Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

More documents

Recommendations

Info

epresentation as follows: ⎧ ⎨ ⎩ ˙q ¨q ⎫ ⎬ ⎭ = z = A ⎡� �� 0 I ⎣ −Ω2 ⎤ ⎧ ⎨ q ⎦ −2ZΩ ⎩ ˙q ⎧ ⎫ � � ⎨ q ⎬ CzˆxΦ 0 � �� ⎩ ˙q ⎭ C ⎫ ⎬ ⎭ + B ⎡� �� ⎤� ⎣ 0 ⎦ w (2.6) Φ T Bˆxw where q are modal coordinates, w is white noise, z is the output, and Czˆx is a mapping matrix from physical states to the output. Damping is introduced into the system through the matrix Z, a diagonal matrix of modal damping ratios. In the development model, modal damping is set to 0.001 for all modes. In interferometer design, the optical performance determines the success of the instrument. There are many optical metrics that are of interest such as the angle of the wave-front at the combiner, beam shear and optical path-length difference. Since the SCI development model is a low-fidelity model <strong>with</strong>out true optics, the output, z, is a linearized geometric approximation of the optical path difference (OPD) between the two arms of the interferometer and is based only on the translations of the mirror nodes. If the star is located at a distance R from the interferometer in the Y -axis direction (Figure 2-1), then the linearized optical path lengths from the star to the combiner through the two interferometer arms are: OP1 = R − y1 + xc − x1 + BI 2 OP2 = R − y2 − xc − x2 + x3 + BI 2 (2.7) (2.8) where BI is the interferometric baseline, x1 and y1 are the negative-x collector translations, x2 and y2 are the positive-x collector translations, and xc is the combiner x-translation. The linearized OPD is the difference of Equations 2.7 and 2.8: z = −x1 − y1 +2xc − x2 + y2 44 (2.9)
The frequency response functions from disturbance inputs to the performance are computed from the state-space matrices as follows: Gzw (s) =C (sI − A) −1 B (2.10) where s = jω and, Gzw is a matrix of transfer functions. The resulting transfer functions from the SCI model are plotted in Figure 2-2. The transfer function from Fx and Tz to OPD are drawn in solid and dash-dotted lines, respectively. The transfer function from Fy to OPD does not appear on the plot because the force in this direction does not couple to the output due to the symmetry in the model. Forcing the structure at the center node in the y-direction only excites the symmetric bending modes. It is seen from Equation 2.9 that symmetric y motion of the two collector nodes results in zero OPD. The dominant transfer function is that from torque at the FRF Magnitude 10 0 10 −5 10 −1 10 0 10 1 Frequency [Hz] F x [µm/N] T z [µm/Nm] Figure 2-2: Frequency response functions (FRF) of SCI: disturbance input to performance. center to OPD. The first flexible mode is the first asymmetric bending mode at 0.708 Hz, followed by the higher order asymmetric bending modes. In contrast, the Fx transfer function has no low-frequency modes because the beam elements are much stiffer axially than in bending. The first observable axial mode is at 80.2 Hz. 45 10 2
Page 1: Dynamic Tailoring and Tuning for Sp
Page 4 and 5: Acknowledgments This work was suppo
Page 6 and 7: 3.2 RPT Formulation . . . . . . . .
Page 9 and 10: List of Figures 1-1 Timeline of Ori
Page 11 and 12: List of Tables 1.1 Effect of simula
Page 13 and 14: Nomenclature Abbreviations ACS atti
Page 15: dk optimization search direction f
Page 18 and 19: 1.1 Space-Based Interferometry NASA
Page 20 and 21: unfettered by the Earth’s atmosph
Page 22 and 23: the SCI, both the size and flexibil
Page 24 and 25: maybethatitbecomescertain that the
Page 26 and 27: Table 1.1: Effect of simulation res
Page 28 and 29: Table 1.2: Effect of simulation res
Page 30 and 31: has been found that structural desi
Page 32 and 33: precision telescope structure for m
Page 34 and 35: attractive, and more conservative a
Page 36 and 37: to solve the performance tailoring
Page 39 and 40: Chapter 2 Performance Tailoring A c
Page 41 and 42: ometer (SCI). In the following sect
Page 43: The equations of motion of the unda
Page 47 and 48: the output covariance matrix, Σz,
Page 49 and 50: where the subscript indicates the i
Page 51 and 52: 2.3.3 Design Variables The choice o
Page 53 and 54: and then, by inspection, the inerti
Page 55 and 56: algorithms begin at an initial gues
Page 57 and 58: at least locally optimal, and the s
Page 59 and 60: initial design variable state, x =
Page 61 and 62: and the RMS OPD is computed using E
Page 63 and 64: # Designs 25 20 15 10 5 Accepted, b
Page 65 and 66: does not provide information on why
