Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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attractive, and more conservative alternative, when there is insufficient data to build an accurate statistical model of parametric uncertainty. When these models are used in uncertainty propagation, bounds on the response are obtained in lieu of statistical output distributions. These uncertainty models and propagation techniques are ultimately combined with structural optimization to produce designs that can meet performance require- ments despite the model uncertainty. This practice is known as robust design and has been popular among control experts for some time [110, 10, 47, 48]. Robust design optimization is a relatively new field in structural design, but has generated a lot of interest due to the rising complexity of systems [40]. Anderson considers the problem of robust actuator and damping placement for structural control using a model with known errors [5]. Park, Hwang and Lee use Taguchi methods to post- process gradient-based structural optimization to find discrete sizing variables [99]. The Taguchi method [107, 102] is a quality-control technique originally developed for circuit design that has recently found application in structural optimization. In a later paper, Park et al. apply the Taguchi method to unconstrained structural optimization to find robust optimal designs of three and ten-bar trusses subjected to applied loads [100]. The areas of the truss members are optimized with conventional methods first to minimize mass and then with the Taguchi method to minimize the sensitivity of the displacement of a given node to variations in the member areas. Constrained robust structural optimization problems are considered as well. Sand- gren and Cameron suggest a two-stage hybrid approach that combines the use of genetic algorithms for topology optimization with Monte Carlo uncertainty propagation to determine the statistics of the objective function and/or constraints [103]. The method is computationally expensive, but is successfully applied to a ten-bar truss problem and a higher-fidelity automobile inner panel using probabilistic models of uncertainty. Elishakoff, Haftka and Fang use convex uncertainty models to perform “anti-optimization” or min-max style robust design [42]. They demonstrate their technique on a ten-bar truss problem subjected to uncertain loading as well as stress and displacement constraints. 34
Although structural optimization and integrated modeling do much to enable the design of complex structures like space-based interferometers there is a lack of application of robust design to this problem in the literature. Structural optimization has been applied to the opto-mechanical jitter problem presented by interferometers, but robust design techniques have not. Instead, the robust design literature is generally concerned with truss-sizing problems for mass minimization or static displacement reduction and not with dynamic cost functions such as optical path jitter. If space-based interferometers are to be designed and launched based on simulation predictions alone, model uncertainty must be considered during the design process. However, in order to achieve robustness, a system sacrifices nominal performance. Therefore, robust design techniques reach their limitation when performance require- ments are aggressive and uncertainty is high. In this thesis both the application of robust optimization to SCI design and a technique that can be applied to problems that lie beyond the limits of robust design are presented. 1.4 Thesis Roadmap A design methodology that extends robust design techniques for application to high- performance, high-risk systems such as space-based interferometers is developed. In this chapter, background information on space-based interferometry is presented and motivates the need for an extension of current design techniques. A review of relevant literature is included. In Chapter 2, Performance Tailoring, or the optimization of a structure to minimize performance variance alone, is discussed. A simple model of a structurally- connected interferometer, referred to throughout as the development model, is in- troduced and discussed in detail. Included in the model are the structure, optical performance, and disturbance sources. The performance metric is RMS OPD, and methods for obtaining its gradient with respect to the design parameters are discussed. In addition, an overview of two popular optimization methods, sequential quadratic programming and simulated annealing, is given. These algorithms are used 35
Page 1: Dynamic Tailoring and Tuning for Sp
Page 4 and 5: Acknowledgments This work was suppo
Page 6 and 7: 3.2 RPT Formulation . . . . . . . .
Page 9 and 10: List of Figures 1-1 Timeline of Ori
Page 11 and 12: List of Tables 1.1 Effect of simula
Page 13 and 14: Nomenclature Abbreviations ACS atti
Page 15: 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 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 and 68: energy is distributed almost evenly
Page 69 and 70: also symmetric as seen in the figur
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,
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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
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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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