Nonlinear Control and Observer Design for Dynamic ... - NTNU

Nonlinear Control and Observer Design for Dynamic
Positioning using Contraction Theory
Guttorm Torsetnes
Master Thesis
Submitted to the Department of Engineering Cybernetics at NTNU
22 January 2004

Contents
Abstract
Acknowledgments
Abbreviations
v
vii
ix
1 Introduction 1
1.1 Dynamicpositioning ....................................... 1
1.2 Motivation ............................................ 1
1.3 Contributions........................................... 3
2 Contraction theory 5
2.1 Abasicconvergenceresult.................................... 5
2.2 Generalizedcontractionanalysis ................................ 7
2.3 Interpretation of contraction and stability . . ......................... 8
2.4 Combinationsofcontractingsystems.............................. 10
2.4.1 Parallelcombination................................... 12
2.4.2 Feedbackcombination .................................. 12
2.4.3 Hierarchicalcombination ................................ 12
3 Observer design 15
3.1 Acontraction-basedobserverdesign .............................. 15
3.1.1 Observerincompactform................................ 16
3.1.2 Observerintheframeworkofcontraction ....................... 17
3.1.3 Eliminatingtherotationmatricesfromtheanalysis.................. 18
3.1.4 Contractiontheoremfortheobserver.......................... 19
3.1.5 Observerinfeedbackform................................ 21
3.1.6 Biasmodel ........................................ 22
3.2 Acontraction-basedobserverdesignwithaccelerationmeasurements............ 23
3.2.1 Simplifiedobserverincompactform .......................... 24
3.2.2 Simplifiedobserverintheframeworkofcontraction.................. 25
3.2.3 Contraction theorem for the simplifiedobserver.................... 27
3.3 Gain-scheduled wave filtering .................................. 27
3.3.1 Observer model for gain-scheduled wave filtering ................... 28
3.3.2 Observer with gain-scheduled wave filteringincompactform ............ 29
3.3.3 Stability of LPV systems ................................ 30
3.3.4 Contraction theorem for observer with gain-scheduled wave filtering ........ 31
3.3.5 Measuring ω 0 ....................................... 32
3.4 Gain-scheduled wave filteringwithaccelerationmeasurements................ 33
3.4.1 Simplified observer with gain-scheduled wave filteringincompactform....... 33
iii

iv
CONTENTS
3.4.2 Contraction theorem for simplified observer with gain-scheduled wave filtering and
accelerationmeasurements................................ 35
3.5 Concludingremarks ....................................... 36
4 Controller design 37
4.1 Acontractionbasednonlinearseparationprinciple ...................... 37
4.2 Controlallocation ........................................ 38
4.3 Observer-feedbackcontrol .................................... 38
4.3.1 Observer-feedback control with gain-scheduled wave filtering ............ 38
4.3.2 Observer-feedback control with gain-scheduled wave filtering and acceleration feedback............................................
39
5 Simulation results 43
5.1 Observer-feedback control with gain-scheduled wave filtering................. 44
5.1.1 Moderateseastate.................................... 44
5.1.2 Veryroughseastate................................... 48
5.2 Observer-feedback control with gain-scheduled wave filtering and acceleration feedback . . 53
5.2.1 Moderateseastate.................................... 53
5.2.2 Veryroughseastate................................... 56
5.3 Concludingremarks ....................................... 60
6 Conclusion 63
A Observer data 65
B Controller data 69

Abstract
This thesis contains new results on the design of dynamic positioning (DP) systems for marine surface
vessels.
A positioned ship is continuously exposed to environmental disturbances, and the objective of the
DP system is to maintain the desired position and heading by the use of thrusters and propellers, or
to perform low-speed maneuvering. In a DP system, an observer is needed to produce low-frequency
(LF) estimates of the vessel motion. The observer includes a model of the waves, and is therefore able
to filter out the wave-frequency (WF) motion from the feedback loop. This is known as wave filtering.
The oscillatory WF motion should not enter the feedback loop because this will only cause unnecessary
wear and tear on the actuators. However, the sea state is constantly changing, and the observer should
therefore be able to automatically adjust the estimation of the LF motion to the slowly-varying wave
model parameters. This can be achieved by measuring the wave model parameters on-line, and is referred
to as gain-scheduled wave filtering.
The main contributions of this thesis are:
• Observer design: Two observers, with and without acceleration measurements, with gain-scheduled
wave filtering, both based on existing observers, have been developed and analysed. The gainscheduled
wave filtering is achieved by measuring the slowly-varying wave model parameters online,
and therefore require no a priori knowledge of the sea state.
• Stability analysis of the resulting observer-controller system by using results on nonlinear cascades:
It is shown that a PID controller (with acceleration feedback when acceleration measurements are
available) can be implemented with the estimates of the proposed observers, and that the overall
system is globally contracting, and consequently UGES.
The performance of the DP system is demonstrated in computer simulations for different environmental
disturbances.
v

vi
ABSTRACT

Acknowledgments
First and foremost I would like to thank my supervisor Professor Thor I. Fossen, and my advisors, Post
DocJeromeJouffroy and PhD student Ivar Ihle, for good advice and feedback. I am especially grateful
to Jerome Jouffroy for teaching me (almost) everything there is to know about contraction theory. Also,
I would like to thank Post Doc Lars Imsland for good advice on LMIs, PhD Karl-Petter Lindegaard
for feedback regarding his observer design and PhD student Øyvind Smogeli for helping me to run a
realistic simulation in MCSim. And last, but not least I would like to thank my (former) fellow students
Johannes Tjønnås, PhD student Morten Breivik and PhD student Jostein Hansen for countless good
advice during the last five years.
Guttorm Torsetnes
Trondheim, January 2004
vii

viii
ACKNOWLEDGMENTS

Abbreviations
UGAS
UGES
u.n.d.
LF
WF
DOF
AFB
NED
LMI
PDS
PID
LTI
LTV
LPV
Uniformly Globally Asymptotically Stable
Uniformly Globally Exponentially Stable
Uniformly Negative Definite
Low Frequency
Wave Frequency
Degree Of Freedom
Acceleration Feedback
North-East-Down coordinate system
Linear Matrix Inequality
Power Density Spectrum
Proportional Integral Derivative
Linear Time-Invariant
Linear Time-Varying
Linear Parameter-Varying
ix

x
ABBREVIATIONS

Chapter 1
Introduction
1.1 Dynamic positioning
The subject of this thesis is the design of DP systems for marine surface vessels. DP systems are designed
for station-keeping or low speed maneuvering by means of thrusters and propellers. In the systems
considered, the pitch and roll motion of the vessel is neglected, such that the mathematical model used
for control is only a description in 3 DOF as shown in figure 1.1. That is, only the horizontal-plane
position and heading is controlled. For background information on mathematical modelling of marine
vessels, the reader is referred to (Fossen 2002).
1.2 Motivation
One of the most important issues to take into account when designing DP systems for ships, is wave
filtering, and dates back to the work of Balchen and co-workers. That is, the total vessel motion is
modelled as the superposition of the low-frequency (LF) vessel motion and the wave induced wavefrequency
(WF) motion as illustrated in figure 1.2, and an observer or a filter is used to reconstruct the
LF motion from noisy position, heading and in some cases also velocity and acceleration measurements.
In (Grøvlen & Fossen 1996), UGAS of a DP system (simple ship model, observer and control law)
was proven by using observer backstepping. A revised version of this paper is found in (Fossen &
Grøvlen 1998). These papers can be regarded as the first attempt to design a fully nonlinear DPcontrol
system, since wave filtering and bias estimation was not included. In (Strand 1999), an UGES
observer with wave filtering capabilities and bias estimation was designed, which resulted in (Fossen
& Strand 1999). An extension of this observer with adaptive wave filtering appeared in (Strand &
Fossen 1999). This will be commented in detail in chapter 3.3. In (Loria, Fossen & Panteley 2000) the
same observer was used with a PD-type control law, and by using the separation principle of (Panteley
& Loria 1998), it was shown that the DP system was UGAS. Recently, in (Lindegaard 2003), this result
has been extended to the case where acceleration measurements are available, and it is argued that
the resulting observer-controller system consisting of a PID controller with acceleration feedback, and a
UGES observer, is UGAS by using the separation principle of (Panteley & Loria 1998).
However, all these designs assume that the DP system is operating in a static sea state, and/or
that the wave model parameters are known a priori. This is of course not a realistic assumption. The
sea state is constantly changing, and therefore the observer should be able to automatically adjust the
reconstruction of the LF motion in accordance with the varying sea state. To the author’s knowledge,
no observers with gain-scheduled wave filtering that require no a priori knowledge of the sea state exist
in the literature. Thus, the main motivation for this thesis has been to design such an observer for use
in a DP system for marine surface vessels.
1

2 INTRODUCTION
Figure 1.1: Earth-fixed and vessel-fixed coordinate frames.
3.5
3
2.5
2
Total motion, LF+WF
1.5
LF motion
1
WF motion
0.5
0
-0.5
1 2 3 4 5 6 7 8 9 10
Time (seconds)
Figure 1.2: The total ship motion as the sum of LF-motion and WF-motion

1.3. CONTRIBUTIONS 3
Figure 1.3: DP-system based on position and heading measurements y, with gain-scheduled wave filtering
1.3 Contributions
The main contributions of this thesis are
• Observer design: Two observers, with and without acceleration measurements (chapter 3.3 and
3.4 respectively), with gain-scheduled wave filtering, both based on existing observers, have been
developed and analysed. The gain-scheduled wave filtering is achieved by measuring the slowlyvarying
wave model parameters on-line, and therefore require no a priori knowledge of the sea
state.
• Stability analysis of the resulting observer-controller system by using results on nonlinear cascades:
It is shown that a PID controller (with acceleration feedback when acceleration measurements are
available) can be implemented with the estimates of the proposed observers, and that the overall
system is globally contracting, and consequently UGES. The total system is illustrated in figure
1.3
Besides, all the proofs given here, both for observer and controller design, are done using contraction
theory, which is a new tool to study stability of nonlinear systems, and to a small extent has been used in
the design of DP systems. Contraction theory is very powerful, and combined with the simplicity of the
design, it will be shown that this is a good alternative to the classical Lyapunov approach. The results
given herein can be regarded as an extension of the work of Lindegaard to include gain-scheduled wave
filtering by using contraction theory, and the main results will be summarized in a paper to be submitted
to the IEEE Conference on Decision and Control (CDC), December 2004 at Atlantis, Paradise Island,
the Bahamas.

4 INTRODUCTION

Chapter 2
Contraction theory
Contraction theory is a recent tool to study incremental stability of systems of differential equations, and
was brought to attention in the control engineering community by the work of Slotine and Lohmiller.
Incremental stability means stability of trajectories with respect to one another, rather than with respect
to some attractor. For more information on the history of contraction theory, the reader is referred to
(Jouffroy 2003a) and(Jouffroy 2002, section 2.3).
In this chapter, the main results of contraction theory are summarized. The summary is mainly
based on (Lohmiller & Slotine 1998), to which the reader is referred for further details.
2.1 A basic convergence result
The systems considered is of the form
ẋ = f(x, t) (2.1)
where f(x, t) is an n × 1 nonlinear vector function and x is the n × 1 state vector. Note that (2.1)
implicitly also represent closed-loop dynamics of systems with state feedback u(x, t). All quantities in
this thesis are assumed to be real and smooth, by which is meant that any required derivative or partial
derivativesexistsoriscontinuous.
The system in equation (2.1) can be thought of as an n-dimensional fluid flow, where ẋ is the
n-dimensional "velocity" vector at the n-dimensional position x at time t. Assuming that f(x, t) is
continuously differentiable, equation (2.1) yields the exact differential relation
δẋ= ∂f (x, t)δx, (2.2)
∂x
where δx is the virtual displacement, which is an infinitesimal displacement at a fixed time. Consider
now two neighboring trajectories in the flow field ẋ = f(x, t), and the virtual displacement δx between
them. This is illustrated in figure 2.1. The squared distance between these two trajectories can be
defined as δx T δx, leading from (2.2) to the rate of change of length
d
dt (δxT δx) =2δx T δẋ =2δx T ∂f δx (2.3)
∂x
Denote by λ max (x, t) the largest eigenvalue of the symmetric part of the Jacobian ∂f
∂x
(i.e. the largest
eigenvalue of 1 2 ( ∂f
∂x + ∂f
∂x
T
) ), then
5

6 CONTRACTION THEORY
Figure 2.1: Virtual dynamics of two neighboring trajectories
and hence,
d
dt (δxT δx) ≤ 2λ max δx T δx (2.4)
kδxk ≤ kδx 0 k e R t
0 λ max(x,t)dt
(2.5)
Assume now that λ max (x, t) is uniformly strictly negative (i.e. ∃ β>0, ∀x, ∀ t ≥ 0, λ max (x, t) ≤
−β 0, ∀x,∀t ≥ 0, 1 ∂f
2 ∂x + ∂f T
≤−βI < 0. (2.6)
∂x
Consider now a ball of constant radius centered about a given trajectory, such that given this trajectory,
the ball remains within a contracting region at all times (i.e.,∀ t ≥ 0). Because any length
within the ball decreases exponentially, any trajectory starting in the ball, remains in the ball (since
by definition the center of the ball is a particular system trajectory) and converges exponentially to
the given trajectory. Thus, as in stable linear time-invariant (LTI) systems, the initial conditions are
exponentially "forgotten." This leads to the following theorem:
Theorem 2.1 Given the system equations ẋ = f(x, t) any trajectory, which starts in a ball of constant
radius centered about a given trajectory and contained at all times in a contraction region, remains in
that ball and converges exponentially to this trajectory.
Furthermore, global exponential convergence to the given trajectory is guaranteed if the whole statespace
is a contraction region.

