Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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Multiple Model The multiple model RPT optimization is run on the SCI sample problem using both SA, SQP and MC SQP algorithms. The optimization performance metrics and resulting designs are listed in Table 3.4. The optimal costs listed in the table are much Table 3.4: Algorithm performance: multiple model, βi =1/npv. J ∗ iter fevals time x ∗ [m] x ∗ [kg] Alg. [µm] # # [min] d1 d2 m1 m2 SA 273.51 64 2059 12.79 0.0440 0.0515 0.8464 19.5131 SQP 271.78 20 48 3.83 0.0432 0.0529 0.0 22.613 MC SQP 271.78 238 545 41.06 0.0432 0.0529 0.0 22.613 lower than those of the AO designs (Table 3.3), since the multiple model objective is the weighted sum of the performance values at each of the uncertainty vertices, and not the worst-case performance. In the results presented here the weighting βi is the same at each vertex and equals 1/npv, so that the cost is the average of the performance values at the uncertainty vertices. The SQP and SA designs are nearly equivalent as the SA cost is only 0.6% higher than SQP. The individual runs in the MC SQP algorithm result in a few different optimal solutions with slight variations in the diameter of the inner array segments and the design mass values. All designs, however, are at the maximum mass con- straint, and the best design is found with either MC SQP or the combination of SA and SQP. As seen previously with AO, using SA in conjunction with SQP finds the global optimum much more quickly (16.62 minutes) than performing ten randomly started MC SQP searches (41.06 minutes). The optimal cross-sectional diameters are similar to those in the anti-optimization designs, in that the diameters of the inner truss segments are larger than those of the outer segments. However, the MM design has 22 kg of design mass on the positive-x arm of the interferometer while the AO design has no lumped mass at all. 90
Statistical Robustness The statistical robustness RPT optimization is run with the SA, SQP, and MC SQP algorithms and two measures of standard deviation. The algorithm performance metrics and resulting designs for are listed in Table 3.5. For the cases with vertex uncertainty propagation the standard deviation is calculated by solving Equation 3.12 using only the designs at the uncertainty vertices. This method is a conservative measure of the standard deviation since only the extremes of the distribution are considered. In the cases labelled “MC,” a Monte Carlo uncertainty analysis with 500 samples is conducted to obtain an output distribution and then the standard deviation of that distribution is computed from Equation 3.12 at each iteration. This method is closer to a true standard deviation measure. Table 3.5: Algorithm performance: statistical robustness, α =0.5. Unc. J ∗ iter fevals time x ∗ [m] x ∗ [kg] Alg. Prop. [µm] # # [min] d1 d2 m1 m2 SA vertex 137.27 72 2223 15.54 0.0415 0.0543 0.0426 0.0477 SQP vertex 137.22 14 30 1.55 0.0423 0.0540 0.0 0.0 MC SQP vertex 137.22 238 895 25.19 0.0423 0.0540 0.0 0.0 SA MC 114.75 68 2043 75.53 0.0344 0.0597 0.0683 0.0197 SQP MC 114.63 16 38 75.83 0.0360 0.0485 0.0 0.0 Recall from Equation 3.11 that the statistical robustness objective function in- cludes both the nominal performance and its standard deviation over the uncertainty space. In the results presented here the relative weighting on the nominal performance is α =0.5, so that it is as important as the robustness metric. The first three rows in the table compare the results of the SA, SQP and MC SQP algorithms with the vertex standard deviation measure. The results are similar to those seen in the anti-optimization and multiple model studies in that the performance of all three algorithms is similar with the SA-SQP combination obtaining the optimal design in less time than MC SQP. The Monte Carlo standard deviation results for both SQP and SA have optimal costs that are significantly lower than the vertex method counter-parts. This differ- 91
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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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Page 41 and 42: ometer (SCI). In the following sect
Page 43 and 44: The equations of motion of the unda
Page 45 and 46: The frequency response functions fr
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Page 51 and 52: 2.3.3 Design Variables The choice o
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Page 59 and 60: initial design variable state, x =
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Page 63 and 64: # Designs 25 20 15 10 5 Accepted, b
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Page 73 and 74: through careful and experienced mod
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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
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Page 89: Table 3.3: Algorithm performance: a
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
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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
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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)
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Page 139 and 140: Data: initial iterate, p0, performa
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the new tuning configuration is ver
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Performing an AO tuning optimizatio
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Uncertainty Bounds Test �y [kg] S
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Table 4.6: Tuning results on fifty
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eters are discussed. The optimizati
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Chapter 5 Robust Performance Tailor
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MPC optimization by allowing a diff
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where the notation yij indicates th
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(Table 4.1), and the uncertainty pa
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Table 5.2: Performance and design p
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it in the worst-case uncertainty re
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The data in Figure 5-2 indicate tha
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configuration. The tuned configurat
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same requirement. The effect become
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indicating that this requirement is
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E 2 [Pa] 7.8 7.6 7.4 7.2 7 6.8 6.6
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than the RPT design, 155.45µm to 5
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have a very small nominal performan
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of these simulations fail to meet r
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and that it is the only design meth
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Chapter 6 Focus Application: Struct
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optical path differences between th
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Table 6.1: RWA disturbance model pa
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FRF Magnitude 10 1 10 0 10 −1 10
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Y Z Z X Y (a) w (c) w Y h Z Figure
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Table 6.6: Primary mirror propertie
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OP1 STAR Z OP2 OP3 Coll 1 Coll 2 Bu
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PSD OPD14 [m 2 /Hz] CumulativeOPD14
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6.2 Design Parameters In order to a
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6.2.2 Tuning The tuning parameters
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complex and the normal modes analys
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does not change with the design par
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(a) (b) (c) Figure 6-11: SCI TPF PT
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Table 6.14: Performance predictions
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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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