Page 67 and 68: energy is distributed almost evenly
Page 69 and 70: also symmetric as seen in the figur
Page 71 and 72: Chapter 3 Robust Performance Tailor
Page 73 and 74: through careful and experienced mod
Page 75 and 76: described above. However, one can r
Page 77 and 78: ic, σz(�x, �p), that is depend
Page 79 and 80: Magnitude, OPD/F x [µm/N] Magnitud
Page 81 and 82: % Energy 100 90 80 70 60 50 40 30 2
Page 83 and 84: metric to the cost function. Note,
Page 85 and 86: tion: ∂hi (z,�x, �pi) ∂�x
Page 87 and 88: values are chosen from their statis
Page 89 and 90: Table 3.3: Algorithm performance: a
Page 91 and 92: Statistical Robustness The statisti
Page 93 and 94: Performance [µm] 1400 1200 1000 80
Page 95 and 96:
(Figure 3-6(b)). The nominal perfor
Page 97 and 98:
Norm. Cum. Var. [µm 2 ] PSD [µm 2
Page 99 and 100:
energy by mode for easy comparison.
Page 101 and 102:
Y−coordinate [m] Y−coordinate [
Page 103 and 104:
RMS performance, [µm] 400 350 300
Page 105 and 106:
The requirement chosen here is some
Page 107 and 108:
Chapter 4 Dynamic Tuning Robust Per
Page 109 and 110:
on a physical truss. Since tailorin
Page 111 and 112:
Table 4.1: Tuning parameters for SC
Page 113 and 114:
m 2 [kg] J ∗ # # time y ∗ [kg]
Page 115 and 116:
m 2 [kg] 800 700 600 500 400 300 20
Page 117 and 118:
configuration than the untuned, but
Page 119 and 120:
Norm. Cum. Var. [µm 2 ] PSD [µm 2
Page 121 and 122:
Performance Requirement [µm] 400 3
Page 123 and 124:
is considered. 4.2.1 Hardware-only
Page 125 and 126:
and added to the objective function
Page 127 and 128:
using either a decreasing step-size
Page 129 and 130:
for tailoring, but tuning parameter
Page 131 and 132:
tained by randomly choosing paramet
Page 133 and 134:
p [GPa] y ∗ [kg] Performance [µm
Page 135 and 136:
# Func. Evals Performance RMS (µm)
Page 137 and 138:
tion changes in the updated solutio
Page 139 and 140:
Data: initial iterate, p0, performa
Page 141 and 142:
the new tuning configuration is ver
Page 143 and 144:
Performing an AO tuning optimizatio
Page 145 and 146:
Uncertainty Bounds Test �y [kg] S
Page 147 and 148:
Table 4.6: Tuning results on fifty
Page 149 and 150:
eters are discussed. The optimizati
Page 151 and 152:
Chapter 5 Robust Performance Tailor
Page 153 and 154:
MPC optimization by allowing a diff
Page 155 and 156:
where the notation yij indicates th
Page 157 and 158:
(Table 4.1), and the uncertainty pa
Page 159 and 160:
Table 5.2: Performance and design p
Page 161 and 162:
it in the worst-case uncertainty re
Page 163 and 164:
The data in Figure 5-2 indicate tha
Page 165 and 166:
configuration. The tuned configurat
Page 167 and 168:
same requirement. The effect become
Page 169 and 170:
indicating that this requirement is
Page 171 and 172:
E 2 [Pa] 7.8 7.6 7.4 7.2 7 6.8 6.6
Page 173 and 174:
than the RPT design, 155.45µm to 5
Page 175 and 176:
have a very small nominal performan
Page 177 and 178:
of these simulations fail to meet r
Page 179 and 180:
and that it is the only design meth
Page 181 and 182:
Chapter 6 Focus Application: Struct
Page 183 and 184:
optical path differences between th
Page 185 and 186:
Table 6.1: RWA disturbance model pa
Page 187 and 188:
FRF Magnitude 10 1 10 0 10 −1 10
Page 189 and 190:
Y Z Z X Y (a) w (c) w Y h Z Figure
Page 191 and 192:
Table 6.6: Primary mirror propertie
Page 193 and 194:
OP1 STAR Z OP2 OP3 Coll 1 Coll 2 Bu
Page 195 and 196:
PSD OPD14 [m 2 /Hz] CumulativeOPD14
Page 197 and 198:
6.2 Design Parameters In order to a
Page 199 and 200:
6.2.2 Tuning The tuning parameters
Page 201 and 202:
complex and the normal modes analys
Page 203 and 204:
does not change with the design par
Page 205 and 206:
(a) (b) (c) Figure 6-11: SCI TPF PT
Page 207 and 208:
Table 6.14: Performance predictions
Page 209 and 210:
performance trends similar to those
Page 211 and 212:
Chapter 7 Conclusions and Recommend
Page 213 and 214:
a statistical robustness measure su
Page 215 and 216:
and the worst-case performance is a
Page 217 and 218:
- Consider uncertainty analysis too
Page 219 and 220:
Appendix A Gradient-Based Optimizat
Page 221 and 222:
such that the gradient direction is
Page 223 and 224:
the descent direction. In some case
Page 225 and 226:
Bibliography [1] Jpl planet quest w
Page 227 and 228:
[24] Nightsky Systems Carl Blaurock
Page 229 and 230:
[48] S. C. O. Grocott, J. P. How, a
Page 231 and 232:
[74] M. Lieber. Development of ball
Page 233 and 234:
AIAA/ASME/ASCE/AHS/ASC Structures,
show all

Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?