Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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Table 5.1: Algorithm performance: RPTT, α =0.0, ∆ = 0.1. J ∗ iter fevals time x ∗ [m] y ∗ WC [kg] Alg. Form [µm] # # [min] d1 d2 m1 m2 SA Eq 5.3 216.03 201 8304 5.26 0.0451 0.0388 175.63 0.0774 SQP Eq 5.5 215.94 23 53 1.77 0.0451 0.0388 175.88 0.0 SQP Eq 5.3 215.97 13 1001 1.62 0.0452 0.0388 175.66 0.028 MC SQP Eq 5.5 215.94 414 906 31.47 0.0451 0.0388 175.90 0.0 MC SQP Eq 5.3 215.94 661 5843 51.82 0.0451 0.0388 175.93 0.0 Since the weighting on robustness is set to zero the optimal cost, J ∗ , is the tuned performance at the worst-case uncertainty vertex. SA provides a starting point that is very close to the optimal design after 201 temperature iterations and 8304 function evaluations. The SQP algorithm performs nicely when started at the SA design and applied to the problem of Equation 5.5. It converges to a design in only 23 iterations and 53 function evaluations. In contrast, the SQP solution obtained from the formulation in Equation 5.3 is slightly sub-optimal. Note that the algorithm ran for only 13 iterations, but 1001 function evaluations. The algorithm fails to properly converge because it becomes stuck in the line search and the maximum number of function evaluations (1000) is reached. One possible reason for this behavior is that the objective function gradients required for the SQP algorithm are not well-posed for this problem. The MC SQP results are equivalent to those obtained by starting from the SA design indicating that a globally optimal design has been found. MC SQP does find the optimal design when applied to Equation 5.3 since five out of ten trials do converge successfully. The time results show that the combination of SA and SQP finds the optimal design much more quickly (7 minutes) than MC SQP (31.5 minutes). 5.2.2 Comparison to PT and RPT Designs The optimal tailoring parameters for the RPTT design and its performance in the nominal, and worst-case (untuned) uncertainty configurations are listed in Table 5.2 along with the corresponding design variables and performances for the PT and RPT 158
Table 5.2: Performance and design parameters for optimal designs. Tailoring, �x Performance, [µm] Total Mass d1 [m] d2 [m] m1 m2 σ0 σWC [kg] PT 0.030 0.030 0.0 1.934 100.53 1355.50 673.18 RPT 0.0486 0.0581 0.0 0.0 263.87 306.86 827.04 RPTT 0.0451 0.0388 N/A N/A 158.53 572.66 740.05 (AO) designs. The RPTT design is significantly different from both the PT and RPT designs. In the PT design all of the truss segments are of equal diameter and are at the lower bound, while in the RPT design they are tailored so that the mass is concentrated towards the center of the array with the inner truss diameters are about a centimeter larger that those of the outer truss segments. The RPTT design is opposite the RPT as the inner segments have a smaller cross-sectional diameter than the outer segments, so mass is distributed towards the ends of the array. The tuning parameters are used as tailoring parameters in the PT and RPT optimizations so that the same design variables are used to ensure a fair comparison among the methods. The PT design has the lowest nominal cost by far (100.53µm), but is very sen- sitive to uncertainty with a worst case performance of 1355µm. The RPT design is much more robust to uncertainty, and has a much lower worst-case performance value (306.86µm), but sacrifices nominal performance (263.87µm) to achieve this robustness. The RPTT design lies somewhere in between the PT and RPT designs. It has a nominal performance of 158.53µm, higher than that of the PT design, but lower than that of the RPT design. It is not quite as robust as the RPT design, but has a much lower worst-case performance (572.66µm) than the PT design. Table 5.3: Performance and parameters for tuned worst-case realizations. Tuning, �y [kg] Performance Total Mass m1 m2 σt [µm] [kg] PT 175.06 0.0 303.17 848.25 RPT 81.04 0.0 273.32 908.08 RPTT 0.0 175.88 215.94 915.94 The tuned performance of the designs is obtained by applying the tuning op- 159
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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
Page 107 and 108: Chapter 4 Dynamic Tuning Robust Per
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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
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Page 119 and 120: Norm. Cum. Var. [µm 2 ] PSD [µm 2
Page 121 and 122: Performance Requirement [µm] 400 3
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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
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Page 153 and 154: MPC optimization by allowing a diff
Page 155 and 156: where the notation yij indicates th
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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
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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
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Page 181 and 182: Chapter 6 Focus Application: Struct
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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
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Page 205 and 206: (a) (b) (c) Figure 6-11: SCI TPF PT
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performance trends similar to those
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Chapter 7 Conclusions and Recommend
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a statistical robustness measure su
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and the worst-case performance is a
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- Consider uncertainty analysis too
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Appendix A Gradient-Based Optimizat
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such that the gradient direction is
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the descent direction. In some case
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Bibliography [1] Jpl planet quest w
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[24] Nightsky Systems Carl Blaurock
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[48] S. C. O. Grocott, J. P. How, a
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[74] M. Lieber. Development of ball
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AIAA/ASME/ASCE/AHS/ASC Structures,
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