reduces the latency in case of many modules.
The communications bridge will recognize a combined UDP message by its size and splits
it into multiple LVDS messages before transmitting them sequentially over the LVDS connection.
The limit of eight burst messages per UDP message is due to the protocol limit of
the UDP message size.
Register write
A register write message can be used to modify a specific register on one of the modules.
The message identifier for this message is 2 and the command data field is defined as:
Register read
byte index number of bits description
0 16 register number
2 16 new register value
A register read message can be used to read back a specific register on one of the modules.
The message identifier for this message is 3 and the command data field consists only of a
16-bit register number.
The module will respond over the LVDS connection in the form specified in appendix D.
This is wrapped into the standard UDP form by the communication bridge. The module and
message identifiers will be identical to those bytes of the request message and the command
data field is identical to that of a register write message.
Diagnostic messages
Finally, the communications bridge counts the number of burst messages that it has properly
received since start-up. This 16-bit number can be read back by reading register 255 of
module 255. This counter can also be set by writing to this same register.

ÔÔÒÜ
ËÔØÐÚÖØÓÒÓØÙØÓÖ
ÔÖÓÔÖØ×
The two figures below illustrate that there is no obvious correlation between the values
of the resonance frequency and motor constant of the actuators measured and their location
in the grid.
Figure F.1: The value of the resonance frequencies,
represented proportionally by the size of the
dots for the modules 1...7 from left to right and
then top to bottom.
213
Figure F.2: The value for the motor constants,
represented proportionally by the size of the dots
for the modules 1...7 from left to right and then
top to bottom.

ÔÔÒÜ
ÉÙÒØÞØÓÒ
In chapter 6 a choice was made to use a 16 bit PWM voltage signal generator to drive the
actuators, leading to sufficiently small quantization errors. At that point, the consideration
was based on general design parameters of the system, which is here performed in more
detail to validate the choice.
The Root Mean Square (RMS) wavefront error σquant due to quantization errors has been
specified in chapter 2 as at most 5nm. Due to quantization, the actual command vector
˜VPWM will be the sum of the intended actuator voltage VPWM and quantization noise n.
Since wavefront correction is only possible for frequencies below the control bandwidth,
which is generally below the first system resonance, the static DM system model from (7.7)
on page 164 with influence matrixBf will here be used:
ˆzf = Bf ˜ VPWM = Bf(VPWM +n),
where ˆzf denotes the actual facesheet deflection in contrast to the facesheet deflection for
the intended command VPWM. According to the design requirements, the variance σ 2 quant
of the quantization noisenshould be such thatσquant = 〈BfnF〉 ≤ 5nm, where
σ 2 quant
1
= Tr
Na
T
Bf nn B T f . (G.1)
To derive the covariance matrix nnT , let the elementsni of n be uncorrelated stochastic
values with a square probability density functionPn(ni):
q
Pn(ni) =
−1 for |ni| ≤ q/2,
0 otherwise,
where q is the command quantization step size in Volt. Each diagonal element of nn T
can thus be expressed as [132]:
2
ni =
∞
−∞
n 2 iPn(ni)dni =
q/2
−q/2
n 2 i
q dni = q2
12
hence nn T = q
12 I. Using (G.1) the q for which σquant is smaller than 5nm can now be
214

Appendix G Quantization 215
derived as:
1
Na
2 q
Tr
12 BfB T
f ≤ (5·10 −9 ) 2 ,
q2 12 Tr(BfB T f ) ≤ (5·10−9 ) 2 Na,
q ≤
12(5·10 −9 ) 2Na Tr(BfBT f )
.
Using the influence matrix identified in section 7.3.2 for the 61 actuator DM system this
yields q ≈ 190µV, which is only slightly more than the 100µV accuracy provided by the
realized 16 bit PWM voltage source with a supply voltage of 3.3V. The quantization value
of the driver electronics was thus properly chosen.
Finally, note that incorporation of noise shaping techniques [78, 158, 162] into the controller
design may enlarge the required quantization step. Such techniques could push the
quantization effects to (high) temporal frequencies for which the wavefront disturbance has
a low magnitude, so reducing the effect on the optical performance.

ÐÓÖÔÝ
[1] www.goodfellow.com.
[2] C. A. Aerts. Optimizing and implementing leaf springs for the reluctance motor.
Technical Report DCT 2005.94, Technische Universiteit Eindhoven, 2005.
[3] S. T. Alexander. Adaptive signal processing. Texts and monographs in computer
science. Springer-Verlag, New York, 1986.
[4] P. Alriksson and A. Rantzer. Distributed Kalman filtering using weighted averaging.
In Proceedings of the 17 th International Symposium on Mathematical Theory of
Networks and Systems, Kyoto, Japan, july 2006.
[5] P. Alriksson and A. Rantzer. Experimental evaluation of a distributed Kalman filter
algorithm. In Proceedings of the 46 th IEEE Conference on Decision and Control,
New Orleans, LA, December 2007.
[6] T. Andersen, Mette-Owner-Petersen, and H. Riewaldt. An integrated simulation
model of the euro50. In T. Andersen, editor, Proceedings of SPIE: Integrated Modeling
of Telescopes, volume 4757, pages 84–92, July 2002.
[7] V. Apollonov, V. Borodin, A. Brynskikh, G. Vdovin, and S. Murav. Cooled adaptive
mirror with magnetostrictive spring-type actuators. Soviet Journal of Quantum
Electronics, 20(11):1403–1406, 1990.
[8] M. Arcak and E. D. Sontag. A passivity-based stability criterion for a class of interconnected
systems and applications to biochemical reactor networks. In Proceedings
of the 46 th IEEE Conference on Decision and Control, pages 4477–4482, New Orleans,
LA, USA, December 2007.
[9] R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo,
R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau. A deformable
secondary mirror for the VLT. In Proceedings of SPIE, volume 6272, 2006.
[10] F. Assémat, E. Gendron, and F. Hammer. The FALCON concept: multi-object adaptive
optics and atmospheric tomography for integral field spectroscopy Ű principles
and performance on an 8-m telescope. Monthly Notices of the Royal Astronomical
Society, 376(1):287–312, March 2007.
[11] N. Assimakis. Optimal distributed kalman filter. In Nonlinear analysis, volume 47,
pages 5367–5378. Elsevier Science Ltd., 2001.
[12] H. Babcock. The possibility of compensating astronomical seeing. Publications of
the Astronomical Society of the Pacific, 65(386):229–236, October 1953.
[13] B. Bamieh, F.Paganini, and M. A. Dahleh. Optimal control of distributed arrays with
spatial invariance. Lecture Notes in Control and Information Sciences, 245:329–343,
1999.
[14] B. Bamieh, F.Paganini, and M. A. Dahleh. Distributed control of spatially invariant
systems. IEEE Transactions on Automatic Control, 47:1091–1107, July 2002.
216