2.2. GENERALIZED CONTRACTION ANALYSIS 7
Figure 2.2: Two trajectories in a contraction region
2.2 Generalized contraction analysis
Theaboveconsiderationscanalsobedoneusingthedifferential coordinate transformation
δz = Θ(x, t)δx (2.7)
where Θ(x, t) is a square matrix. The time derivative of the local coordinates δz can be written as
d
dt δz = ˙Θδx + Θδẋ (2.8)
= (˙Θ + Θ ∂f
∂x )Θ−1 δz = Fδz (2.9)
where F =(˙Θ + Θ ∂f
∂x )Θ−1 is called the generalized Jacobian. Therateofchangeofsquaredlength
can be written as
d
dt (δzT δz) =2δz T d dt δz =2δzT Fδz (2.10)
Hence, the system ẋ = f(x, t) is in a contraction region if the generalized Jacobian is u.n.d. in that
region. The previous rate of change can also be expressed in δx coordinates, and is given by
d
dt (δxT Mδx)=δx T ( ∂f T
M +
∂x Ṁ + M ∂f )δx (2.11)
∂x
where M = Θ T Θ, so that exponential convergence to a single trajectory can be concluded in regions
where
Ã
!
T
∂f
M +
∂x Ṁ + M ∂f ≤−β
∂x
M M (2.12)
where β M > 0. It is immediate to verify that these regions are of course exactly the regions of
uniformly negative definite F in equation (2.9). If M is restricted to be a constant, this exponential
convergence result represents a generalization and strengthening of Krasovskii’s generalized asymptotic
global convergence theorem. It may also be regarded as an extension of the Lyapunov matrix equation
in LTI systems due to the term Ṁ.

8 CONTRACTION THEORY
The above leads to the following generalized definition, superseding Definition 2.1 (which corresponds
to Θ = I and M = I),
Definition 2.2 Given the system equations ẋ = f(x, t), a region of the state space is called a contraction
region with respect to a uniformly positive definite metric M(x, t) =Θ T Θ, if equivalently F in equation
(2.9) or
T
∂f
M +
∂x Ṁ + M ∂f
∂x
are uniformly negative definite in that region.
(2.13)
and the generalized convergence result can be stated as:
Theorem 2.2 Given the system equation ẋ = f(x, t), any trajectory, which starts in a ball of constant
radius with respect to the metric M(x, t), centered at a given trajectory and contained at all times in
a contraction region with respect to M(x, t), remains in that ball and converges exponentially to this
trajectory.
Furthermore global exponential convergence to the given trajectory is guaranteed if the whole state
space is a contraction region with respect to the metric M(x, t).
It can be shown conversely that the existence of a uniformly positive definite metric with respect
to which the system is contracting is also a necessary condition for global exponential convergence of
trajectories. In the linear time-invariant case, a system is globally contracting if and only if it is strictly
stable, with F simply being a normal Jordan form of the system and Θ the coordinate transformation
to that form (Lohmiller & Slotine 1998),(Slotine 2003).
2.3 Interpretation of contraction and stability
To better understand how stability considerations for linear systems can be extended to nonlinear systems
by using virtual displacements as in contraction theory, consider the following autonomous LTI system 1 :
ẋ = Ax, x ∈ R 2 (2.14)
Asufficient condition for exponential convergence of the system states to the origin, is that the rate
of change of squared length converges exponentially to zero:
Thus, a sufficient condition is that
d
dt (xT x) = ẋ T x + x T ẋ (2.15)
= x T (A T + A)x (2.16)
A = λ max I

2.3. INTERPRETATION OF CONTRACTION AND STABILITY 9
x T x ≤ x(0) T x(0)e 2λ maxt
(2.18)
and consequently
kxk ≤ kx(0)k e λ maxt
(2.19)
which means that x converges exponentially to the origin with the convergence rate determined by
λ max . Evidently, the same result is obtained by using Lyapunov’s direct method with the Lyapunov
function V = x T x. A geometrical interpretation of the above is that
which from elementary calculus can be rewritten as
ẋ T x

10 CONTRACTION THEORY
Thus, if there exists a matrix M = Θ T T
Θ > 0 such that A (t)M(t)+Ṁ(t)+M(t)A(t) < 0, z(and
consequently x) converges exponentially to the origin.
Finally, consider the nonlinear time-varying system
ẋ = f(x, t) (2.31)
The differential dynamics for this system is given by
δẋ = ∂f (x, t)δx (2.32)
∂x
and the rate of change of squared length is given by
This means that in the region where
d
dt (δxT δx) = δẋ T δx + δx T δẋ (2.33)
= δx T ( ∂f
∂x
T
T
+ ∂f )δx (2.34)
∂x
∂f
+ ∂f
∂x ∂x < 0 (2.35)
the distance between the trajectories converges exponentially to zeros. Hence, all trajectories in
this region converges to the same trajectory as summarized in theorem 2.1. However, (2.35) is not
a necessary condition for exponential incremental stability. Consider the time and state-dependent
differential coordinate transformation
δz = Θ(x, t)δx (2.36)
Therateofchangeofsquaredlengthinδx-coordinates can be written as
d
dt (δxT M(x, t)δx) = δẋ T M(x, t)δx + δx T Ṁ(x, t)δx + δx T M(x, t)δẋ (2.37)
= δx T ∂f
∂x
Ã
= δx T
T
(x, t)M(x, t)δx + δx T
∂f
∂x
Ṁ(x, t)δx + δx T M(x, t) ∂f (x, t)δx (2.38)
∂x
!
T
(x, t)M(x, t)+Ṁ(x, t)+M(x, t)∂f
∂x (x, t) δx (2.39)
Thus, in the region where there exists a metric M = Θ T Θ > 0 such that
T
∂f
M +
∂x Ṁ + M ∂f < 0 (2.40)
∂x
the distance between the trajectories converges exponentially to zeros. If this region is the whole
state-space, then all trajectories in the state-space converge exponentially to the same trajectory, and
this is called global contraction.
2.4 Combinations of contracting systems
One of the main features of nonlinear contraction, is that it is automatically preserved through a variety of
system combinations. In this section, three combination properties of contracting systems are presented
(Lohmiller & Slotine 1998),(Slotine 2003).
T

2.4. COMBINATIONS OF CONTRACTING SYSTEMS 11
Figure 2.3: Exponential convergence of the system states to the origin with no overshoot
Figure 2.4: Exponential convergence of the system states to the origin with overshoot
Figure 2.5: Exponential convergence of the system states to the origin in z-coordinates

12 CONTRACTION THEORY
2.4.1 Parallel combination
Consider two systems of the same dimension
with virtual dynamics
ẋ 1 = f 1 (x 1 ,t) (2.41)
ẋ 2 = f 2 (x 2 ,t) (2.42)
δż 1 = F 1 δz (2.43)
δż 2 = F 2 δz (2.44)
If both systems are contracting in the same metric, then so is any uniformly positive superposition
α 1 (t)δż 1 + α 2 (t)δż 2 where ∃ α>0, ∀t ≥ 0, α i (t) ≥ α(t) (2.45)
By recursion, this property can be extended to any number of systems.
2.4.2 Feedback combination
Consider two systems, of possibly different dimensions and metrics
ẋ 1 = f 1 (x 1 ,x 2 ,t) (2.46)
ẋ 2 = f 2 (x 1 ,x 2 ,t) (2.47)
which are connected in feedback, in such a way that the overall virtual dynamics is of the form
· ¸ · ¸· ¸
d δz1 F1 G δz1
=
dt δz 2 −G T (2.48)
F 2 δz 2
then the augmented system is contracting if the separated plants are contracting. The result can be
extended to any number of systems.
2.4.3 Hierarchical combination
Consider two contracting systems, of possibly different dimensions and metrics, connected in series with
virtual dynamics of the form
· ¸ · ¸· ¸
d δz1 F11 0 δz1
=
(2.49)
dt δz 2 F 21 F 22 δz 2
The first equation does not depend on the second, so that exponential convergence of δz 1 to zero
can be concluded for uniformly negative definite F 11 . In turn for bounded F 21 , F 21 δz 1 represents an
exponentially decaying disturbance in the second equation. Thus, uniform negative definitiveness of F 22
implies exponential convergence of δz 2 to zero. The augmented system is contracting as well, and by
recursion, the result extends to hierarchies or cascades of contracting systems of arbitrary depths.
Note that the hierarchical combination (2.49) corresponds to the same kind of combination considered
in the work of Loria and Panteley with coworkers, referred to as cascades of nonlinear systems. In
(Panteley & Loria 1998) conditions for UGAS for systems of the form
ẋ 1 = f 1 (t, x 1 ) (2.50)
ẋ 2 = f 2 (t, x 2 )+g(t, x)x 1 (2.51)

2.4. COMBINATIONS OF CONTRACTING SYSTEMS 13
are given. One necessary condition is that the interconnection term g(t, x) is bounded. Also note
the results established by Sontag on cascades of input-to-state stable (ISS) systems. In (Sontag 1989),
systems of the form
ẋ = f(x)+
mX
u i g(x), ∈ R n (2.52)
i=1
are considered, and conditions are given for the existence of a smooth map K : R n → R m such that
is GAS.
ẋ = f(x)+G(x)K(x)+G(x)u (2.53)

14 CONTRACTION THEORY

Chapter 3
Observer design
This chapter consists of two parts: In the first part, it is shown that the observer of (Fossen & Strand
1999) and the simplified observer of (Lindegaard 2003) are globally contracting (section 3.1 and 3.2
respectively). In the second part, a new version of the above mentioned observers with gain-scheduled
wave filtering is presented (section 3.3 and 3.4). These observers are particular contributions of this
thesis.
3.1 A contraction-based observer design
In this section, the observer of (Fossen & Strand 1999), which is based on position and heading measurements
only, is analysed in the framework of contraction theory, and it is shown that this observer
is globally contracting with respect to a metric M 1 . First we recapitulate the observer equations. The
model of which the observer is based (neglecting the white noise terms) is given by
˙ξ = A w ξ (3.1)
˙η = R(ψ)ν (3.2)
ḃ = −T −1
b
b (3.3)
M ˙ν + Dν = τ + R T (ψ)b (3.4)
y = η + C w ξ (3.5)
where η =[x, y, ψ] T is the vector of the NED position (x, y) and heading (ψ) The yaw angle rotation
matrix is defined as
⎡
⎤
cos(ψ) − sin(ψ) 0
R(y 3 )=R(ψ) = ⎣sin(ψ) cos(ψ) 0⎦ (3.6)
0 0 1
Further, the following assumptions are made (low speed application)
M = M T > 0, Ṁ =0, D > 0 (3.7)
and the time bias matrix T b > 0. The first-order WF motion is modelled as a second order linear
system for each of the 3 degrees of freedom:
h i (s) = η i
w w
(s) =
k wi s
s 2 +2ζ i ω 0i s + ω 2 0i
(3.8)
15

