Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

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The first constraint, Equation 5.6, requires the tailoring for tuning dummy variable, z1, to be greater than or equal to the maximum tuned performance over the uncertainty vertices. Equation 5.7 is equivalent to Equation 3.6 from the AO formulation, and requires that z2 be greater than or equal to the maximum un-tuned performance over the uncertainty space. Since the objective is to minimize a weighted sum of z1 and z2, the constraints ensure that these variables are effectively the worst-case tuned and worst-case untuned performances, respectively. In this form, the analytical gradients of the cost function are easily derived by inspection: ∂JRP T T ∂�x = �0 ∂JRP T T ∂�yi = �0 ∂JRP T T ∂z1 =1− α ∂JRP T T ∂z2 = α (5.8) Constraint gradients are also required for a constrained gradient-based optimization. The gradients of the parameter constraints, �g, depend on the particular design problem under consideration. The augmented constraint gradients include the gradients of the performance with respect to the tailoring and tuning parameters: ∂h1i ∂�x ∂h2i ∂�x ∂f (�x, �yi,�pi) = ∂�x ∂f (�x, �y0,�pi) = ∂�x ∂h1i ∂�yi ∂h2i ∂�yi = ∂f (�x, �yi,�pi) ∂�yi = �0 ∂h1i ∂z1 ∂h2i ∂z1 = −1 =0 ∂h1i ∂z2 ∂h2i ∂z2 =0 (5.9) = −1 (5.10) In the following section the RPTT optimization is applied to the design of a structurally connected interferometer using the SCI development model. A variety of optimization algorithms are used to solve the problem and the performance of the different formulations are compared for efficiency. 5.2 SCI Development Model Application of the RPTT methodology to the SCI development model requires all of the elements presented in the previous chapters. The tailoring parameters are the two cross-sectional areas (Table 2.3), the tuning parameters are the design masses 156
(Table 4.1), and the uncertainty parameters are the Young’s Moduli of the two truss arms (Table 3.1), so that the design variable vector is: x T dv = � d1 d2 m11 m12 m21 m22 m31 m32 m41 m42 � (5.11) The design masses are no longer considered as tailoring parameters in this formulation since they are accounted for by the tuning component. The performance metric is the RMS of the OPD between the two collectors and is a function of all the parameters. A constraint equation limiting the total mass of the tailored and tuned system is enforced at each uncertainty vertex: ⎧ ⎪⎨ g (�x, �yi) = ⎪⎩ π 2 ρL �2 j=1 d2j + �2 j=1 m1j − ¯ M . π 2 ρL �2 j=1 d2j + �2 j=1 m4j − ¯ M ∀i =1...4 (5.12) This constraint couples the tailoring and tuning parameters into the mass equation allowing the optimization algorithm to spend mass where it is most beneficial. The results presented in this section are for the case of bounded uncertainty within ±10% of the nominal uncertainty values. The weighting on robustness, α, inthe RPTT formulations is set to zero to produce a design with maximum tuning authority. 5.2.1 Optimization Algorithms RPTT designs of the SCI development problem are generated by running the opti- mizations formulated in Equations 5.3 and 5.5 through the simulated annealing and sequential quadratic programming algorithms. The resulting designs are presented in Table 5.1 along with algorithm performance data. Simulated annealing is run first on the formulation in Equation 5.3 to provide an initial guess to the SQP algorithm. SA is not run with the simple minimization formulation (Equation 5.5) as it is difficult to obtain feasible guesses due to the augmented constraints. SQP is also run with ten randomly-chosen initial guesses to assess the convexity of the solution space. The results show that nearly all of the SQP runs converged on the same design. 157
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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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Chapter 2 Performance Tailoring A c
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ometer (SCI). In the following sect
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The equations of motion of the unda
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The frequency response functions fr
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the output covariance matrix, Σz,
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where the subscript indicates the i
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2.3.3 Design Variables The choice o
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and then, by inspection, the inerti
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algorithms begin at an initial gues
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at least locally optimal, and the s
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initial design variable state, x =
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and the RMS OPD is computed using E
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# Designs 25 20 15 10 5 Accepted, b
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does not provide information on why
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energy is distributed almost evenly
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also symmetric as seen in the figur
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Chapter 3 Robust Performance Tailor
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through careful and experienced mod
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described above. However, one can r
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ic, σz(�x, �p), that is depend
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Magnitude, OPD/F x [µm/N] Magnitud
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% Energy 100 90 80 70 60 50 40 30 2
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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
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
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)
Page 137 and 138: tion changes in the updated solutio
Page 139 and 140: Data: initial iterate, p0, performa
Page 141 and 142: the new tuning configuration is ver
Page 143 and 144: Performing an AO tuning optimizatio
Page 145 and 146: Uncertainty Bounds Test �y [kg] S
Page 147 and 148: Table 4.6: Tuning results on fifty
Page 149 and 150: eters are discussed. The optimizati
Page 151 and 152: Chapter 5 Robust Performance Tailor
Page 153 and 154: MPC optimization by allowing a diff
Page 155: where the notation yij indicates th
Page 159 and 160: Table 5.2: Performance and design p
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Page 163 and 164: The data in Figure 5-2 indicate tha
Page 165 and 166: configuration. The tuned configurat
Page 167 and 168: same requirement. The effect become
Page 169 and 170: indicating that this requirement is
Page 171 and 172: E 2 [Pa] 7.8 7.6 7.4 7.2 7 6.8 6.6
Page 173 and 174: than the RPT design, 155.45µm to 5
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Page 181 and 182: Chapter 6 Focus Application: Struct
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Page 185 and 186: Table 6.1: RWA disturbance model pa
Page 187 and 188: FRF Magnitude 10 1 10 0 10 −1 10
Page 189 and 190: Y Z Z X Y (a) w (c) w Y h Z Figure
Page 191 and 192: Table 6.6: Primary mirror propertie
Page 193 and 194: OP1 STAR Z OP2 OP3 Coll 1 Coll 2 Bu
Page 195 and 196: PSD OPD14 [m 2 /Hz] CumulativeOPD14
Page 197 and 198: 6.2 Design Parameters In order to a
Page 199 and 200: 6.2.2 Tuning The tuning parameters
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Page 205 and 206: (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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