Bibliography 217
[15] F. Bazán. Eigensystem realization algorithm (ERA): reformulation and system pole
perturbation analysis. Journal of Sound and Vibration, 274:433–444, 2004.
[16] C. Béchet, M. Tallon, and E. Thiébaut. FRIM: minimum-variance reconstructor with
a FRactal Iterative Method. In B. L. Ellerbroek and D. C. B. Calia, editors, Proceedings
of SPIE: Astronomical Telescopes and Instrumentation, volume 6272, May
2006.
[17] A. Bejan. Heat transfer. John Wiley and Sons Ltd, 1990.
[18] R. Biasi, M. Andrighettoni, D. Veronese, V. Biliotti, L. Fini, A. Riccardi, P. Mantegazza,
and D. Gallieni. LBT adaptive secondary electronics. In P. L. Wizinowich
and D. Bonaccini, editors, Proceedings of SPIE: Astronomical Telescopes and Instrumentation,
volume 4839, pages 772–782, Waikoloa, Hawaii, USA, August 2002.
[19] K. Birch and M. Downs. Correction to the updated Edlen equation for the refractive
index of air. Metrologia, 31(4):315–316, 1994.
[20] M. Born and E. Wolf. Principles of Optics: Electromagnetic Theory of Propagation,
Interference, and Diffraction of Light. Cambridge University Press,7 th edition, 1999.
[21] A. H. Bouchez, R. G. Dekany, J. R. Angione, C. Barane, K. Bui, R. S. Burruss,
J. R. Crepp, E. E. Croner, J. L. Cromer, S. R. Guiwits, D. D. Hale, J. R. Henning,
D. Palmer, J. E. Roberts, M. Troy, T. N. Truong, and J. Zolkower. Status of the
PALM-3000 high-order adaptive optics system. In N. Hubin, C. E. Max, and P. L.
Wizinowich, editors, Proceedings of SPIE: Adaptive Optics Systems, volume 7015,
2008.
[22] J. Brenner and L. Cummings. The hadamard maximum determinant problem. American
Mathematics Montly, 79(6):626–630, 1972.
[23] J. W. Brewer. Kronecker Products and Matrix Calculus in System Theory. IEEE
Transactions on Circuits and Systems, 25(9):772–781, September 1978.
[24] G. Brusa, A. Riccardi, V. Biliotti, C. D. Vecchio, P. Salinari, P. Stefanini, P. Mantegazza,
R. Biasi, M. Andrighettoni, C. Franchini, and D. Gallieni. Adaptive secondary
mirror for the 6.5m conversion of the Multiple Mirror Telescope: first laboratory
testing results. In R. K. Tyson and R. Q. Fugate, editors, Proceedings of SPIE:
Adaptive Optics Systems and Technology, volume 3762, September 1999.
[25] G. Brusa-Zappellini, A. Riccardi, S. D. Ragland, S. Esposito, C. D. Vecchio, L. Fini,
P. Stefanini, V. Biliotti, P. Ranfagni, P. Salinari, D. Gallieni, R. Biasi, P. Mantegazza,
G. Sciocco, G. Noviello, and S. Invernizzi. Adaptive secondary P30 prototype: laboratory
results. In D. Bonaccini and R. K. Tyson, editors, Proceedings of SPIE: Adaptive
Optical System Technologies, volume 3353, pages 764–775, September 1998.
[26] D. Buscher. A Thousand and One Nights of Seeing on Mt. Wilson. In J. B. Breckinridge,
editor, Proceedings of SPIE: Amplitude and Intensity Spatial Interferometry
II, volume 2200, pages 260–271, 1994.
[27] Y. Carmon and E. N. Ribak. Phase retrieval by demodulation of a Hartmann-Shack
sensor. Optics Communications, 215:285–288, January 2003.
[28] R. Changhai and S. Lining. Hysteresis and creep compensation for piezoelectric
actuator in open-loop operation. Sensors and Actuators A, 122:124–130, 2005.
[29] A. Chiuso. The role of vector autoregressive modeling in predictor-based subspace
identification. Automatica, 43:1034–1048, 2007.