16 OBSERVER DESIGN
where
ω 0i (i=1,2,3)
ζ i (i=1,2,3)
k wi (i=1,2,3)
dominating wave frequency
relative damping ratio
parameter related to the wave intensity
A minimal state-space realization with x w ∈ R 6 is given by
with
A w =
ẋ w = A w x w + E w w w (3.9)
y = C w x w (3.10)
·
03×3 I 3×3
−Ω 3×3 −Λ 3×3¸
E w =
¸ ·03×1
w w
(3.11)
C w = [0 3×3 ,I 3×3 ] (3.12)
where
Ω
Λ
= diag{ω 2 01,ω 2 02,ω 2 03}
= diag{2ζ 1 ω 01 , 2ζ 2 ω 02 , 2ζ 3 ω 03 }
w w ∈ R 3 = vector of zero-mean Gaussian white noise
The observer is given by:
·
ˆξ = Awˆξ + K1 (y − ŷ) (3.13)
·
ˆη = R(ψ)ˆν + K 2 (y − ŷ) (3.14)
·
ˆb = −T
−1
b
ˆb + K3 (y − ŷ) (3.15)
M ˆν ·
+ Dˆν = τ + R T (ψ)ˆb + R T (ψ)K 4 (y − ŷ) (3.16)
ŷ = ˆη + C wˆξ (3.17)
Note: The observer is based on a priori knowledge of the sea state, which means that the parameters
of A w are assumed to be known. This means that the observer gains can be tuned according to the
peak wave frequency ω 0 to give a notch filter effect which removes 1st order wave components from the
feedback-loop. The reader is referred to (Fossen & Strand 1999) for the observer tuning rules.
3.1.1 Observer in compact form
The observer can be compactly written as
where ˆx =[ˆξ T , ˆη T , ˆb T , ˆν T ] T , y = η + C w ξ and
·
ˆx = A(ψ)ˆx + Bτ + K y y (3.18)

3.1. A CONTRACTION-BASED OBSERVER DESIGN 17
A(ψ) =
⎡
⎢
⎣
A w − K 1 C w −K 1 0 0
−K 2 C w −K 2 0 R(ψ)
−K 3 C w −K 3 −T −1
b
0
−M −1 R T (ψ)K 4 C w −M −1 R T (ψ)K 4 M −1 R T (ψ) −M −1 D
⎤
⎥
⎦ , (3.19)
B = £ 0 12×3 M −1¤ T
, (3.20)
K y = [K1 T ,K2 T ,K3 T ,K4 T R(ψ)M −1 ] T (3.21)
3.1.2 Observer in the framework of contraction
From chapter 2 we conclude that there are generally two ways of verifying that a system of the form
ẋ = f(x, t) is contracting:
1. The Jacobian of the differential dynamics is uniformly negative definite (u.n.d.):
∂f
(x, t) is u.n.d. (3.22)
∂x
2. The Generalized Jacobian of the differential dynamics is u.n.d. This implies that there exists a
positive definite metric M(x, t) =Θ T Θ such that
∂f
∂f
(x, t)M + Ṁ + M (x, t) is u.n.d. (3.23)
∂x ∂x
Note that the first case is a special of the second with M = I. The differential dynamics of (3.18) is
given by
·
δˆx = ∂ ˆf
δˆx = A(ψ)δˆx (3.24)
∂ˆx
Thus, for the observer to be contracting with respect to the metric M 1 = I, A(ψ) must be u.n.d.,
that is
A T (ψ)+A(ψ) < 0 (3.25)
When inspecting A(ψ) it is seen from equation (3.11) and (3.12) that the upper left element (A w −
K 1 C w ) has zeros in the upper left corner for any K 1 , which from Sylvester’s criterion imply that A(ψ)
can only be negative semidefinite. Hence, the above condition is not fulfilled. Thus, we have to look for
ametricM 1 6= I. We therefore introduce the following differential coordinate transformation
δẑ = Θδˆx (3.26)
where Θ is a constant matrix. This leads to the new differential dynamics
δ ẑ ·
= ΘA(ψ)Θ −1 δẑ (3.27)
hence contraction requires the new Jacobian to be u.n.d.
which can be rewritten as
ΘA(ψ)Θ −1 < 0 (3.28)
A T (ψ)M 1 + M 1 A(ψ) < 0 (3.29)

18 OBSERVER DESIGN
where M 1 = Θ T Θ > 0. From a theoretical point of view, one may say that if equation (3.29) is
fulfilled, then the observer is contracting. However, equation (3.29) is very difficult to solve in practise.
Since A(ψ) includes the rotation matrix R(ψ), the equation must be solved for every possible measured
heading ψ. In other words, this result is not suited for practical purposes. Since equation (3.29) is
too complex to be solved with standard software, no conclusion can be made regarding the contraction
properties of the observer.
Evidently, if the rotation matrices (which is the only term making the model nonlinear) could be
disregarded, the contraction condition would be a LMI, and thus solved with standard software.
3.1.3 Eliminating the rotation matrices from the analysis
In this section it is shown that the rotation matrices can be disregarded in the stability analysis by
utilizing the observer error-dynamics-form of (Lindegaard 2003, section 4.4.2 p.53). First we recapitulate
the necessary matrix property from (Lindegaard 2003):
Property 1 A matrix A ∈ R 3 is said to commute with the rotation matrix R(α) if
AR(α) =R(α)A (3.30)
Examples of matrices A satisfying Property 1 are linear combinations A = a 1 R(θ) +a 2 I + a 3 k T k
for scalars a i ,θ and k =[0, 0, 1] T , the axis of rotation. Also note that since R(α) is orthogonal, that is,
R T (α) =R −1 (α), Property 1 implies that
A = R T (α)AR(α) =R(α)AR T (α) (3.31)
Furthermore, if A is nonsingular, A −1 commutes with R(α) too. That is
AR(α) =R(α)A
A is nonsingular
⇐⇒ A −1 R(α) =R(α)A −1 (3.32)
Thus, given the following assumption
Assumption A1 The bias time constant matrix T b , the observer gains K 2 and K 3 and each 3 × 3
sub-block of A w , and K 1 satisfies Property 1
the observer of (Fossen & Strand 1999), can be compactly written as
A =
⎢
⎣
·
ˆx = T T (ψ)AT (ψ)ˆx + Bτ + K y y (3.33)
where ˆx =[ˆξ T , ˆη T , ˆb T , ˆν T ] T , T (ψ) =diag{R T (ψ),R T (ψ),R T (ψ),I 3x3 },y= η + C w ξ and
⎡
A w − K 1 C w −K 1 0 0
⎤
−K 2 C w −K 2 0 I
−K 3 C w −K 3 −T −1
b
0
−M −1 K 4 C w −M −1 K 4 M −1 −M −1 D
The differential dynamics of the observer can therefore be written as
Hence, it is required that J is u.n.d. which corresponds to
⎥
⎦ (3.34)
δ ·
ˆx = T T (ψ)AT (ψ)δˆx = Jδˆx (3.35)
J T + J = T T (ψ)A T T (ψ)+T T (ψ)AT (ψ) < 0 (3.36)

3.1. A CONTRACTION-BASED OBSERVER DESIGN 19
However, since A

20 OBSERVER DESIGN
For the sake of comparison, we will use the same parameters as in (Fossen & Strand 1999). That is,
ω 0i =0.8976 rad/s, ζ i =0.1, and the cut-off frequency in the notch filter is chosen as ω ci =1.1 rad/s
and the ’notch width’ ζ ni =1. By using the tuning rules of (Fossen & Strand 1999) this gives values for
K 1 and K 2 . In addition
K 4 = diag{0.1, 0.1, 0.01} (3.45)
K 3 = 0.1K 4 (3.46)
T b = diag{1000, 1000, 1000} (3.47)
This set of parameters gives the system matrix A in equation (??), and by using YALMIP (Løfberg
2003) with Matlab TM ,themetricM 1 can be found. The calculations were performed with the SeDuMi
(Sturm 2001) solver, and it can be verified in Matlab TM that M 1 is a positive definite, symmetric matrix
M 1 = M1
T > 0 that fulfills equation (3.42). Since M 1 ∈ R 15×15 ,M 1 is not stated due to lack of space,
however it can be mentioned that for the given parameters, the resulting calculations gave
max{eig(A T M 1 + M 1 A)} = −1.0396 × 10 −4 (3.48)
min{eig(A T M 1 + M 1 A)} = −1.2649 (3.49)
¯σ(M 1 ) = 4.1771 (maximum singular value) (3.50)
σ(M 1 ) = 0.007195 (minimum singular value) (3.51)
which clearly satisfies the contraction condition. Thus, based on these calculations we conclude that
the observer of (Fossen & Strand 1999) is globally contracting for the given set of parameters. In order to
visualize that the observer is contracting with respect to a metric M 1 6= I, we can simulate the observer
for two different initial conditions, and plot the Euclidean norm of the difference of the states:
where
kˆx 1 (t) − ˆx 2 (t)k , ˆx 1 (0) = ˆx 10 , ˆx 2 (0) = ˆx 20 (3.52)
ˆx 1 =ˆx 2 =
h
ˆξT
Figure 3.1 shows the incremental state norm with the initial conditions
ˆη T ˆbT ˆν T i T
(3.53)
ˆx 10 = £ 0 1×3 0 1×3 0 1×3 0 1×3
¤ T
(3.54)
ˆx 20 = £ 0 1×3 0 1×3 0 1×3
£ −1 01×2
¤¤ T
(3.55)
from a simulation in Matlab TM without environmental disturbances. The only difference in the initial
condition is the forward velocity which is estimated to be û 20 = −1 m/s in one observer and zero in the
other. The input to the ship model (with initial conditions ˆx 10 ) is a constant force of 100kN in forward
direction, that is τ =[1× 10 5 , 0, 0] T . The observer gains were chosen as stated above. From figure 3.1
it can be seen that the curve has an overshoot., The reason for this is summarized in theorem 2.2 in
(Jouffroy & Lottin 2002):

Incremental state norm
3.1. A CONTRACTION-BASED OBSERVER DESIGN 21
4
3.5
3
2.5
2
1.5
1
0.5
0
0 50 100 150 200 250 300
time [s]
Figure 3.1: Incremental state norm kˆx 1 (t) − ˆx 2 (t)k of the observer with initial conditions ˆx 10 and ˆx 20
Theorem 3.2 If the system ẋ = f(x, t) is globally contracting with respect to a constant Θ, then it is
also UGES in the incremental sense, i.e. ∃ k, b > 0 such that ∀t ≥ 0, ∀ x r0 ,x p0 ∈ R n ,
kx r − x p k ≤ k kx r0 − x p0 k e −bt (3.56)
where x r =(x r0 ,t), x p =(x p0 ,t) and k•k denotes the Euclidean norm.
Thus, the observer is not contracting with respect to M 1 = I, because this would correspond to a
curve bounded by an exponential which starting point is the norm of the initial value of the difference
kx 1 (0) − x 2 (0)k. The overshoot is bounded by the minimum and maximum singular value of M 1 .Also
note that kx 1 (t) − x 2 (t)k is converging quite slowly to zero. This is due to the choice of observer gains.
In this case, the choice of K 2 and K 4 will have significant influence on the convergence rate, but both
gains are quite low. The reason for this is that K 2 is used to determine the cut-off frequency of the
notch filter ω c >ω 0 , and the choice of K 4 is based on the tuning rule (Fossen & Strand 1999, section 5)
1/T bi ¿ K 3i /K 4i

22 OBSERVER DESIGN
where F 1 ,F 2 < 0 for Ṁ =0. The feedback form is very attractive because the cross-terms are
cancelled, and this form is therefore used in Backstepping techniques etc. In (Jouffroy & Lottin 2002),
the feedback form was used to show that the observer design in (Fossen & Grøvlen 1998) could be
performed using contraction theory. By performing the same approach as in (Jouffroy & Lottin 2002)
for the observer of (Fossen & Strand 1999), the resulting conditions are given by
A T 11M 11 + M 11 A 11 < 0 (3.59)
A T 22 M 22 + M 22 A 22 < 0 (3.60)
M11A T 12 + A T 21M 22 = 0 (3.61)
where M 11 = M11 T > 0, M 22 = M22 T > 0 is to be found and
A 11 =
A 21 =
¸
·Aw + K 1 C w K 1
·0 0
, A
K 2 C w K 12 =
2¸
0 I
·
¸
−1
K 3 C w K 3
·−T
M −1 K 4 C w M −1 ,A
K 22 =
b
0
4¸
M −1 −M −1 D
(3.62)
(3.63)
That is, if equations (3.59)-(3.61) are fulfilled, the observer (3.13)-(3.17) is contracting with respect
to M 11 and M 22 and has the feedback form. When comparing the general condition (equation (3.42))
with the conditions for the system to have the feedback form (equations (3.59)-(3.61)), it is seen that the
conditions for the observer to have the feedback form is much stricter. This can be seen by the equality
constraint (3.61), which means that the equations can no longer be solved as a LMI (the search for M 11
and M 22 must be performed with semidefinite programming), and therefore the choice of M 11 and M 22
is severely limited as opposed to the case where there are no equality constraint.
Equations (3.59)-(3.61) were calculated with Yalmip (Løfberg 2003) in Matlab TM using the SeDuMi
(Sturm 2001) solver for the given set of parameters, however no solution could be found.
3.1.6 Bias model
From a physical point of view, one should use ḃ =0+w w in the model instead of the low-pass filtered
ḃ = −T −1
b
b + w w , which means that the bias is modelled as integration of white noise (random walk).
When calculating equation (3.42) for T b =0with Yalmip (Løfberg 2003) in Matlab TM using the SeDuMi
(Sturm 2001) solver for the given set of parameters, the following results were obtained:
max{eig(A T M 1 + M 1 A)} = −2.1948 × 10 −12 (3.64)
min{eig(A T M 1 + M 1 A)} = −0.8622 (3.65)
¯σ(M 1 ) = 3.0373 (maximum singular value) (3.66)
σ(M 1 ) = 0.0054476 (minimum singular value) (3.67)
Hence, the observer is also globally contracting (which from Theorem 3.2 implies UGES) for T b =0.
Also note that ¯σ(M 1 ) is less in this case than when using T b = diag{1000, 1000, 1000} as above, which
means that the overshoot as illustrated in figure 3.1 is less.

