Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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6.1.4 Attitude Control System The model of the attitude control system (ACS) is created with the docs_acs.m function, a part of the DOCS suite of analysis tools. This function implements a rigid body angular position controller with a bandwidth, facs, specified as a fraction of the first flexible mode. The bandwith fraction of the ACS is set at facs =0.001 to keep it well below the first flexible mode of the structure and avoid coupling with the flexible modes. The controller is a proportional-derivative controller with inertia decoupling. The inputs to the controller are three ACS sensor outputs from the plant which are simply the angular motions of the bus node. The controller outputs are torque commands for the RWA and are fed back to the structural plant through a summer element that combines the commanded torques with the disturbance torques from the disturbance model (see Figure 6-1). Since the RWA disturbance model is broadband and therefore not a function of the instantaneous wheel speed the ACS commands simply add to the disturbances and do not drive the vibrations. 6.1.5 Integrated Model Nominal Performance The individual model components are integrated within MATLAB using the DOCS toolbox. The plant, ACS and RWA isolator component models are transformed to state-space form (Equation 2.6) and integrated to obtain the system model. Since the RWA disturbance model is provided as PSDs, a frequency domain disturbance analysis is run using Equation 2.12 to obtain the RMS of the optical performance metrics over the frequency band of interest. The OPD RMS is the performance used in the PT, RPT and RPTT objective functions. For simplicity, only the RMS OPD between collectors 1 and 4, σ14, is considered in the following sections. The output PSD from the nominal TPF SCI model is plotted in Figure 6-7 along with the cumulative variance. The RMS OPD between collectors one and four is 15.88 nm. It is clear from the cumulative variance plot that the energy is largely accumulated in the mode at 3.06 Hz. This mode is one of the second bending modes and is shown in Figure 6-8(b). The first bending mode (Figure 6-8(a)) is just under 194
PSD OPD14 [m 2 /Hz] CumulativeOPD14 [m 2 ] 2 1 0 10 −15 10 −20 10 −25 10 −30 x 10 −16 10 −2 10 −1 3.06 Hz 10 0 Frequency [Hz] 10 1 6 Hz 7.55 Hz Figure 6-7: Output PSD and cumulative variance from nominal TPF SCI design. Table 6.9: Critical modes of nominal TPF SCI design. Mode # Freq. [Hz] % Energy (app.). Description 4 1.07 < 1 truss 1 st bending 7 3.06 88.00 truss 2 nd bending 13 5.99 2.42 collector 1 st bending (potato-chip) 18 7.55 16.7 truss 3 rd bending 2 Hz, but does not contribute appreciably to the OPD since it is symmetric and the path lengths change equally on each arm of the interferometer. Additional critical modes under 10 Hz are listed in Table 6.9. The cluster of modes around 6Hz are due to local bending modes of the primary mirrors as shown in Figure 6-8(c). The third bending mode at 7.55 Hz (Figure 6-8(d)) is also a significant contributor. It is interesting to note that, although this model is higher-fidelity than the development model presented in Chapter 2, the global behavior is similar in that the second bending mode is the most critical. 195 10 2
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Dynamic Tailoring and Tuning for Sp
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Acknowledgments This work was suppo
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3.2 RPT Formulation . . . . . . . .
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List of Figures 1-1 Timeline of Ori
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List of Tables 1.1 Effect of simula
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Nomenclature Abbreviations ACS atti
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dk optimization search direction f
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1.1 Space-Based Interferometry NASA
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unfettered by the Earth’s atmosph
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the SCI, both the size and flexibil
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maybethatitbecomescertain that the
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Table 1.1: Effect of simulation res
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Table 1.2: Effect of simulation res
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has been found that structural desi
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precision telescope structure for m
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attractive, and more conservative a
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to solve the performance tailoring
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Chapter 2 Performance Tailoring A c
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ometer (SCI). In the following sect
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The equations of motion of the unda
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The frequency response functions fr
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the output covariance matrix, Σz,
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where the subscript indicates the i
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2.3.3 Design Variables The choice o
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and then, by inspection, the inerti
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algorithms begin at an initial gues
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at least locally optimal, and the s
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initial design variable state, x =
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and the RMS OPD is computed using E
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# Designs 25 20 15 10 5 Accepted, b
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does not provide information on why
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energy is distributed almost evenly
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also symmetric as seen in the figur
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Chapter 3 Robust Performance Tailor
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through careful and experienced mod
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described above. However, one can r
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ic, σz(�x, �p), that is depend
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Magnitude, OPD/F x [µm/N] Magnitud
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% Energy 100 90 80 70 60 50 40 30 2
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metric to the cost function. Note,
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tion: ∂hi (z,�x, �pi) ∂�x
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values are chosen from their statis
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Table 3.3: Algorithm performance: a
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Statistical Robustness The statisti
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Performance [µm] 1400 1200 1000 80
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(Figure 3-6(b)). The nominal perfor
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Norm. Cum. Var. [µm 2 ] PSD [µm 2
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energy by mode for easy comparison.
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Y−coordinate [m] Y−coordinate [
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RMS performance, [µm] 400 350 300
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The requirement chosen here is some
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Chapter 4 Dynamic Tuning Robust Per
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on a physical truss. Since tailorin
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Table 4.1: Tuning parameters for SC
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m 2 [kg] J ∗ # # time y ∗ [kg]
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m 2 [kg] 800 700 600 500 400 300 20
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configuration than the untuned, but
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Norm. Cum. Var. [µm 2 ] PSD [µm 2
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Performance Requirement [µm] 400 3
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is considered. 4.2.1 Hardware-only
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and added to the objective function
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using either a decreasing step-size
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for tailoring, but tuning parameter
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tained by randomly choosing paramet
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p [GPa] y ∗ [kg] Performance [µm
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# Func. Evals Performance RMS (µm)
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tion changes in the updated solutio
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Data: initial iterate, p0, performa
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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
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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: OP1 STAR Z OP2 OP3 Coll 1 Coll 2 Bu
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,
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Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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