218 Bibliography
[30] A. Chiuso, R. Muradore, and E. Marchetti. Dynamic calibration of adaptive optics
systems: a system identification approach. In Proceedings of the Conference on
Decision and Control, 2008.
[31] A. Chiuso and G. Picci. Consistency analysis of some closed loop subspace identification
methods. Automatica, 41(3):377–391, 2005.
[32] P. Ciddor. Refractive index of air: new equations for the visible and near infrared.
Applied Optics, 35:1566–1573, 1996.
[33] J.-M. Conan, G. Rousset, and P.-Y. Madec. Wavefront temporal spectra in highresolution
imaging through turbulence. Journal of the Optical Society of America A,
12:1559–1570, July 1995.
[34] R. Conan, R. Avila, and L. Sanchez. Wavefront outer scale and seeing measurements
at San Pedro Martir Observatory. Astronomy and Astrophysics, 396:723–730, 2002.
[35] D. T. Corporation. SigLab Users Guide, 1994.
[36] A. Cox, editor. AllenŠs Astrophysical Quantities. AIP Press., 2000.
[37] L. Cuellar, P. Johnson, and D. Sandlar. Performance tests of a 1500 degree-offreedom
adaptive optics system for atmspheric compensation. In Proceedings of
SPIE: Active and adaptive optical systems, volume 1542, pages 468–476, 1991.
[38] O. Cugat, S. Basrour, C. Divoux, P. Mounaix, and G. Reyne. Deformable magnetic
mirror for adaptive optics: technological aspects. Sensors and Actuators A: Physical,
89:1–9, 2001.
[39] O. Cugat, P. Mounaix, S. Basrour, C. Divoux, and G. Reyne. Deformable magnetic
mirror for adaptive optics: first results. In The13 th Annual International Conference
on Micro Electro Mechanical Systems, 2000.
[40] J. Dainty, D. Hennings, and K. O’Donnel. Space-time correlation of stellar speckle
patterns. Journal of the Optical Society of America, 71:490–492, April 1981.
[41] R. D’Andrea and G. E. Dullerud. Distributed Control Design for Spatially Interconnected
Systems. IEEE Transactions on Automatic Control, 48(9):1478–1495,
September 2003.
[42] W. de Bruijn. Low power deformable mirror actuator controller. Master’s thesis,
Technische Universiteit Eindhoven (TU/e), Den Dolech 2, 5600MB, Eindhoven, The
Netherlands, 2009.
[43] P. Dierickx and R. Gilmozzi. Progress of the OWL 100-m Telescope Conceptual
Design. In T. A. Sebring and T. Andersen, editors, Proceedings of SPIE: Telescope
Structures, Enclosures, Controls, Assembly/Integration/Validation, and Commissioning,
volume 4004, pages 290–299, Munich, March 2000.
[44] N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen. Adaptive and real-time
optimal control for adaptive optics systems. European Journal of Control, 15:480–
502, 2009.
[45] N. Doelman, K. Hinnen, F. Stoffelen, and M. Verhaegen. Optimal control strategy to
reduce the temporal wavefront error in AO systems. In D. B. Calia, B. L. Ellerbroek,
and R. Ragazzoni, editors, Proceedings of SPIE: Advancements in Adaptive Optics,
volume 5490, pages 1426–1437, June 2004.
[46] M. Ealey. Deformable mirrors at Litton/Itek: A historical persepective. In D. Vukobratovich,
editor, Proceedings of SPIE: Precision Engineering and Optomechanics,
volume 1167, November 1989.

Bibliography 219
[47] M. Ealey and J. Washeba. Continous facesheet low voltage deformable mirrors. Optical
Engineering, 29:1191–1198, October 1990.
[48] F. Eaton and G. Nastrom. Preliminary estimates of the vertical profiles of inner and
outer scales from White Sands Missile Range. Radio Science, 33:895–903, 1998.
[49] Edlén. The refractive index of Air. Metroligia, 2:71–80, 1966.
[50] S. E. Egner. Multi-Conjugate Adaptive Optics for LINC-NIRVANA. PhD thesis, Max
Planck Institute for Astronomy, University of Heidelberg, 2006.
[51] R. Ellenbroek and R. Hamelinck. Adaptief deformeerbare spiegel voor telescopen.
Precisietechnologie jaarboek, 16:112–118, 2009.
[52] R. Ellenbroek, M. Verhaegen, R. Hamelinck, N. Doelman, M. Steinbuch, and
N. Rosielle. Distributed control in Adaptive Optics - Deformable mirror and turbulence
modeling. In B. L. Ellerbroek and D. C. B. Calia, editors, Proceedings of
SPIE: Astronomical telescopes and instrumentation - Advances in Adaptive Optics,
volume 6272, May 2006.
[53] B. Ellerbroek. Efficient computation of minimum-variance wave-front reconstructors
with sparse matrix techniques. Journal of the Optical Society of America A, 19:1803–
1816, September 2002.
[54] B. Ellerbroek and C. Vogel. Simulations of closed-loop wavefront reconstruction for
multiconjugate adaptive optics on giant telescopes. In Proc. SPIE 5169-23, Adaptive
Optics System Technologies II, pages 206–217, 2003.
[55] B. L. Ellerbroek. First-order performance evaluation of adaptive-optics systems for
atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes.
In Journal of the Optical Society of America A, volume 11, pages 783–805,
Februari 1994.
[56] B. L. Ellerbroek and T. Rhoadarmer. Real time adaptive optimization of wavefront
reconstruction algorithms for closed loop adaptive optical systems. In D. Bonaccini
and R. K. Tyson, editors, Proceedings of SPIE: Adaptive Optical System Technologies,
volume 3353, pages 1174–1185, March 1998.
[57] S. Fassois. MIMO LMS-ARMAX identification of vibrating structures - part I: the
method. Mechanical systems and signal processing, 15(4):723–735, 2001.
[58] E. Fedrigo and R. Donaldson. Architecture of the MAD realtime computer. In P. L.
Wizinowich and D. Bonaccini, editors, Proceedings of SPIE: Adaptive Optical System
Technologies II, volume 4839, pages 600–611, February 2003.
[59] A. E. Fitzgerald, C. K. Jr., and S. D. Umans. Electric Machinery. McGraw-Hill, 6 t h
edition, 2003.
[60] A. Florakis and S. Fassois. MIMO LMS-ARMAX identification of vibrating structures
- part II: a critical assessment. Mechanical systems and signal processing, 15,
2001.
[61] R. Fraanje and M. Verhaegen. A spatial canonical approach to multidimensional
state-space identification for distributed parameter systems. In Proceedings of the
fourth international workshop on multidimensional systems, 2005.
[62] G. F. Franklin, J. Powell, and A. Emami-Naeini. Feedback control of dynamic systems.
Addison-Wesley, 3rd edition, 1994.
[63] D. Fried. Statistics of a Geometric Representation of Wavefront Distortion. Journal
of the Optical Society of America, 55:1427–1435, November 1965.