3.2. A CONTRACTION-BASED OBSERVER DESIGN WITH
ACCELERATION MEASUREMENTS 23
3.2 A contraction-based observer design with acceleration measurements
In this section, the simplified observer of (Lindegaard 2003), which is based on position and acceleration
measurements, is analysed in the framework of contraction theory, and it is shown that this observer is
globally contracting with respect to a metric M a . The model of which the observer is based, neglecting
the white noise terms and without mooring forces (G =0), is given by
ṗ w = A pw p w , ∈ R 6 (3.68)
˙η = R(ψ)ν, ∈ R 3 (3.69)
ḃ = −T −1
b
b, ∈ R 3 (3.70)
˙ν w = A νw ν w , ∈ R nνw (3.71)
ȧ w = A aw a w , ∈ R naw (3.72)
˙ν = −M −1 Dν + M −1 τ + M −1 R T (ψ)b + K a a f , ∈ R 3 (3.73)
ȧ f = −T −1
f
a f , a f (0) = 0 (3.74)
y 1 = η + C pw p w , ∈ R n y1
(3.75)
y 2 = Υ 2 ν + C νw ν w , ∈ R n y2
(3.76)
y 3 = Υ 3 ˙ν + C aw a w , ∈ R n y3
(3.77)
where η =[x, y, ψ] T is the vector of the NED position (x, y) and heading (ψ). The yaw angle rotation
matrix is defined as
⎡
⎤
cos(ψ) − sin(ψ) 0
R(y 3 )=R(ψ) = ⎣sin(ψ) cos(ψ) 0⎦ (3.78)
0 0 1
Υ 2 and Υ 3 are projections isolating the components of the low frequency (LF) model that are actually
measured.
Remark: Note that equation (3.74) is added to the model as opposed to (Lindegaard 2003). This
means that the model belongs to a bigger class of systems than the model of (Lindegaard 2003), however
for the initial condition a f (0) = 0, the models are identical. This relies on the concept of general
systems, which is described in (Jouffroy 2003b). The reason for adding equation (3.74) to the model, is
to assure that the observer model is a particular solution of the observer, and thus global contraction of
the observer will imply exponential convergence of the estimates to the actual system trajectories.
Also note that the acceleration measurement (equation (3.77)) is a simplification which is only valid
for small angular rates (Lindegaard 2003, chapter 4.4, p.52).
Further, the following assumptions are made (low speed application)
M = M T > 0, Ṁ =0, D > 0 (3.79)
The measurements can be written compactly as
⎡
y = ⎣ y ⎤
1
y 2
⎦ = C y (ψ)x + D y τ (3.80)
y 3
where x =[p T w,η T ,b T ,ν T w,a T w,ν T ] and

24 OBSERVER DESIGN
⎡
C y (ψ) = ⎣ C ⎤
pw I 0 0 0 0
0 0 0 C νw 0 Υ 2
⎦ (3.81)
0 0 Υ 3 M −1 R T (ψ) 0 C aw −Υ 3 M −1 D
D y = £ ¤
0 0 M −T Υ T T
3
(3.82)
The simplified observer is given by:
˙ˆp w = A pw ˆp w + K 11 (y 1 − ŷ 1 ) (3.83)
·
ˆη = R(ψ)ˆν + K 21 (y 1 − ŷ 1 ) (3.84)
·
ˆb = −T
−1
b
ˆb + K31 (y 1 − ŷ 1 ) (3.85)
·
ˆν w = A νwˆν w + K 42 (y 2 − ŷ 2 ) (3.86)
·
â w = A aw â w + K 53 (y 3 − ŷ 3 ) (3.87)
·
ˆν = −M −1 Dˆν + M −1 τ + M −1 R T (ψ)ˆb +
R T (ψ)K 61 (y 1 − ŷ 1 )+K 62 (y 2 − ŷ 2 )+K a a f (3.88)
ȧ f = −T −1
f
a f + T −1
f
(y 3 − ŷ 3 ) (3.89)
ŷ 1 = ˆη + C pw ˆp w (3.90)
ŷ 2 = Υ 2ˆν + C νwˆν w (3.91)
ŷ 3 = Υ 3ˆ˙ν + C aw â w (3.92)
Note the introduction of a low-pass filter in equation (3.89). The observer is based on a priori
knowledge of the sea state, which means that the parameters of the wave models are assumed to be
known. This means that the observer gains can be tuned according to the peak wave frequency ω 0 to
give a notch filter effect which removes 1st order wave components from the feedback-loop. The reader
is referred to (Lindegaard 2003) for the observer tuning rules.
3.2.1 Simplified observer in compact form
Given the following assumption
Assumption A2 The bias time constant matrix T b ,andeach3 × 3 sub-block of A pw ,K 11 ,K 21 and
K 31 commutes with the rotation matrix R(ψ) (Property 1).
the observer can be compactly written as
˙ẑ = T T z (ψ)A z T z (ψ)ẑ + B τ τ + K(ψ)y (3.93)
where ẑ =[ˆp T w, ˆη T , ˆb T , ˆν T w, â T w, ˆν T ,a T f ]T ∈ R nz , y as given in equation (3.80) and
T z (ψ) =diag{R T (ψ),R T (ψ),R T (ψ),R T (ψ),I 6+nνw +n aw
} (3.94)

3.2. A CONTRACTION-BASED OBSERVER DESIGN WITH
ACCELERATION MEASUREMENTS 25
B τ = £ 0 0 0 0 0 M −1 0 ¤ T
⎡
⎤
K 11 0 0
K 21 0 0
K 31 0 0
K(ψ) =
0 K 42 0
⎢
0 0 K 53
⎥
⎣K 61 R(ψ) K 62 0 ⎦
0 0 T −1
f
(3.95)
(3.96)
· ¸
A0 K f
A z =
T −1
f
C 3 −T −1
(3.97)
f
C 3 = £ 0 0 Υ 3 M −1 0 C aw −Υ 3 M −1 D ¤ (3.98)
K f = £ ¤
0 0 0 0 0 Ka
T T
(3.99)
A 0 =
·A11 A 12
A 21 A 22¸
(3.100)
⎡
⎤
A pw − K 11 C pw −K 11 0
A 11 = ⎣ −K 21 C pw −K 21 0 ⎦ (3.101)
−K 31 C pw −K 31 −T −1
b
⎡
A 12 = ⎣ 0 0 0 ⎤
0 0 I⎦ (3.102)
0 0 0
⎡
⎤
0 0 0
A 21 = ⎣ 0 0 −K 53 Υ 3 M −1 ⎦ (3.103)
−K 61 C pw −K 61 M −1
⎡
A 22 = ⎣ A ⎤
νw − K 42 C νw 0 −K 42 Υ 2
0 A aw − K 53 C aw K 53 Υ 3 M −1 D ⎦ (3.104)
−K 62 C νw 0 −M −1 D − K 62 Υ 2
3.2.2 Simplified observer in the framework of contraction
The differential dynamics of (3.93) is given by
δ ˙ẑ = Tz T (ψ)A z T z (ψ)δẑ = Jδẑ (3.105)
Thus, for the observer to be contracting with respect to the metric M a = I, J must be u.n.d., that is
J T + J = Tz T (ψ)A T z T z (ψ)+Tz T (ψ)A z T z (ψ) < 0 (3.106)
However, since A z < 0 ⇒ Tz T (ψ)A z T z (ψ) < 0, the above condition is equal to
A T z + A z < 0 (3.107)
When inspecting A z it is seen from equation (3.11) and (3.12) (C pw = C w ), that the upper left element
(A w − K 11 C pw ) has zeros in the upper left corner for any K 11 , which from Sylvester’s criterion imply

26 OBSERVER DESIGN
that A z canonlybenegativesemidefinite. Hence, the above condition is not fulfilled. Thus, we have to
look for a metric M a 6= I. We therefore introduce the following differential coordinate transformation
δˆϑ = Θ a δẑ (3.108)
where Θ a is a constant matrix. This leads to the new differential dynamics
·
δˆϑ = Θ a A z Θ −1
a δˆϑ (3.109)
Contraction requires the new Jacobian to be u.n.d.
which can be rewritten as
where M a = Θ T a Θ a > 0.
Θ a A z Θ −1
a < 0 (3.110)
A T z M a + M a A z < 0 (3.111)
Limitation on the angular rate
The above calculations assume that ˙ψ =0. As noted previously, the acceleration measurements (equation
(3.77)) are a simplification of the real measurements. The low-frequency (LF) linear dynamics that are
being measured is not ˙ν as claimed in (3.77), since ¨η 6= ˙ν when the Earth-fixed frame is considered as
being the inertial frame. More specifically, considering the LF dynamics
y 3,LF =¨η = ˙ψSR(ψ)ν + R(ψ)˙ν (3.112)
where S is a skew symmetric matrix
⎡
S = ⎣ 0 −1 0 ⎤
1 0 0⎦ (3.113)
0 0 0
This means that (3.77) is approximately correct for small angular rates (Lindegaard 2003, chapter
4.4, p.52). However, for ˙ψ 6= 0, we must ensure that the mapping ˆξ = T z (ψ)ẑ (Lindegaard 2003, chapter
4.4, p.55) is stable. More precisely, we must ensure that this mapping is contracting:
.
·
δˆξ = Tz (ψ)δ ẑ ·
+ T ˙ z (ψ)δẑ (3.114)
= T z (ψ)Tz T (ψ)A z T z (ψ)δẑ + ˙ψS z T z (ψ)δẑ (3.115)
= (A z + ˙ψS z )δˆξ = ∂ ˆf
∂ˆξ δˆξ (3.116)
which is contracting if A z + ˙ψS z is u.n.d.. S z is a skew symmetric matrix which appears when the
rotation T z (ψ) is differentiated:
˙ T z (ψ) = d dt (T z(ψ)) = ˙ψS z T z (ψ) = ˙ψT z (ψ)S z , (3.117)
S z = diag{S T ,S T ,S T ,S T , 0 nz −12} (3.118)
where S is given by (3.113). Thus, we must ensure that ˙ψ is bounded: −r max ≤ ˙ψ ≤ r max .

