Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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method are shown with dashed lines. The transfer functions from axial force and torque to the output do not show much change between the two models. However, there is a significant increase in FRF magnitude between the nominal and worst- case designs for the y-direction force input. Recall, that in the PT configuration, the design is nearly symmetric and that the y-force disturbance has little effect on the output. When the system asymmetry is increased the symmetric modes become asymmetric and are observable in the OPD. The normalized cumulative variance and output PSD plots are shown in Fig- ure 3-2(d). It is clear from this plot that nearly all of the RMS in the worst-case PT model is attributed to one mode. This mode is the first bending mode at 0.023 Hz, and does not contribute significantly to the output in the nominal PT design, as evidenced by the modal energy distributions plotted and tabulated in Figure 3-3. The modal energy breakdown for the nominal uncertainty PT system is repeated here for comparison. In the nominal PT model the first bending mode contributes only 3.50µm to the total RMS performance. However, at the worst-case uncertainty vertex this mode accounts for 99.1% (1343µm) of the output energy. The reason for the dramatic degradation in performance is found by comparing the shape of the first bending modes of the two systems (Figure 3-4). Note that the nominal PT mode shape (solid line) is nearly symmetric about the center so that there is no discernible relative y-motion between the collectors and no resulting OPD contribution. However, at the worst-case uncertainty vertex the system is highly asymmetric, and this bending mode is not symmetric about the center of the array (dotted line). In that case, the collector nodes do not move the same amount and OPD results. The addition of this mode to the output is significant since it is excited by both the force in the y-direction and torque about the z-axis. The uncertainty propagation analysis of the PT optimized design shows that this design is only optimal if the model represents the physical system exactly. Small changes in the model parameters have drastic effects on the performance. This result is common in optimization problems as the design is often pushed to sensitive areas of the solution space in order to achieve maximum performance. In order to increase 80
% Energy 100 90 80 70 60 50 40 30 20 10 0 Nominal PT Worst−Case PT 1 2 3 4 5 11 Mode # (a) Nominal Uncertainty Worst-case Uncertainty Mode fn energy σ 2 z σz fn energy σ 2 z σz # [Hz] % [µm 2 ] [µm] [Hz] % [µm 2 ] [µm] 1 0.013 38.46 3886 38.66 0.013 0.61 11144 8.22 2 0.023 3.48 352 3.50 0.023 99.09 1820600 1343.10 3 0.281 26.01 2628 26.15 0.281 0.14 2641 1.95 4 0.302 1.26 127 1.26 0.300 0.01 119 0.09 5 1.08 3.47 351 3.49 1.04 0.01 140 0.10 11 31.7 25.85 2612 25.98 31.7 0.14 2554 1.88 Total: 98.53 9956 99.04 100.00 1837198 1355.34 (b) Figure 3-3: % Total energy by mode (∆ = 0.10): PT with nominal uncertainty (dark), PT with worst-case uncertainty (light). 81
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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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Page 39 and 40: Chapter 2 Performance Tailoring A c
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
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
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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
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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: Magnitude, OPD/F x [µm/N] Magnitud
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
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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
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
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Page 127 and 128: using either a decreasing step-size
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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
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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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