220 Bibliography
[64] D. Fried. Time delay induced mean square error in adaptive optics. Journal of the
Optical Society of America A, 7(7):1224–1225, 1990.
[65] M. E. Furber and D. Jordan. Optimal design of wavefront sensors for adaptive optical
systems: part 1, controllability and observability analysis. Optical Engineering,
36:1843–1855, July 1997.
[66] M. E. Furber and D. Jordan. Optimal design of wavefront sensors for adaptive optical
systems: part 2, optimization of sensor subaperture locations. Optical Engineering,
36:1856–1871, July 1997.
[67] A. Fuschetto. Three-actuator deformable water-cooled mirror. Journal of Optical
Engineering, 20(2):310–315, March/April 1981.
[68] D. Gallieni, E. Anaclerio, P. Lazzarini, A. Ripamonti, R. Spairani, C. DelVecchio,
P. Salinari, A. Riccardi, P. Stefanini, and R. Biasi. LBT adaptive secondary units final
design and construction. In P. L. Wizinowich and D. Bonaccini, editors, Proceedings
of SPIE: Adaptive Optical System Technologies II, volume 4839, pages 765–771,
2003.
[69] A. Gelb. Applied optimal estimation. The MIT press, 15th edition, 1999.
[70] D. Geng, S. J. Goodsell, A. G. Basden, N. A. Dipper, R. M. Myers, and C. D. Saunter.
FPGA cluster for high-performance AO real-time control system. In B. L. Ellerbroek
and D. C. B. Calia, editors, Proceedings of the SPIE: Astronomical telescopes and
instrumentation - Advances in Adaptive Optics, volume 6272, 2006.
[71] S. Gerschgorin. Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk.
SSSR Ser. Mat., 1:749–754, 1931.
[72] L. Gilles. Order-N sparse minimum-variance open-loop reconstructor for extreme
adaptiveoptics. Optics Letters, 28(20):1927–1929, October 2003.
[73] L. Gilles, B. Ellerbroek, and C. Vogel. Preconditioned conjugate-gradient wave-front
reconstructors for multiconjugate adaptive optics. Applied Optics, 42:5233–5250,
2003.
[74] L. Gilles, C. R. Vogel, and B. Ellerbroek. Multigrid predonditioned conjugategradient
method for large-scale wave-frontreconstruction. Journal of the Optical
Society of America A, 19:1817–1822, September 2002.
[75] R. Gilmozzi and J. Spyromilio. The European Extremely Large Telescope. The
Messenger (Telescopes and Instrumentation) 127, European Southern Observatory,
March 2007.
[76] A. Glecker, D. Markason, and G. Ames. PAMELA; Control of a segmented mirror
via wavefront tilt and segment piston sensing. In M. A. Ealey, editor, Proceedings
op SPIE: Active and Adaptive Optical Components, volume 1543, pages 176–189,
January 1991.
[77] G. H. Golub and C. F. V. Loan. Matrix computations. Johns Hopkins University
Press, Baltimore, MD, USA, 3rd edition, 1996.
[78] G. Goodwin, D. Quevedo, and D. McGrath. Moving-horizon optimal quantizer for
audio signals. Journal of the Audio Engineering Society, 51(3):138–149, 2003.
[79] D. Gorinevsky, S. Boyd, and G. Stein. Design of Low-bandwidth Spatially Distributed
Feedback. IEEE Transactions on Automatic Control, June 2005.
[80] D. Greenwood. Bandwidth specification for adaptive optics systems. Journal of the
Optical Society of America, 67:390, 1977.

Bibliography 221
[81] R. Hamelinck. Ontwerp van een adaptieve deformeerbare spiegel voor het corrigeren
van atmosferische turbulentie. Master’s thesis, Technische Universiteit Eindhoven,
Den Dolech 2, 5600MB, Eindhoven, The Netherlands, 2003.
[82] R. Hamelinck, R. Ellenbroek, N. Rosielle, M. Steinbuch, M. Verhaegen, and N. Doelman.
Validation of a new adaptive deformable mirror concept. In N. Hubin, C. E.
Max, and P. L. Wizinowich, editors, Proceedings of SPIE: Astronomical telescopes
and instrumentation, volume 7015, Marseille, France, June 2008.
[83] R. Hamelinck, N. Rosielle, and N. Doelman. Large adaptive deformable membrane
mirror with high actuator density: design and first actuator tests. In Proceedings of
the5 th Workshop on Adaptive Optics for Industry and Medicine, August 2005.
[84] R. Hamelinck, N. Rosielle, and N. Doelman. Large adaptive deformable mirror:
design and first prototypes. In M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles,
editors, Proceedings of SPIE: Advanced Wavefront Control: Methods, Devices, and
Applications III, volume 5894, San Diego, August 2005.
[85] R. Hamelinck, N. Rosielle, J. Kappelhof, B. Snijders, and M. Steinbuch. Large adaptive
deformable membrane mirror with high actuator density. In D. B. Calia, B. L.
Ellerbroek, and R. Ragazzoni, editors, Proceedings of the SPIE: Astronomical Telescopes
and Instrumentation, volume 5490, pages 1482–1492, Glasgow, UK, June
2004.
[86] R. Hamelinck, N. Rosielle, and M. Steinbuch. Modular adaptive deformable mirror
technology based on electromagnetic actuators. In 8 th International Workshop on
Adaptive Optics for Industry and Medicine, Shatura, Russia, 2009.
[87] R. Hamelinck, N. Rosielle, M. Steinbuch, and N. Doelman. Electromagnetic dm
technology meets future ao demands. In AO4ELT, Paris, France, 2009.
[88] R. Hamelinck, N. Rosielle, M. Steinbuch, N. Doelman, R. Ellenbroek, and M. Verhaegen.
Actuator grid design for an adaptive deformable mirror. In Proceedings of
the 7 th EUSPEN International Conference, Bremen, Germany, May 2007.
[89] R. Hamelinck, N. Rosielle, M. Steinbuch, R. Ellenbroek, M. Verhaegen, and N. Doelman.
Actuator tests for a large deformable membrane mirror. In B. L. Ellerbroek and
D. C. B. Calia, editors, Proceedings of SPIE: Astronomical telescopes and instrumentation
- Advances in Adaptive Optics, volume 6272, May 2006.
[90] R. Hamelinck, N. Rosielle, M. Steinbuch, R. Ellenbroek, M. Verhaegen, and N. Doelman.
Deformable membrane mirror with high actuator density and distributed control.
In 6 th International Workshop on Adaptive Optics for Industry and Medicine,
Galway, Ireland, June 2007.
[91] R. Hamelinck, N. Rosielle, M. Steinbuch, R. Ellenbroek, M. Verhaegen, and N. Doelman.
Test results of an adaptive deformable mirror for future large telescopes. In
Proceedings of the 8 th EUSPEN International Conference, 2008.
[92] P. J. Hampton, R. Conan, O. Keskin, C. Bradley, and P. Agathoklis. Selfcharacterization
of linear and nonlinear adaptive optics systems. Applied Optics,
47(2):126–134, Januari 2008.
[93] Handshake Solutions. Haste-Programming Language Manual, 2007.
[94] D. S. Hansen. Digital controller design for alised sampled-data systems. PhD thesis,
Brigham-Young university, department of mechanical engineering, August 2000.

