Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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Y−coordinate [m] Y−coordinate [m] 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 0.1 0.05 0 −0.05 −0.1 Nominal Design PT Design −15 −10 −5 0 5 10 15 X−coordinate [m] (a) −15 −10 −5 0 5 10 15 X−coordinate [m] (c) Y−coordinate [m] Y−coordinate [m] 0.2 0.1 0 −0.1 −0.2 0.2 0.1 0 −0.1 −0.2 −15 −10 −5 0 5 10 15 X−coordinate [m] (b) −15 −10 −5 0 5 10 15 X−coordinate [m] Figure 2-8: Comparison of critical mode shapes for nominal (–) and PT (- -) designs: (a) Mode #1: ACS (b) Mode #3: second bending (c) Mode #2: first bending (d) Mode #4: third bending. 68 (d)
also symmetric as seen in the figure. It is obvious from inspection of the OPD equation (Equation 2.9) that symmetric y-translation of the collectors does not result in OPD. It is only relative motion that is important. The PT design, however, is slightly asymmetric due to the small mass added to the negative-x arm of the interferometer. The mode shapes of the PT design show definite asymmetry resulting in a small relative displacement between the end points and a slight energy accumulation in these modes. The asymmetry is slight however, and does not have a large affect on the total OPD. The final mode of interest is the second observable axial mode, Mode #11, listed in the table. The mode shape is not pictured here because the motion is only axial and difficult to discern on a linear plot. In both designs the positive x-motion of the collectors together with the negative x-displacement of the combiner node increase the OPD (Equation 2.9). The main difference between the two systems is that in the nominal design this mode is much higher in frequency and therefore contributes very little to the overall OPD, while the axial stiffness in the PT case is decreased significantly so that this mode plays a major role in the accumulation of output energy. The increase is only relevant to the distribution of energy; the total PT energy is still much lower than that of the nominal design. In summary, the optimized design achieves better performance by choosing very small truss elements that result in lower natural frequencies overall and move the nodal points of the first two asymmetric bending modes to the ends of the array, where the collector optics are located. The large mass of the collectors on a very flexible truss effectively pin the collectors in place, so that the truss isolates the optics from the disturbances entering at the center. 2.6 Summary In this chapter Performance Tailoring (PT) is introduced and formalized. A sim- ple model of structurally-connected interferometer is presented in detail and used to step through the process of applying the PT formalization to a structural model. The 69
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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
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 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
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: energy is distributed almost evenly
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
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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
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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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