3.3. GAIN-SCHEDULED WAVE FILTERING 27
3.2.3 Contraction theorem for the simplified observer
To verify that the observer estimate ẑ converges to the actual system trajectory z, we compare the
observer model (equation (3.68)-(3.74)) with the observer (equation (3.83)-(3.89)). It is seen that for
ẑ = z which implies that ŷ = y, the systems are identical. That is, the observer model is a particular
solution of the observer. Thus, from Theorem 2.2 we conclude that if ∂ ˆf is u.n.d (for the general case
∂ˆξ
˙ψ 6= 0), all trajectories in the state space converge exponentially to the same trajectory. Hence, ẑ
converges exponentially to z if ∂ ˆf is u.n.d., and leads to the following theorem:
∂ˆξ
Theorem 3.3 Contraction of the simplified observer
The simplified observer (3.93) which is based on position and acceleration measurements, is globally
contracting and the estimation error converges exponentially to zero if, given the assumption A2, and
for an explicitly given maximum angular rate ˙ψ = r max , there exists a metric M a = Ma
T > 0 such that
M a A z + A T z M a < r max (M a S z + Sz T M a ) (3.119)
M a A z + A T z M a < −r max (M a S z + Sz T M a ) (3.120)
where A z is given in equation (3.97) and S z in equation.(3.118).
The proof follows directly from the above deduction.
Remark: The LMI in the above theorem is identical to the LMI of theorem 4.2 in (Lindegaard 2003),
and merely adds the fact that for the given conditions the observer is also globally contracting.
3.3 Gain-scheduled wave filtering
In this section a new design method to automatically adjust the estimation of the LF motion to the
slowly-varying wave model parameters, for the observer of (Fossen & Strand 1999), is presented. One
possible method is suggested in (Strand & Fossen 1999), however this design has two main drawbacks:
Firstly, only the wave model parameters (not the observer gains) are adapted. This means that a priori
knowledge of the sea state is required to choose observer gains, such that a notch filtering effect is
achieved to remove 1st order wave components. Secondly, the observer tuning can be quite difficult.
This is due to the stability analysis which requires one LMI (for a given set of observer gains) with two
corresponding equality constraints to be fulfilled. Since there exists no tuning rules to ensure that these
equations are fulfilled, there are two possible ways to tune the observer:
1. Trial and error: Choose a set of observer gains, and then check if the stability conditions are
fulfilled. However, since there is no evident choice for the observer gains, this may take a lot of
time.
2. Solve the stability constraints using an advanced computer algorithm. The constraints have the
form (Strand & Fossen 1999, section 4.3)
A T (K)P + PA(K) < 0 (3.121)
B1 T P = C 1 (K) (3.122)
B2 T P = C 2 (3.123)

28 OBSERVER DESIGN
where K is the set of observer gains to be computed, given that there exists a matrix P = P T > 0
such that the above equations are fulfilled. This problem may be solved using computer algorithms
for BMIs (bilinear matrix inequalities) with semi-definite constraints. Evidently, the drawback of this
approach is the complexity of the problem. Besides, the computed observer gains will have no physical
meaning.
Due to these problems, a new approach is suggested in which the wave model parameters are measured
on-line. This simplifies the design considerably, because the parameters are not estimated. Therefore, it is
not necessary to require the system to be persistently exciting (PE) to guarantee parameter convergence.
By doing so, it will be shown that both the wave model parameters and the observer gains can be
adjusted on-line to the present sea state. This means that all the properties of the observer of (Fossen
& Strand 1999) will be sustained, and that no observer tuning is required. The stability proof for the
overall system is done with contraction theory using LMIs and parameter dependent Lyapunov functions.
3.3.1 Observer model for gain-scheduled wave filtering
First we state the observer model of (Fossen & Strand 1999) when ω 0 isassumedtobeanunknown
parameter. Neglecting the white noise terms, the model is given by
˙ξ = A w (ω 0 )ξ (3.124)
˙η = R(ψ)ν (3.125)
ḃ = −T −1
b
b (3.126)
M ˙ν + Dν = τ + R T (ψ)b (3.127)
y = η + C w ξ (3.128)
And the corresponding observer of (Fossen & Strand 1999) where the wave peak frequency ω 0 is
measured on-line is given by
·
ˆξ = Aw (¯ω 0 )ˆξ + K 1 (¯ω 0 )(y − ŷ) (3.129)
·
ˆη = R(ψ)ˆν + K 2 (¯ω 0 )(y − ŷ) (3.130)
·
ˆb = −T
−1
b
ˆb + K3 (y − ŷ) (3.131)
M ˆν ·
+ Dˆν = τ + R T (ψ)ˆb + R T (ψ)K 4 (y − ŷ) (3.132)
ŷ = ˆη + C wˆξ (3.133)
where ¯ω 0 is the measured wave peak frequency, ˆη =[ˆx, ŷ, ˆψ] T is the vector of the NED position (x,y)
and heading (ψ) and
· ¸ 0 I
A w (¯ω 0 )=
(3.134)
−Ω −Λ
where
Ω = diag(¯ω 2 01 , ¯ω2 02 , ¯ω2 03 ) ( measured dominating wave frequency)
Λ = diag(2ζ 1 ¯ω 01 , 2ζ 2 ¯ω 02 , 2ζ 3¯ω 03 )
The design method relies on the following assumptions

3.3. GAIN-SCHEDULED WAVE FILTERING 29
Assumption B1
• ω 0i is the only unknown parameter in the wave model. For standard wave spectra such as JON-
SWAP, the damping parameter ζ i is approximately constant for a wide range of ω 0 . Hence, if the
sea state corresponds to such a wave spectrum, this is a good assumption.
• ˙ω 0i =0(slowly varying)
• ¯ω 0i is calculated from the position measurements using standard digital signal processing techniques.
Since ω 0i is slowly varying, it will be sufficient to update ¯ω 0i every T minutes (e.q.
T=15). It is assumed that the measured dominating wave frequency ¯ω 0 converges to true value
ω 0 :¯ω 0 → ω 0
• 0 < ¯ω 0min ≤ ¯ω 0 ≤ ¯ω 0max < ∞
3.3.2 Observer with gain-scheduled wave filtering in compact form
By performing the same procedure as in section 3.1.1 and 3.1.3, and given the following assumption
Assumption A3 The bias time constant matrix T b , the observer gains K 2 (¯ω 0 ) and K 3 and each
3 × 3 sub-block of A w (ω 0 ),and K 1 (¯ω 0 ) commutes with the rotation matrix (Property 1).
the observer in equation (3.129) to (3.133) can be compactly written as
·
ˆx = T T (ψ)A(¯ω 0 )T (ψ)ˆx + Bτ + K y (¯ω 0 )y (3.135)
where ˆx =[ˆξ T , ˆη T , ˆb T , ˆν T ] T , T (ψ) =diag{R T (ψ),R T (ψ),R T (ψ),I 3x3 },y= η + C w ξ and
A(¯ω 0 ) =
⎡
⎤
A w (¯ω 0 ) − K 1 (¯ω 0 )C w −K 1 (¯ω 0 ) 0 0
⎢ −K 2 (¯ω 0 )C w −K 2 (¯ω 0 ) 0 I
⎥
⎣ −K 3 C w −K 3 −T −1
b
0 ⎦ (3.136)
−M −1 K 4 C w −M −1 K 4 M −1 −M −1 D
K y (¯ω 0 ) = [K T 1 (¯ω 0 ),K T 2 (¯ω 0 ),K T 3 ,K T 4 R(ψ)M −1 ] T (3.137)
The differential dynamics of the observer can therefore be written as
δ ˆx ·
= T T (ψ)A(¯ω 0 )T (ψ)δˆx = J(¯ω 0 )δˆx (3.138)
Hence, it is required that J(¯ω 0 ) is u.n.d. which corresponds to
J T (¯ω 0 )+J(¯ω 0 )=T T (ψ)A T (¯ω 0 )T (ψ)+T T (ψ)A(¯ω 0 )T (ψ) < 0 (3.139)
However, since A(¯ω 0 ) < 0 ⇒ T T (ψ)A T (¯ω 0 )T (ψ) < 0 the above condition is equal to
A T (¯ω 0 )+A(¯ω 0 ) < 0 (3.140)
That is, the rotation matrices can be removed from the analysis. Instead of (3.138) we therefore
consider the linear system
δ ˆx ·
= A(¯ω 0 )δˆx (3.141)
Hence, it must be proven that A(¯ω 0 ) fulfills property (3.140) for 0 ≤ ¯ω 0min < ¯ω 0 ≤ ¯ω 0max < ∞.

30 OBSERVER DESIGN
3.3.3 Stability of LPV systems
Linear parameter dependent (LPV) systems are linear systems where the system matrices depend on
an unknown parameter, and affine LPV systems are linear systems where the system matrices depend
affinely on the unknown parameter By considering ¯w 0 and ¯w 0 2 as two different parameters which are
measured, the differential dynamics (3.141) can be rewritten as an autonomous, affine LPV system. We
therefore need to use some tools for LPV systems. Consider the linear system defined by
ẋ = A(δ)x (3.142)
where the state matrix A(δ) is a function of a real valued parameter vector δ =col(δ 1 ,δ 2 , ..., δ k )
∈ R k . A sufficient condition for x ∗ =0to be an asymptotically stable equilibrium point of (3.142) is the
existence of a quadratic Lyapunov function
with K = K T > 0 such that
V (x) =x T Kx (3.143)
dV (x, t)
˙V = < 0 (3.144)
dt
along all trajectories of x of (3.142) that originate in a neighborhood of the equilibrium x ∗ =0.
Definition 3.1 The system (3.142) is said to be quadratically stable for perturbations ∆ if there
exists a matrix K = K T such that
A(δ(t)) T K + KA(δ(t)) < 0 (3.145)
for all perturbations δ ∈ ∆ (Weiland & Scherer 2000).
The function ’quadstab’ inMatlab TM canbeusedtofind a parameter dependent Lyapunov function
which fulfills Definition 3.1. The main disadvantage in searching for one quadratic Lyapunov function
for a class of parameter dependent systems is the conservatism of the test to prove stability of a class
of models. Indeed, the test of quadratic stability does not discriminate between systems that have slow
time-varying parameters and systems whose dynamical characteristics quickly vary in time. To reduce
conservatism of the quadratic stability test, parameter dependent Lyapunov functions will be considered
instead.
Definition 3.2 The system (3.142) is called affinely quadratically stable if there exists matrices
K 0 = K0 T , ...., K k = Kk
T such that
K(δ) =K T (δ) :=K 0 + δ 1 K 1 + ... + δ k K k > 0 (3.146)
for all perturbations δ ∈ ∆.
A(δ) T K(δ)+K(δ)A(δ)+ dK(δ)
dt
< 0 (3.147)
Definition 3.2 is a modification of the definition of affine quadratic stability used in (Weiland &
Scherer 2000), since it is required that K(δ) is symmetric. In Matlab TM , the function ’pdlstab’ is
designed to find parameter dependent Lyapunov functions of the form given in Definition 3.2.

3.3. GAIN-SCHEDULED WAVE FILTERING 31
3.3.4 Contraction theorem for observer with gain-scheduled wave filtering
In our case equation (3.147) reduces to A(δ) T K(δ)+K(δ)A(δ) < 0, since the parameters are assumed to
be time-invariant (slowly varying). Thus, if this requirement is fulfilled, the proposed observer structure
will be contracting.
Now it must be ensured that the observer estimate ˆx converges to the actual system trajectory x. By
comparing the observer model (equation (3.124)-(3.128)) with the observer (equation (3.129)-(3.133)),
it is seen that for ˆx = x which implies that ŷ = y, the systems are identical. That is, the observer model
is a particular solution of the observer. From Theorem 2.2, we therefore conclude that if ∂ ˆf
∂ ˆx
is u.n.d,
all trajectories in the state space converge exponentially to the same trajectory. Hence, ˆx converges
exponentially to x if ∂ ˆf
∂ ˆx
is u.n.d This leads to the following theorem:
Theorem 3.4 Contracting observer with gain-scheduled wave filtering based on position
and heading measurements The observer in equation (3.129)-(3.133) is globally contracting and the
observer error converges exponentially to zero if there exists a metric M(¯ω 0 )=M T (¯ω 0 ) > 0 such that
A(¯ω 0 ) T M(¯ω 0 )+M(¯ω 0 )A(¯ω 0 ) < 0 (3.148)
given the assumptions B1 and A3, andwhereA(¯ω 0 ) is given in (3.136).
The proof follows directly from the above deduction.
Note that the M(¯ω 0 ) corresponds to a parameter dependent metric used in contraction theory. To
verify that Theorem 3.4 can be fulfilled, we rewrite A(¯ω 0 ) as
where
A(¯ω 0 )=A 0 + A 1¯ω 0 + A 2 ¯ω 2 0 (3.149)
A 0 =
A 011 =
A 012 =
⎡
⎤
A 011 A 012 0 6×3 0 6×3
⎢ 0 3×6 0 3×3 0 3×3 I 3×3
⎥
⎣ −K 3 −K 3 −T −1
b
0 3×3
⎦ (3.150)
−M −1 K 4 C w −M −1 K 4 M −1 −M −1 D
¸
·03×3 (1 + 2(ζ d − ζ)k)I 3×3
(3.151)
0 3×3 0 3×3
¸
·2(ζd − ζ)kI 3×3
(3.152)
0 3×3
A 1 =
A 111 =
A 112 =
⎡
⎤
A 111 A 112 0 6×3 0 6×3
⎢−kI 3×3 C w −kI 3×3 0 3×3 0 3×3
⎥
⎣ 0 3×6 0 3×3 0 3×3 0 3×3
⎦ (3.153)
0 3×6 0 3×3 0 3×3 0 3×3
·03×3 0 3×3
(3.154)
0 3×3 −2ζ d I 3×3¸
·
0 3×3
(3.155)
−2(ζ d − ζ)I 3×3¸