222 Bibliography
[95] C. M. Harding, R. A. Johnston, and R. G. Lane. Fast simulation of a Kolmogorov
phase screen. Applied Optics, 38(11):2161–2170, April 1999.
[96] J. Hardy. Analog data processor. US Patent 3921080, November 1975.
[97] J. Hardy. Instrumental limitations in adaptive optics for astronomy. In Proceedings
of SPIE: Active telescope systems, volume 1114, pages 2–13, November 1989.
[98] J. Hardy. Adaptive optics for Astronomical Telescopes. Oxford University Press,
New York, 1998.
[99] T. Henningsson and A. Rantzer. Scalable Distributed Kalman Filtering for Mass-
Spring Systems. In Proceedings of the 46 th IEEE Conference on Decision and Control,
New Orleans, LA, December 2007.
[100] K. J. Hinnen. Data-Driven Optimal Control for Adaptive Optics. Phd dissertation,
Delft University of Technology, January 2007. ISBN 978-90-9021188-6.
[101] P. Hinz, M. Kenworthy, D. Miller, V. Vaitheeswaran, G. Brusa, and G. Zappellini.
Riding the hub: characterization of the MMT adaptive secondary performance. In
B. L. Ellerbroek and D. C. B. Calia, editors, Proceedings of SPIE: Astronomical
telescopes and instrumentation, volume 6272, 2006.
[102] J. Hois and S. Fassois. Stochastic vector identification and uncertain modal parameter
estimation for a smart composite beam. In Proceedings of the 3 rd international
conference smart materials structures systems, Acireale, Italy, June 2008.
[103] N. Hubin. Technical specification for the Conceptual Design, Prototyping, Preliminary
Design of the M4 adaptive unit for the E-ELT. Technical Report E-ESO-SPE-
106-0037 1.0, European Southern Observatory, 2007.
[104] B. Hulburd and D. Sandler. Segmented mirrors for atmospheric compensation. Optical
Engineering, 29(10):1186, 1990.
[105] R. Irwan. Wavefront estimation in astronomical imaging. PhD thesis, Department of
electrical engineering, University of Canterbury, March 1999.
[106] R. Irwan and R. G. Lane. Analysis of optimal centroid estimation applied to Shack-
Hartmann sensing. Applied Optics, 38:6737–6743, November 1999.
[107] S. Jiang, P. G. Voulgaris, L. E. Holloway, and L. A. Thompson. Distributed Control of
Large Segmented Telescopes. In Proceedings of 2006 American Control Conference,
Minneapolis, Minnesota, USA, June 2006.
[108] F. Jones. The refractivity of air. Journal of Research of the National Bureau of
Standards, 86:27–32, 1981.
[109] M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti. Fast
calibration of high-order adaptive optics systems. Journal of the Optical Society of
America A, 2:1004–1008, 2004.
[110] A. G. Kelkar and S. M. Joshi. Dissipativity and passivity, chapter 3. Springer Berlin
/ Heidelberg, 1996.
[111] H. Kharaghani and B. Tayfeh-Rezaie. A hadamard matrix of order 428. Journal of
Combinatorial Designs, 13:435–440, 2005.
[112] A. Kolmogorov. Dissipation of energy in a locally isotropic turbulence, volume 3
of Turbulence, Classic Papers on Statistical Theory. Wiley-Interscience, New York,
October 1961.

