Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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optical metric. Disturbance analysis methods for evaluating performance RMS and sensitivity analyses that provide performance gradients are discussed. Both gradient based and stochastic search techniques are applied to the sample problem to obtain a performance tailored design. 2.1 PT Formulation The objective of performance tailoring is to design a system using a set of design variables, �x, such that a desired performance metric, f(�x), is below some required value, freq. One approach to this problem is to formulate an optimization that minimizes the performance metric: min f (�x) �x (2.1) s.t. �g (�x) ≤ �0 where f (�x) is the performance, or cost, function, �x is a vector of design variables that affect the performance, and �g (�x) are constraints on the design variables. Examples of performance metrics include first mode natural frequency and the variance of the displacement of a particular structural element. For example, consider the design of a truss that must achieve a specified level of stability at a given point, such as the tip, when subjected to a dynamic disturbance environment. A PT optimization would minimize the variance of the tip motion within a given frequency range by varying the cross-sectional areas of the truss bars. Due to practical considerations, the cross-sectional areas may be constrained by a lower bound set by manufacturing and local buckling considerations and an upper bound on the mass of the structure. 2.2 SCI Development Model The PT formulation is applied to the problem of high-performance systems through a simple sample structure that is representative of a structurally-connected interfer- 40
ometer (SCI). In the following section the development model is presented in detail. This model is used throughout the thesis to develop and demonstrate the design methodology and provide a comparison among problem formulations and optimization techniques. The model is two-dimensional and consists of three optical elements, two collectors and a combiner, modeled by lumped masses rigidly connected to a truss structure as shown in Figure 2-1. The truss model is broken into four segments, each with its 000 111 000 111 000 111 000 111 M coll1 00 11 00 11 00 11 00 11 1 L1 optical mass design mass 2 3 4 000 111 000 111 000 111 000 111 000 111 M comb B I Y θ z y x 000 111 000 111 000 111 000 111 000 111 θ z 1 2 M coll2 Figure 2-1: Schematic of SCI development model. own material and geometric properties, to represent an articulated truss. Each truss segment is modeled with one Bernoulli-Euler beam finite element. The beam elements have two nodes each with three degrees of freedom: x-translation, y-translation, and z-rotation (See Fig. 2-1), for a total of four elements, five nodes and fifteen degrees 41 y x X
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 34 and 35: attractive, and more conservative a
Page 36 and 37: to solve the performance tailoring
Page 39: Chapter 2 Performance Tailoring A c
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,
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
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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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