32 OBSERVER DESIGN
A 2 =
A 211 =
⎡
⎤
A 211 0 6×3 0 6×3 0 6×3
⎢0 3×6 0 3×3 0 3×3 0 3×3
⎥
⎣0 3×6 0 3×3 0 3×3 0 3×3
⎦ (3.156)
0 3×6 0 3×3 0 3×3 0 3×3
·03×3 0 (3.157)
−I 0 3×3¸
where ζ d is the desired damping ratio (typically 1) and ω c = kω 0 ,k>1, is the cut-off frequency
of the notch filter. By using the LMI Toolbox in Matlab TM (see ’pdlstab’), a parameter dependent
Lyapunov function
M(¯ω 0 )=M 0 + M 1 ¯ω 0 + M 2¯ω 2 0 (3.158)
which satisfies equation (3.149) can be found. Since M 0 ,M 1 ,M 2 ∈ R 15×15 the matrices are not
stated due to lack of space. Calculations show that the conditions are easily satisfied for the bounds:
¯ω 0min =0.01 rad/s and ¯ω 0max = 100 rad/s , which practically cover every possible sea state, for the
choice of T b ,K 3 and K 4 stated in (Fossen & Strand 1999) with ω c =1.22ω 0 and ζ d =1for the same
supply vessel which is also found in (Matlab GNC Toolbox 2003).
In other words, this means that all the vital properties of the observer which are
• wave filtering capabilities: The observer estimates the wave frequency components η WF
these frequencies can be removed from the feedback loop
so that
• notch filtering: The choice of K 1 and K 2 (see Fossen & Strand 1999) are done such that the 1st
order wave components (frequencies around the dominating wave frequency ω 0 )arenotchfiltered.
• exponential converges to zero of the estimation error
• velocity and bias estimation
remain intact as the observer adapts to the changing sea state, and there is no need for observer
tuning (K 3 and K 4 are simply chosen so that 1/T bi ¿ K 3i /K 4i

3.4. GAIN-SCHEDULED WAVE FILTERING WITH ACCELERATION
MEASUREMENTS 33
1. High-pass filter the position and heading measurements to remove the vessel drift
2. Sample and buffer a time series of the position and heading measurements
3. Calculate the PDS and identify the peak frequency
4. Use a rate limiter to assure that the assumption ˙ω 0 =0is maintained
In the observer the same ¯ω 0 is used in all degrees of freedom. Also, the median of the estimated peak
frequencies should be used in case of loss of position measurements. Figure 3.2 shows how this can be done
in Simulink TM in one degree of freedom. The blocks are found in the DSP (Digital Signal Processing)
Toolbox in Simulink TM . The power spectrum estimation method used here is the parametric method
of Burg, which has several advantages compared to nonparametric methods. Parametric methods is
recommended since they give better estimates for shorter finite-length data records. For more information
on the Burg method and digital signal processing, the reader is referred to (Proakis & Manolakis 1996).
Figure 3.2: Measurement of ω 0 inonedegreeoffreedom
3.4 Gain-scheduled wave filtering with acceleration measurements
In this section, the result of section 3.3 is extended to the simplified observer of (Lindegaard 2003). Since
the observer of (Fossen & Strand 1999) and the simplified observer of (Lindegaard 2003) have the same
form, the procedure is identical.
3.4.1 Simplified observer with gain-scheduled wave filtering in compact form
The observer model is given by

34 OBSERVER DESIGN
ṗ w = A pw (¯ω 0 )p w , ∈ R 6 (3.159)
˙η = R(ψ)ν, ∈ R 3 (3.160)
ḃ = −T −1
b
b, ∈ R 3 (3.161)
˙ν w = A νw (¯ω 0 )ν w , ∈ R n νw
(3.162)
ȧ w = A aw (¯ω 0 )a w , ∈ R n aw
(3.163)
˙ν = −M −1 Dν + M −1 τ + M −1 R T (ψ)b + K a a f , ∈ R 3 (3.164)
ȧ f = −T −1
f
(¯ω 0 )a f , a f (0) = 0 (3.165)
y 1 = η + C pw p w , ∈ R n y1
(3.166)
y 2 = Υ 2 ν + C νw ν w , ∈ R n y2
(3.167)
y 3 = Υ 3 ˙ν + C aw a w , ∈ R n y3
(3.168)
and given the following assumption
Assumption B4 The bias time constant matrix T b ,andeach3×3 sub-block of A pw (¯ω 0 ),K 11 (¯ω 0 ),K 21 (¯ω 0 )
and K 31 commutes with the rotation matrix R(ψ)
the observer can be written compactly as
·
ẑ = T T z (ψ)A z (¯ω 0 )T z (ψ)ẑ + B τ τ + K(ψ, ¯ω 0 )y (3.169)
where ẑ =[ˆp T w, ˆη T , ˆb T , ˆν T w, â T w, ˆν T ,a T f ]T ∈ R nz , y as given in equation (3.80) and
T z (ψ) =diag{R T (ψ),R T (ψ),R T (ψ),R T (ψ),I 6+nνw+n aw
} (3.170)
B τ = £ 0 0 0 0 0 M −1 0 ¤ T
⎡
⎤
K 11 (¯ω 0 ) 0 0
K 21 (¯ω 0 ) 0 0
K 31 0 0
K(ψ, ¯ω 0 ) =
0 K 42 (¯ω 0 ) 0
⎢
0 0 K 53
⎥
⎣K 61 R(ψ) K 62 (¯ω 0 ) 0 ⎦
0 0 T −1
f
(¯ω 0 )
(3.171)
(3.172)
· ¸
A0 (¯ω 0 ) K f
A z (¯ω 0 ) =
T −1
f
(¯ω 0 )C 3 −T −1
(3.173)
f
(¯ω 0 )
C 3 = £ 0 0 Υ 3 M −1 0 C aw −Υ 3 M −1 D ¤ (3.174)
K f = £ ¤
0 0 0 0 0 Ka
T T
(3.175)
¸
·A11 (¯ω
A 0 (¯ω 0 )= 0 ) A 12
A 21 A 22 (¯ω 0 )
(3.176)
⎡
A pw (¯ω 0 ) − K 11 (¯ω 0 )C pw −K 11 (¯ω 0 ) 0
⎤
A 11 (¯ω 0 )= ⎣ −K 21 (¯ω 0 )C pw −K 21 (¯ω 0 ) 0 ⎦ (3.177)
−K 31 C pw −K 31 −T −1
b

3.4. GAIN-SCHEDULED WAVE FILTERING WITH ACCELERATION
MEASUREMENTS 35
⎡
A 12 = ⎣ 0 0 0 ⎤
0 0 I⎦ (3.178)
0 0 0
⎡
⎤
0 0 0
A 21 = ⎣ 0 0 −K 53 Υ 3 M −1 ⎦ (3.179)
−K 61 C pw −K 61 M −1
⎡
A 22 (¯ω 0 )= ⎣ A ⎤
νw(¯ω 0 ) − K 42 (¯ω 0 )C νw 0 −K 42 (¯ω 0 )Υ 2
0 A aw (¯ω 0 ) − K 53 C aw K 53 Υ 3 M −1 D ⎦ (3.180)
−K 62 (¯ω 0 )C νw 0 −M −1 D − K 62 (¯ω 0 )Υ 2
3.4.2 Contraction theorem for simplified observer with gain-scheduled wave
filtering and acceleration measurements
Theorem 3.5 Contracting observer with gain-scheduled wave filtering based on position,
heading and acceleration measurements. The observer in equation (3.169) is globally contracting
and the estimation error converges exponentially to zero if, given the assumptions B1 and B4, andfor
an explicitly given maximum angular rate ˙ψ = r max , there exists a metric M a (¯ω 0 )=Ma T (¯ω 0 ) > 0 such
that
M a (¯ω 0 )A z (¯ω 0 )+A T z (¯ω 0 )M a (¯ω 0 ) 0 such that the generalized Jacobian is
u.n.d.:
∂ ˆf(¯ω
T
0 )
M(¯ω 0 )+M(¯ω 0 ) ∂ ˆf(¯ω 0 )
< 0
∂ˆξ
∂ˆξ
∀ˆξ,∀t ≥ 0 (3.186)
Inserting ˙ψ = ±r max this corresponds to the two equations

36 OBSERVER DESIGN
which can be rewritten as
M(¯ω 0 )(A z (¯ω 0 )+r max S z )+(A z (¯ω 0 )+r max S z ) T M(¯ω 0 ) < 0 (3.187)
M(¯ω 0 )(A z (¯ω 0 ) − r max S z )+(A z (¯ω 0 ) − r max S z ) T M(¯ω 0 ) < 0 (3.188)
M(¯ω 0 )A z (¯ω 0 )+A T z (¯ω 0 )M(¯ω 0 ) < −r max (M(¯ω 0 )S z + Sz T M(¯ω 0 )) (3.189)
M(¯ω 0 )A z (¯ω 0 )+A T z (¯ω 0 )M(¯ω 0 )

Chapter 4
Controller design
In order to take advantage of the observers from the previous chapter, the control law must be implemented
using the estimated states instead of the true one, due to the absence of required measurements,
and because of the wave filtering. Hence, it must be shown that such a combination yields stability of
the closed loop.
In (Panteley & Loria 1998), a separation principle for nonlinear time-varying systems in cascades
was established, and in (Loria et al. 2000), this separation principle was used to prove UGAS of the
closed loop for a PD-controller with bias feed-forward using the observer of (Fossen & Strand 1999).
This separation principle is also referred to in (Lindegaard 2003), to argue that a PID controller with
acceleration feedback can be implemented using the estimates from the observer in section 3.2.
Likewise, the observers with gain-scheduled wave filtering from section 3.3 and 3.4 can be used with
the corresponding controllers as their static wave filtering counterparts, using the separation principle of
(Panteley & Loria 1998) or (Loria & Morales 2002), since the observers with gain-scheduled wave filtering
are UGES. However, in contraction theory there exists a simpler and more powerful separation principle,
which is used in this chapter to prove UGES of the closed loop when the controller is implemented using
the state estimates.
4.1 A contraction based nonlinear separation principle
The following separation principle is cited from (Lohmiller & Slotine March 2000): Consider a plant
dynamics in terms of an explicit state vector z
combined with the observer
ż = f(z, t)+G(z, t)u(ẑ,t) (4.1)
·
ẑ = f(ẑ, t) − [e(ẑ) − e(z)] + G(z, t)u(ẑ,t) (4.2)
where ẑ is the state estimate, and u(ẑ,t) is the control input. Letting ˜z =ẑ − z, the Lyapunov-like
analysis
Z
d
∂(f − e)
dt (˜zT ˜z) =2˜z T (z − λ˜z)dλ˜z (4.3)
∂z
then shows that the convergence rate of ẑ to z is specified by ∂(f −e)/∂z. For bounded G, this system
is a hierarchy, and thus the convergence rate of the plant dynamics is given by ∂(f +Gu)/∂z. This result
may be viewed as an extension of the standard linear separation principle in Luenberger (1979).
37