Bibliography 223
[113] M. Langlois, R. Angel, M. Lloyd-Hart, F. W. G. Love, and A. Naumov. High Order
Reconstructor-Free Adaptive Optics for 6-8 meter class Telescopes. In ESO Conference
& Workshop Proceedings, Beyond Conventional Adaptive Optics, volume 58,
pages 113–120, Venice, Italy, 2001.
[114] W. Langlois. Isothermal squeeze films. Quarterly applied mathematics, XX:131–
150, 1962.
[115] H. Lev-Ari. Efficient solution of linear matrix equations with application to multistatic
antenna array processing. Communications in information and systems,
5(1):123–130, 2005.
[116] J.-S. Lew, J.-N. Juang, and R. Longman. Comparison of several system identification
methods for flexible structures. Journal of Sound and Vibration, 167(3):461–480,
November 1993.
[117] Y.-T. Liu and S. Gibson. Adaptive Optics with Adaptive Filtering and Control. In
Proceedings of the American Control Conference, pages 3176–3179, Boston, Massachusetts,
USA, June/July 2004.
[118] P. M. Livingston. Proposed method for inner scale measurements in a turbulent atmosphere.
Applied Optics, 11(3):684–687, 1972.
[119] L. Ljung. System identification - theory for the user. Prentice Hall, inc., 1987.
[120] M. Lloyd-Hart, F. Wildi, H. Martin, P. McGuire, M. Kenworthy, R. Johnson, B. Fitz-
Patrick, G. Angeli, S. Miller, and R. Angel. Adaptive optics for the 6.5m MMT. In
P. L. Wizinowich, editor, Proceedings of SPIE: Adaptive Optical Systems Technology,
volume 4007, pages 167–174, 2000.
[121] M. Loktev. Modal wavefront correctors based on nematic liquid crystals. PhD thesis,
Delft University of Technology, 2005.
[122] M. Loktev, D. W. D. L. Monteiro, and G. Vdovin. Comparison study of the performance
of piston, thin plate and membrane mirrors for correction of turbulenceinduced
phase distortions. Optics Communications, 192:91–99, May 2001.
[123] D. P. Looze. Minimum variance control structure for adaptive optics systems. In
Journal of the Optical Society of America A, 2005.
[124] D. G. MacMartin. Local, hierarchic, and iterative reconstructors for adaptive optics.
Journal of the Optical Society of America, 20:1084–1093, June 2003.
[125] E. Manders. Design of a device to measure the out-of-plane, nonlinear stiffness of
circular membranes. Technical report, Technische Universiteit Eindhoven, 2004.
[126] E. Marchetti, R. Brast, B. Delabre, et al. On-sky testing of the multi-conjugate
adaptive optics demonstrator. The Messenger 129, European Southern Observatory,
September 2007.
[127] F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi,
and M. Sarazin. Optical parameters relevant for High Angular Resolution at Paranal
from GSM instrument and surface layer contribution. Astronomy and Astrophysics,
144:39–44, May 2000.
[128] P. Massioni, T. Keviczky, and M. Verhaegen. New Approaches to Distributed Control
of Satellite Formation Flying. In Proceedings of the 3 rd International Symposium on
Formation Flying, Missions and Technologies, ESA/ESTEC, Noordwijk, The Nederlands,
April 2008.

224 Bibliography
[129] P. Massioni and M. Verhaegen. Distributed Control for Identical Dynamically Coupled
Systems: a Decomposition Approach. Submitted paper.
[130] T. Mathworks. Scientific computing software, 2009. http://www.mathworks.com.
[131] Mentor Graphics Corporation. ModelSim User’s Manual, 2007.
[132] D. C. Montgomery and G. C. Runger. Applied statistics and probability for engineers.
John Wiley and Sons, Inc., New York,2 nd edition, 1999.
[133] E. M. Navarro-López. Several dissipativity and passivity implications in the linear
discrete-time setting. Mathematical Problems in Engineering, pages 599–616, 2005.
[134] R. Negenborn. Multi-Agent Model Predictive Control with Applications to Power
Networks. PhD thesis, Delft University of Technology, December 2007. ISBN 978-
90-5584-093-9.
[135] M. Nicolle, T. Fusco, G. Rousset, and V. Michau. Improvement of Shack-
Hartmann wave-front sensor measurement for extreme adaptive optics. Optics Letters,
29(23):2743–2745, 2004.
[136] R. Noll. Zernike polynomials and atmospheric turbulence. Journal of the Optical
Society of America, 66:207–211, March 1976.
[137] R. Olfati-saber and R. M. Murray. Distributed cooperative control of multiple vehicle
formations using structural potential functions. In IFAC World Congress, 2002.
[138] R. Olifati-Saber. Distributed Kalman Filter with Embedded Consensus Filter. In Proceedings
of the 44 th IEEE Conference on Decision and Control 2005, pages 8179–
8184. Dartmouth College, December 2005.
[139] J. Owens. Optical refractive index of air: dependence on pressure, temperature and
composition. Applied Optics, 6:51–59, 1967.
[140] S. Oya, A. Bouvier, O. Guyon, M. Watanabe, Y. Hayano, H. Takami, M. Iye, M. Hattori,
Y. Saito, M. Itoh, S. Colley, M. Dinkins, M. Eldred, and T. Golota. Performance
of deformable mirror for Subaru LGSAO system. In B. L. Ellerbroek and D. C. B.
Calia, editors, Proceedings of SPIE: Astronomical telescopes and instrumentation -
Advances in Adaptive Optics, volume 6272, 2006.
[141] E. Peck and K. Reeder. Dispersion of air. Journal of the Optical Society of America,
62:958–962, 1972.
[142] L. A. Poyneer. Correlation wave-front sensing algorithms for Shack-Hartmann-based
Adaptive Optics using a point source. In Lawrence Livermore National Laboratory
(LLNL), number UCRL-JC-152975, May 2003.
[143] L. A. Poyneer, D. T. Gavel, and J. M. Brase. Fast wave-front reconstruction in large
adaptive optics systems with use of the Fourier transform. Journal of the Optical
Society of America A, 19(10):2100–2111, October 2002.
[144] R. B. Quirino and C. P. Bottura. An approach for distributed Kalman filtering. In
SBA Controle & Automação, volume 12, 2001.
[145] R. Ragazzoni. Pupil plane wavefront sensing with an oscillating prism. Journal of
Modern Optics, 43(2):289–293, 1996.
[146] R. Ragazzoni, E. Marchetti, and F. Rigaut. Modal tomography for Adaptive Optics.
Astronomy and Astrophysics, 342:53–56, 1999.
[147] A. Rantzer. Using Game Theory for Distributed Control Engineering. In Proceedings
of Games 2008, 3rd World Congress of the Game Theory Society, July 2008.

Bibliography 225
[148] F. Roddier. Curvature sensing and compensation: a new concept in adaptive optics.
Applied Optics, 27(7):1223–1225, 1988.
[149] F. Roddier. Adaptive optics in astronomy. Cambridge University Press, 1999.
[150] B. L. Roux and J.-M. Conan. Optimal control law for classical and multiconjugate
adaptive optics. Journal of the Optical Society of America A, 21(7):1261–1276, July
2004.
[151] T. Ruppel, M. Lloyd-Hart, D. Zanotti, and O. Sawodny. Modal Trajectory Generation
for Adaptive Secondary Mirrors in Astronomical Adaptive Optics. IEEE International
Conference on Automation Science and Engineering, pages 430–435, 2007.
[152] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS, 1996.
[153] D. Saint-Jacques and J. E. Baldwin. Taylor’s hypothesis: good for nuts. In P. J.
Lena and A. Quirrenbach, editors, Proceedings of SPIE: Interferometry in Optical
Astronomy, volume 4006, pages 951–962, 2000.
[154] T. Sakai and H. Kawamoto. Improvement in Time-Dependent Displacement Degradation
of Piezoelectrc Ceramics by Manganese/Indium Addition 1427. Journal of
the American Ceramic Society, 83:1423–1427, 2000.
[155] P. Salinari, C. D. Vecchio, and V. Biliotti. A study of an adaptive secondary mirror. In
F. Merkle, editor, ICO-16 Satellite conference on Active and Adaptive Optics, 1993.
[156] V. Samarkin, A. Aleksandrov, and V. Dubikovsky. Water-cooled wavefront correctors.
Advanced Optoelectronics and Lasers, 1:119, 2005.
[157] J. Sauvage, T. Fusco, G. Rousset, and C. Petit. Calibration and precompensation
of noncommon path aberrations for extreme adaptive optics. Journal of the Optical
Society of America A, 24:2334–2346, 2007.
[158] R. Schreier and G. Temes. Delta-Sigma data converters: theory, design and simulation.
IEEE Press, 1997.
[159] T. N. Science and T. R. C. for Computation and Visualization of Geometric Structures.
Qhull software, 2009. http://www.qhull.org.
[160] R. V. Shack and B. C. Platt. Production and use of a lenticular hartmann screen.
Journal of the Optical Society of America, 61:656, 1971.
[161] L. S. Shieh and Y. T. Tsay. Transformations of a class of multivariable control
systems to block companion forms. IEEE Transactions on Automatic Control,
27(1):199–203, February 1982.
[162] E. Silva, G. Goodwin, D. Quevedo, and M. Derpich. Optimal Noise Shaping for
Networked Control Systems. In Proceedings of the European Control Conference,
Kos, Greece, 2007.
[163] J. R. Sims, A. N. Durney, and C. C. Smith. Design of Mechatronic Systems with
Aliased Plant Modes. IEEE/ASME transactions on mechatronics, 3:144–149, 1998.
[164] J. Sinquin, J. Lurçon, and C. Guillemard. Deformable mirrors technologies for astronomy
at CILAS. In N. Hubin, C. E. Max, and P. L. Wizinowich, editors, Proceedings
of SPIE: Adaptive Optics Systems, volume 7015, 2008.
[165] H. Song, R. Fraanje, M. Verhaegen, and G. Vdovin. Hysteresis compensation for
piezo deformable mirror. In Proceedings of the6 th International Workshop on Adaptive
Optics for Industry and Medicine, 2007.

226 Bibliography
[166] D. P. Spanos, R. Olfati-Saber, and R. M. Murray. Approximate distributed kalman
filtering in sensor networks with quantifiable performance. In Proceedings of the4 th
international symposium on Informationprocessing in sensor networks. IEEE Press,
2005.
[167] T. Stalcup and K. Powell. Image motion correction using accelerometers at the MMT
observatory,. In Proceedings of SPIE, volume 7018, 2008.
[168] G. Stein and D. Gorinevsky. Design of surface shape control for large twodimensional
arrays. IEEE Transactions on Control Systems Technology, 13, 2005.
[169] G. Stewart, D. Gorinevsky, and G. Dumont. Feedback controller design for a spatially
distributed system: the paper machine problem. IEEE Transactions on Control
Systems Technology, 11:612–628, September 2003.
[170] P. Stoica and M. Jansson. MIMO system identification: state-space and subspace approximations
versus transfer function and instrumental variables. IEEE Transactions
on Signal Processing, 48:3087–3099, 2000.
[171] K. Szeto, S. Roberts, S. Sun, L. Stepp, J. Nelson, M. Gedig, N. Loewen, and
D. Tsang. TMT telescope structure system - design and development. In L. M.
Stepp, editor, Proceedings of SPIE: Ground-based and airborne telescopes, volume
6267, May 2006.
[172] H. Takami, S. Colley, M. Dinkins, M. Eldred, O. Guyon, T. Golot, M. Hattori,
Y. Hayano, M. Ito, M. Iye, S. Oya, Y. Saito, and M. Watanabe. Status of Subaru
Laser Guide Star AO System. In B. L. Ellerbroek and D. C. B. Calia, editors,
Proceedings of SPIE: Astronomical telescopes and instrumentation - Advances in
Adaptive Optics, volume 6272, 2006.
[173] V. Tatarski. Wavefront Propagation in a Turbulent Medium. McGraw-Hill, 1961.
[174] R.F.M.M. Hamelinck. Adaptive deformable mirror based on electromagnetic actuators.
PhD thesis, Technische Universiteit Eindhoven, 2010. ISBN 978-90-386-2278-
1.
[175] S. Thomas. Optimized centroid computing in a Shack-Hartmann sensor. In D. B.
Calia, B. L. Ellerbroek, and R. Ragazzoni, editors, Proceedings of SPIE: Advancements
in Adaptive Optics., volume 5490, pages 1238–1246, 2004.
[176] M. Tillerson, L. Breger, and J. P. How. Distributed Coordination and Control of
Formation Flying Spacecraft. In Proceedings of the American Control Conference,
2003.
[177] S. Timoshenko and S. Woinowsky-Krieger. Theory of plates and shells. McGraw-
Hill, Auckland, second edition edition, 1989.
[178] A. A. Tokovinin, B. Gregory, H. E. Schwarz, V. Terebizh, and S. Thomas. A visiblelight
AO system for the 4.2-m SOAR telescope. In P. L. Wizinowich and D. Bonaccini,
editors, Proceedings of SPIE: Adaptive Optical System Technologies II, volume
4839, pages 673–680, 2003.
[179] R. K. Tyson. Principles of adaptive optics. Academic Press, 1998.
[180] G. Vdovin and P. Sarro. Flexible mirror micro-machined in silicon. Applied Optics,
34(16):2968–2972, 1995.
[181] C. D. Vecchio. Aluminium reference plate, heat sink, and actuator design for an
adaptive secondary mirror. In D. Bonaccini and R. K. Tyson, editors, Proceedings of
SPIE: Adaptive Optical System Technologies, volume 3353, pages 839–849, 1998.