38 CONTROLLER DESIGN
Figure 4.1: Block diagram showing the control allocation block in a feedback control system
4.2 Control allocation
In this thesis algorithms for control allocation is not considered. In all the equations given here, the
control forces and moments are collected in the generalized control force vector τ. However, for implementation,
the generalized forces must be allocated to the actuators of the ship with an appropriate
algorithm. For more information on control allocation, the reader is referred to the work of Tor. A.
Johansen (Johansen 2003), and (Fossen 2002) and references therein.
Let C A (z, t) represent the control allocation algorithm. See figure 4.1. When comparing equation
(4.2) with the observers from chapter 3, it is seen that the function G(z,t) is given by.
G(z, t) =M −1 C −1
A u (4.4)
Hence, for any appropriate control allocation algorithm C A (z,t), G(z,t) will be bounded, and consequently
the observer-feedback control structure described by (4.1) and (4.2) is a hierarchy.
4.3 Observer-feedback control
In this section, it is shown that a PID-controller implemented with the state estimates from the observer
with gain-scheduled wave filtering in section 3.3, and that a PID-controller with acceleration feedback
implemented with the state estimates from the observer with gain-scheduled wave filtering in section
3.4, yield an overall globally contracting (and consequently UGES) DP system.
Note that the observers from section 3.1 and 3.2 are particular solutions of their gain-scheduled
counterparts, and therefore the theorems given here are also valid for the observers with static wave
filtering. In the static case, simply disregard that ω 0 is a slowly varying parameter, and modify the
theorems thereafter.
For convenience, let us restate the ship model of which the controller design is based:
˙η = R(ψ)ν (4.5)
M ˙ν + Dν = τ (4.6)
4.3.1 Observer-feedback control with gain-scheduled wave filtering
Consider the following control law:
˙ξ = η (4.7)
˙ξ d = η d (4.8)
τ = −K i R T (ψ)(ξ − ξ d ) − K d ν − K p R T (ψ)(η − η d ) (4.9)

4.3. OBSERVER-FEEDBACK CONTROL 39
where η d (t) is the desired position and heading. Thus, if
⎡ ⎤
˙ξ
⎣ ˙η ⎦ =
˙ν
⎡
⎤ ⎡
0 3×3 I 3×3 0 3×3
⎣ 0 3×3 0 3×3 R(ψ) ⎦ ⎣ ξ ⎤
η⎦ + τ d (4.10)
−M −1 K i R T (ψ) −M −1 K p R T (ψ) −M −1 (D + K d ) ν
τ d = £ 0 0 (R T (K i z d + K p η d )) T ¤ T
which can be compactly written as
where x =[ξ T ,η T ,ν T ] T and
(4.11)
ẋ = T T (ψ)A τ T (ψ)x + τ d (4.12)
T (ψ) =diag{R T (ψ),R T (ψ),I 3×3 } (4.13)
⎡
A τ = ⎣ 0 ⎤
3×3 I 3×3 0 3×3
0 3×3 0 3×3 I 3×3
⎦ (4.14)
−M −1 K i −M −1 K p −M −1 (D + K d )
is contracting, the hierarchy of the PID controller implemented with the state estimates
˙ˆξ = ˆη (4.15)
˙ξ d = η d (4.16)
τ = −K i R T (ψ)(ˆξ − ξ d ) − K dˆν − K p R T (ψ)(ˆη − η d ) (4.17)
from the observer (3.135), will be contracting.
Theorem 4.1 Observer-feedback control with gain-scheduled wave filtering based on position
and heading measurements. The hierarchy of the observer (3.135) with gain-scheduled wave filtering
and the PID controller (4.17) is globally contracting if there exists a metric M(¯ω 0 )=M T (¯ω 0 ) > 0 such
that
A(¯ω 0 ) T M(¯ω 0 )+M(¯ω 0 )A(¯ω 0 ) < 0 (4.18)
and a metric M τ = M T τ
> 0 such that
A T τ M τ + M τ A τ < 0 (4.19)
given the assumptions B1 and A3, andwhereA(¯ω 0 ) is given in (3.136) and A τ is given in (4.14).
The proof follows directly from the separation principle.
4.3.2 Observer-feedback control with gain-scheduled wave filtering and acceleration
feedback
Suppose that y a ∈ R ny3 is the measured accelerations in the body-frame and that Π ∈ R ny3×3 is
the projection extracting the available acceleration measurements y a = Π ˙ν. Let the control law be
(Lindegaard 2003, chapter 5.4, p.75)

40 CONTROLLER DESIGN
ȧ f_τ = A f a f_τ + B f (y a − Π{(r − r d )S T R T (ψ − ψ d )ν d + R T (ψ − ψ d )˙ν d }) ∈ R n y3
(4.20)
τ = τ PID − K a a f_τ (4.21)
where a f_τ is the filtered acceleration error and τ PID is to be determined shortly. At low frequencies
a f_τ ≈ y a − Π{(r − r d )S T R T (ψ − ψ d )ν d + R T (ψ − ψ d )˙ν d } (4.22)
which means that the acceleration term K a a f_τ can be regarded as a change in the system’s mass.
This can be seen by inserting (4.21) with (4.22) into the ship model, which results in
˙η = R(ψ)ν (4.23)
(M + K a Π)˙ν + Dν = τ PID − K a Π{(r − r d )S T R T (ψ − ψ d )ν d + R T (ψ − ψ d )˙ν d } (4.24)
Now, let the PID controller be given by
˙ξ = η (4.25)
˙ξ d = η d (4.26)
τ PID = −K i R T (ψ)(ξ − ξ d ) − K p R T (ψ)(η − η d ) − K d (ν − R T (ψ − ψ d )ν d )+τ rff (4.27)
where the reference feed-forward is given by
τ rff = DR T (ψ − ψ d )ν d + M{(r − r d )S T R T (ψ − ψ d )ν d + R T (ψ − ψ d )˙ν d } (4.28)
Thus, if
ẋ τa = Tτa(ψ)A T τa T τa (ψ)x τa + τ ad (4.29)
where
x τa =
h
i T
ξ T η T ν T a T f_τ (4.30)
T τa (ψ) = diag{R T (ψ),R T (ψ),I 3+ny3 } (4.31)
⎡
⎤
0 I 0 0
A τa = ⎢ 0 0 I 0
⎥
⎣ −M −1 K i −M −1 K p −M −1 (D + K p ) −M −1 K a
⎦ (4.32)
−B f ΠM −1 K i −B f ΠM −1 K p −B f ΠM −1 (D + K d ) A f − B f ΠM −1 K a
⎡
⎤
0
τ ad = ⎢
0
⎥
⎣ M −1 (K i R T (ψ)ξ d + K p R T (ψ)η d + K d R T (ψ − ψ d )ν d + τ rff ) ⎦ (4.33)
B f ΠM −1 (K i R T (ψ)ξ d + K p R T (ψ)η d + K d R T (ψ − ψ d )ν d + τ rff )
is contracting, the hierarchy of the control law (4.21) implemented with the state estimates

4.3. OBSERVER-FEEDBACK CONTROL 41
˙ˆξ = ˆη (4.34)
˙ξ d = η d (4.35)
´
ȧ f_τ = −A f a f_τ + B f Π
³ˆ˙ν − {(ˆr − r d )S T R T (ˆψ − ψ d )ν d + R T (ˆψ − ψ d )˙ν d } ∈ R n y3
(4.36)
τ = −K i R T (ψ)(ˆξ − ξ d ) − K p R T (ψ)(ˆη − η d ) − K d (ˆν − R T (ψ − ψ d )ν d ) − K a a f_τ + τ rff (4.37)
from the observer (3.169), will be contracting.
Theorem 4.2 Observer-feedback control with gain-scheduled wave filtering and acceleration
feedback The hierarchy of the observer (3.169) with gain-scheduled wave filtering and the PID controller
(4.27) with acceleration feedback is globally contracting if there exists a metric M a (¯ω 0 )=M T a (¯ω 0 ) > 0
such that
M a (¯ω 0 )A z (¯ω 0 )+A T z (¯ω 0 )M a (¯ω 0 ) 0 such that
A T τaM τa + M τa A τa < 0 (4.40)
given the assumptions B1 and B4, whereA z (¯ω 0 ) is stated in equation (3.173), S z in equation.(3.118)
and A τa in equation (4.32).
The proof follows directly from the separation principle.

42 CONTROLLER DESIGN

Chapter 5
Simulation results
Case study: A model of a 80m supply vessel with two main propellers and three thrusters (one aft and two
in the bow) is used in MCSim to demonstrate the performance of the DP system with gain-scheduled
wave filtering. MCSim is a Simulink TM -model design to test ships when exposed to environmental
disturbances. For details regarding the configuration of the vessel model and MCSim, thereaderis
referred to (Smogeli 2003).
The simulation scenario is illustrated in figure 5.0.1.
Figure 5.0.1: Simulation scenario
The scenario is the following:
1. Station-keeping at position (x 0 ,y 0 ) with heading ψ 0 =45 ◦
43

44 SIMULATION RESULTS
2. Change of position to (x 1 ,y 1 ) and then change of heading to ψ 1 =0 ◦ ,wherex 1 − x 0 = y 1 − y 0 =
20m.
3. Station-keeping at position (x 1 ,y 1 ) with heading ψ 1 =0 ◦
The following features are included in the simulations:
• Waves: The waves are generated from the JONSWAP wave spectrum, and 20 frequency components
are used in the simulations. The significant wave height H s is constant in every simulation.
• Wind: The wind force is generated from the NORSOK wind spectrum based on the expectation
value of H s , and 20 frequency components are used in the gust realization. The maximum variation
in the wind direction is ±5 ◦ .
• Wave elevation and wave induced velocity calculations
• Cross coupling loss for ducted/open propeller, and transverse loss for tunnel thruster
• Thruster control: Hybrid power/torque controller with loss estimation (Smogeli, Sørensen & Fossen
2004)
• Coriolis-centripetal forces
No measurement noise is however included in the simulations, which means that the measurements
are simply the sum of the WF motion and the LF motion. In the simulations, the ship model to be
controlled and the observer ship model have the same initial conditions.
The PID-controller used in all simulations is tuned to yield a critically damped closed-loop system
with natural frequency of ω n =0.8 rad/s. Details regarding the vessel control model and the PID
controller are given in Appendix B, and the observer data is given in Appendix A. The scenario is
simulated in moderate sea state (significant wave height H s =1.4m ) and very rough sea state (significant
wave height H s =5m ), with and without acceleration feedback. Note that when using the method
described in section 3.3.5 for measuring ω 0 , an initial condition ω 0_init 6=0is needed until the first
position and heading time series is buffered and analysed. For the sake of demonstration, the initial
condition chosen here is ω 0_init =0.38 rad/s, which corresponds to H s =15m in the JONSWAP wave
spectrum. That is, the wave filter is initially tuned for Baufort force 12 with hurricane wind.
5.1 Observer-feedback control with gain-scheduled wave filtering
5.1.1 Moderate sea state

Y E
[m]
X E
[m]
5.1. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING 45
25
Measured position X E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.1: Measured and desired position X E (t), H s =1.4m, PID controller
25
Measured position Y E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.2: Measured and desired position Y E (t), H s =1.4m, PID controller

X E
[m]
Course [deg]
46 SIMULATION RESULTS
60
Measured course (solid) and desired (dashed)
50
40
30
20
10
0
-10
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.3: Measured and desired course, H s =1.4m, PID controller
25
Measured position (solid) and desired (dashed)
20
15
10
5
0
-5
-5 0 5 10 15 20 25
Y [m] E
Figure 5.1.1.4: Measured position (X E (t),Y E (t)), H s =1.4m, PID controller

y(1)-y(1)-hat [m]
5.1. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING 47
0.8
Measured ω 0
(t)
0.75
0.7
0.65
Measured ω 0
0.6
0.55
0.5
0.45
0.4
0.35
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.5: Measured ω 0 (t) in rad/s, H s =1.4m, PID controller
0.1
Estimation error in X E
-direction
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.6: Estimation error: y(1) − ŷ(1), H s =1.4m, PID controller

y(3)-y(3)-hat [deg]
y(2)-y(2)-hat [m]
48 SIMULATION RESULTS
1
Estimation error in Y E
-direction
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.7: Estimation error: y(2) − ŷ(2), H s =1.4m, PID controller
2
Course estimation error
1.5
1
0.5
0
-0.5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.1.8: Course estimation error: y(3) − ŷ(3), H s =1.4m, PID controller
5.1.2 Very rough sea state

Y E
[m]
X E
[m]
5.1. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING 49
25
Measured position X E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
-10
-15
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.1: Measured and desired position X E (t), H s =5m, PID controller
30
Measured position Y E
(t) (solid) and desired (dashed)
25
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.2: Measured and desired position Y E (t), H s =5m, PID controller