Bibliography 227
[182] C. D. Vecchio, W. Gallieni, P. Salinari, and P. Gray. Preliminary mechanical design
of an adaptive secondary unit for the MMT-conversion telescope. In M. Cullum,
editor, Proceedings of ESO, volume 54, page 243, 1996.
[183] M. Verhaegen. Subspace model identification part 3. analysis of the ordinary outputerror
state-space model identification algorithm. International Journal of Control,
56:555–586, 1993.
[184] M. Verhaegen. Identification of the deterministic part of state-space models in innovations
form from input-output data. Automatica, 30(1):61–74, 1994.
[185] M. Verhaegen and P. Dewilde. Subspace model identification part I - The outputerror
state-space model identification class of algorithms. International Journal of
Control, 56(5):1187–1210, 1992.
[186] M. Verhaegen and V. Verdult. Filtering and system identification. Cambridge university
press, New York, 1st edition, 2007.
[187] C. Vogel. Sparse matrix methods for wavefront reconstruction revisited. In B. L.
Ellerbroek and R. Ragazzoni, editors, Proceedings of SPIE: Advancements in Adaptive
Optics, volume 5490, pages 1327–1335, 2004.
[188] C. R. Vogel and Q. Yang. Fast optimal wavefront reconstruction for multi-conjugate
adaptive optics using the fourier domain preconditioned conjugate gradient algorithm.
Optics Express, 14:7487–7498, 2006.
[189] C. R. Vogel and Q. Yang. Multigrid algorithm for least-squares wavefront reconstruction.
Applied Optics, 45(4):705–715, 2006.
[190] M. Vorontsov, G. Izakson, A. Kudryashov, G. Kosheleva, S. Nazarkin, Y. F. Suslov,
and V. I. Shmalgauzen. Adaptive cooled mirror for the resonator of an industrial
laser. Soviet Journal of Quantum Electronics, 15(7):888, 1985.
[191] D. M. Wiberg, C. E. Max, and D. T. Gavel. Geometric view of adaptive optics control.
Journal of the Optical Society of America A, 22(5):870–880, May 2005.
[192] W. J. Wild, E. J. Kibblewhite, and R. Vuilleumier. Sparse matrix wave-front estimators
for adaptive-optics systems for largeground-based telescopes. Optics Letters,
20(9):955–957, May 1995.
[193] F. Wildi, G. Brusa, A. Riccardi, R. Allen, M. Lloyd-Hart, D. Miller, B. Martin, R. Biasi,
and D. Gallieni. Progress of the MMT adaptive optics program. In R. K. Tyson,
D. Bonaccini, and M. C. Roggemann, editors, Proceedings of SPIE, volume 4494,
pages 11–18, 2001.
[194] P. Wizinowich. Adaptive optics at the keck oservatory. IEEE Instrumentation and
measurement magazine, 2005.
[195] Xilinx Inc. Xilinx ISE 9.1i Software Manuals and Help.
[196] Q. Yang, C. R. Vogel, and B. L. Ellerbroek. Fourier domain preconditioned conjugate
gradient algorithm for atmospheric tomography. Applied Optics, 45(21):5281–5293,
2006.
[197] D. Young. Iterative solutions for large linear systems. Academic Press, New York,
1971.
[198] A. Zadrozny, M. Chang, D. Buscher, R. Myers, A. Doel, C. Dunlop, R. Sharples, and
R. Arnold. First Atmospheric Compensation With a Linearised High Order Adaptive
Mirror -ELECTRA. In E. D. Bonaccini, editor, ESO Conference Workshop Proceedings,
number 56, pages 459–468. ESO, 1999.

228 Bibliography
[199] K. Zhou and J. C. Doyle. Essentials of robust control. Prentice Hall, 1998.
[200] Y. Zou, P. R. Pagilla, and R. T. Ratcliff. Distributed Formation Flight Control Using
Constraint Forces. Journal of guidance, control, and dynamics, 32(1), Januari 2009.

ÙÖÖÙÐÙÑÚØ
Rogier Ellenbroek was born on Januari 28th 1979 in Heerlen (the Netherlands). He studied
Mechanical Engineering at the Technische Universiteit Eindhoven from 1997 to 2003, where
he completed his master’s degree with a project in collaboration with Philips Centre for
Applied Technologies (CFT) titled time-frequency adaptive iterative learning control with
application to a wafer stage. After graduating, he worked for a year at the department of
mechanical engineering as a visiting scientist, a.o. assisting in a PhD project on optimizing
vehicle fuel economy through intelligent battery charging.
In 2004 he started this PhD project on adaptive optics at the Delft University of Technology.
As of January 2009 he is employed by Mapper Lithography in Delft, where he currently
works as a senior systems engineer.
229