X E
[m]
Course [deg]
50 SIMULATION RESULTS
70
Measured course (solid) and desired (dashed)
60
50
40
30
20
10
0
-10
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.3: Measured and desired course H s =5m, PID controller
25
Measured position (solid) and desired (dashed)
20
15
10
5
0
-5
-10
-15
-5 0 5 10 15 20 25 30
Figure 5.1.2.4: Measured position (X E (t),Y E (t)), H s =5m, PID controller
Y E
[m]

y(1)-y(1)-hat [m]
5.1. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING 51
0.7
Measured ω 0
(t)
0.65
0.6
Measured ω 0
0.55
0.5
0.45
0.4
0.35
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.5: Measured ω 0 (t) in rad/s, H s =5m, PID controller
0.5
Estimation error in X E
-direction
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.6: Estimation error: y(1) − ŷ(1), H s =5m, PID controller

y(3)-y(3)-hat [deg]
y(2)-y(2)-hat [m]
52 SIMULATION RESULTS
2.5
Estimation error in Y E
-direction
2
1.5
1
0.5
0
-0.5
-1
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.7: Estimation error: y(2) − ŷ(2), H s =5m, PID controller
7
Course estimation error
6
5
4
3
2
1
0
-1
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1.2.8: Course estimation error: y(3) − ŷ(3), H s =5m, PID controller

Y E
[m]
X E
[m]
5.2. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING AND ACCELERATION FEEDBACK 53
5.2 Observer-feedback control with gain-scheduled wave filtering
and acceleration feedback
5.2.1 Moderate sea state
25
Measured position X E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.1: Measured and desired position X E (t), H s =1.4m, PID controller with AFB
25
Measured position Y E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.2: Measured and desired position Y E (t), H s =1.4m, PID controller with AFB

X E
[m]
Course [deg]
54 SIMULATION RESULTS
50
Measured course (solid) and desired (dashed)
40
30
20
10
0
-10
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.3: Measured and desired course , H s =1.4m, PID controller with AFB
25
Measured position (solid) and desired (dashed)
20
15
10
5
0
-5
-5 0 5 10 15 20 25
Y [m] E
Figure 5.2.1.4: Measured position (X E (t),Y E (t)), H s =1.4m, PID controller with AFB

y(1)-y(1)-hat [m]
5.2. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING AND ACCELERATION FEEDBACK 55
0.8
Measured ω 0
(t)
0.75
0.7
0.65
Measured ω 0
0.6
0.55
0.5
0.45
0.4
0.35
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.5: Measured ω 0 (t) in rad/s, H s =1.4m, PID controller with AFB
0.08
Estimation error in X E
-direction
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.6: Estimation error: y(1) − ŷ(1), H s =1.4m, PID controller with AFB

y(3)-y(3)-hat [deg]
y(2)-y(2)-hat [m]
56 SIMULATION RESULTS
0.4
Estimation error in Y E
-direction
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.7: Estimation error: y(2) − ŷ(2), H s =1.4m, PID controller with AFB
0.6
Course estimation error
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.1.8: Course estimation error: y(3) − ŷ(3), H s =1.4m, PID controller with AFB
5.2.2 Very rough sea state

Y E
[m]
X E
[m]
5.2. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING AND ACCELERATION FEEDBACK 57
25
Measured position X E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.2.1: Measured and desired position X E (t), H s =5m, PID controller with AFB
25
Measured position Y E
(t) (solid) and desired (dashed)
20
15
10
5
0
-5
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.2.2: Measured and desired position Y E (t), H s =5m, PID controller with AFB

X E
[m]
Course [deg]
58 SIMULATION RESULTS
50
Measured course (solid) and desired (dashed)
40
30
20
10
0
-10
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.2.3: Measured and desired course, H s =5m, PID controller with AFB
25
Measured position (solid) and desired (dashed)
20
15
10
5
0
-5
-5 0 5 10 15 20 25
Y [m] E
Figure 5.2.2.4: Measured position (X E (t),Y E (t)), H s =5m, PID controller with AFB

y(1)-y(1)-hat [m]
Measured ω 0
[rad/s]
5.2. OBSERVER-FEEDBACK CONTROL WITH GAIN-SCHEDULED WAVE
FILTERING AND ACCELERATION FEEDBACK 59
0.6
Measured ω 0
(t)
0.55
0.5
0.45
0.4
0.35
0.3
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.2.5: Measured ω 0 (t) in rad/s, H s =5m, PID controller with AFB
0.3
Estimation error in X E
-direction
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.2.6: Estimation error: y(1) − ŷ(1), H s =5m, PID controller with AFB

y(3)-y(3)-hat [deg]
y(2)-y(2)-hat [m]
60 SIMULATION RESULTS
0.8
Estimation error in Y E
-direction
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0 500 1000 1500 2000 2500
time in seconds
Figure 5.1: Figure 5.2.2.7: Estimation error: y(2) − ŷ(2), H s =5m, PID controller with AFB
2
Course estimation error
1.5
1
0.5
0
-0.5
-1
0 500 1000 1500 2000 2500
time in seconds
Figure 5.2.2.8: Course estimation error: y(3) − ŷ(3), H s =5m, PID controller with AFB
5.3 Concluding remarks
Note that after approximately 800s, the observers are updated with the first measurement of ω 0 , and
the performance of the observer without acceleration measurements then improves noticeably. On the

5.3. CONCLUDING REMARKS 61
other hand, the observer with acceleration measurements is robust against the badly tuned wave filter,
and can compensate for this due to the available acceleration measurements. The acceleration feedback
is very effective, and clearly results in better performance of the DP system.
Also note that the variance of the position in X E -direction is bigger after that the course is changed
to ψ =0 ◦ . The reason for this is not established, but the reason may be that the hydrodynamic damping
of the vessel in X E -direction is less when the vessel is parallel with the mean wave direction, facing the
waves, as opposed to when the vessel course is 45 ◦ relative to the mean wave direction.

62 SIMULATION RESULTS

Chapter 6
Conclusion
A robust control system for marine surface vessels in changing weather conditions has been designed and
analysed. The resulting DP system consists of an observer with gain-scheduled wave filtering and a PID
controller. Extensions of this design to include acceleration feedback is also established. By means of
contraction theory it is shown that resulting DP system is globally contracting, and consequently UGES.
TheperformanceoftheDPsystemisdemonstrated in computer simulations for different environmental
disturbances.
In extreme seas with high wave heights and/or long wave lengths, the assumption of producing control
action from the LF motion signals only, may not be so evident, as the WF motions (due to long wave
lengths and thereby low frequency) will enter the control bandwidth of the DP system. This is outlined
in (Sørensen, Strand & Nyberg 2002). Thus, in extreme seas, one should consider to apply a different
control strategy to improve the performance of the DP system, and is subject for ongoing research.
63

64 CONCLUSION

Appendix A
Observer data
In the observers, a second order wave model was used for position, and fourth order wave models were
used for velocity and acceleration. Only the linear acceleration was measured. The wave model matrices
are given by
and
A w (¯ω 0 ) = A pw (¯ω 0 )=A 0 pw + A 1 pw ¯ω 0 + A 2 pw ¯ω 2 0 (A.1)
·03×3
A 0 I
pw =
3×3
(A.2)
0 3×3 0 3×3¸
·03×3
A 1 0
pw =
3×3
(A.3)
0 3×3 −2ζI 3×3¸
·
A 2 03×3 0
pw =
(A.4)
−I 3×3 0 3×3¸
A νw (¯ω 0 ) = A aw (¯ω 0 )=A 0 νw + A 1 νw¯ω 0 + A 2 νw¯ω 2 0 (A.5)
⎡
⎤
0 3×3 I 3×3 0 3×3 0 3×3
A 0 νw = ⎢0 3×3 0 3×3 0 3×3 I 3×3
⎥
⎣0 3×3 0 3×3 0 3×3 I 3×3
⎦
(A.6)
0 3×3 0 3×3 0 3×3 0 3×3
¸
·A1
A 1 νw = pw 0 6×6
0 6×6 A 1 (A.7)
pw
¸
·A2
A 2 νw = pw 0 6×6
0 6×6 A 2 (A.8)
pw
and
C w = C pw = £ ¤
0 3×3 I 3×3
C νw = £ ¤
I 3×3 0 3×9
C aw = £ ¤
I 2×2 0 2×10
(A.9)
(A.10)
(A.11)
Further
65

66 OBSERVER DATA
⎡
M = 10 9 ⎣ 0.0070 0 0 ⎤
0 0.0110 −0.0130⎦
(A.12)
0 −0.0130 3.1930
⎡
D = ⎣ 200000 0 0 ⎤
0 100000 −700000⎦
(A.13)
0 −700000 63900000
T b = 0 3×3 (A.14)
K 1 (¯ω 0 ) =
¸ ·
·−2(ζd − ζ)kI
K 11 (¯ω 0 )=
3×3 0
+ 3×3
¯ω
0 3×3 2(ζ d − ζ)I 0
3×3¸
(A.15)
K 2 (¯ω 0 ) = K 21 (¯ω 0 )=K 62 (¯ω 0 )=kI 3×3 ¯ω 0 (A.16)
T −1
f
= kI 2×2¯ω 0 (only linear acceleration measured) (A.17)
K 4 = K 61 = diag{0.1, 0.1, 0.01} (A.18)
K 3 = K 31 =0.05K 4 (A.19)
¸
·diag{δ, δ}
K 53 =
, δ =10 (A.20)
0 10×2
where
K 42 (¯ω 0 ) = K42 1 ¯ω 0 + K42¯ω 2 2 0 + K42 3 ¯ω 3 0 (A.21)
⎡ ⎤
2(δ − 1)β
K42 1 = ⎢ 0 3×3
⎥
⎣ 0 3×3
⎦
(A.22)
⎡
⎢−4β ⎣
⎡
0 3×3
⎤
0 3×3
δ +3β 2 + β 2 δ 2 +2βδk − 2βk
⎥
−2β 2 δ + β 2 + β 2 δ 2 ⎦
0 3×3
0 3×3
⎤
K42 2 =
(A.23)
K42 3 =
⎢
0 3×3
⎥
⎣
0 3×3
⎦
(A.24)
2β 3 δ − β 3 − β 3 δ 2 − 2β 2 δk + β 2 k + β 2 δ 2 k
and
β = 2ζI 3×3 (A.25)
ζ = 0.1 (A.26)
ω c = kω 0 (cut-off frequency in notch filter) (A.27)
k = 1.1 (A.28)
For measuring ω 0 (see figure 3.2), the following parameters were used in the simulations

67
• second order highpass, butterworth filter, with passband edge at 0.2 rad/s, implemented in statespace
form in the DSP Blockset in Simulink TM
• sampling frequency: 10Hz
• buffer size: 8192
• Burg Method of fourth order with FFT length of 8192
• rate limiter with slew rate ±0.01
• ω 0 was measured from the position and heading measurements, and the median was selected

68 OBSERVER DATA

Appendix B
Controller data
The following matrices were used for the PID controller design:
⎡
M PID =10 9 × ⎣ 0.0067 0 0 ⎤
0 0.0113 0 ⎦
0 0 3.1931
⎡
D PID = ⎣ 167105 0 0 ⎤
0 141660 0 ⎦
0 0 63862540
(B.1)
(B.2)
where the PID gains were tuned to yield a critically damped closed-loop system with natural frequency
ω n =0.8rad/s
(B.3)
⎡
K p =10 8 × ⎣ 0.0117 0 0 ⎤
0 0.0199 0 ⎦
0 0 3.1515
⎡
K d =10 9 × ⎣ 0.0110 0 0 ⎤
0 0.0188 0 ⎦
0 0 3.9487
(B.4)
(B.5)
K i =0.1ω n K p
(B.6)
The virtually added mass using acceleration feedback was chosen as
⎡
⎤
M(1, 1)0.7 0
K a = ⎣ 0 M(1, 1)0.7⎦
0 0
(B.7)
And the filter gains were chose as
69

70 CONTROLLER DATA
m 11 = M(1, 1) (B.8)
m 22 = M(2, 2) (B.9)
d 11 = D(1, 1) (B.10)
d 22 = D(2, 2) (B.11)
µ
m11
Tf_pid = 0.001min , m
22
(B.12)
d 11 d 22
½
¾
−1
B f = diag
Tf_pid , −1
(B.13)
Tf_pid
A f = −B f (B.14)

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