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43figure (5.9). Clearly the average and worst designs differ significantly from the robustsolutions. Comparing the costs associated with the designs in Φ 1000 against our robustsolution is useful in verifying that the optimization was successful, but less helpful inunderstanding how the robustness improved.12010080# of occurences60402000 0.005 0.01 0.015 0.02 0.025s(x, Σ)Figure 5.10: Histogram of the sensitivity cost term, s(x, Σ U ), for the set Φ 1000 .To demonstrate the robustness of our solution we ran a monte carlo simulationto model an uncertain MEMS fabrication process. To represent the variation in theprocess we generated 400,000 gaussian random vectors with a standard deviation,σ, of 0.1 µm. We then calculated the natural frequencies of the robust design andthe average and worst design from Φ 1000 subject to these variations. The results aredisplayed in figure (5.11). The distribution for the robust design has a much tighterspread than both the average and worst-case designs from Φ 1000 . This is an excellentindication that the optimized design is much less insensitive to geometric uncertaintiesin the design variables.
44Robust Uncorrelated DesignAverage Design from Φ 1000Worst Design from Φ 1000Target Resonant Frequency140 160 180 200 220 240 260Frequency (kHz)Figure 5.11: Distributions of crab-leg resonant frequencies subject to uncorrelateduncertainty.Correlated UncertaintyThe robust design for the correlated uncertainty model, equation (5.14), is givenx ∗ C =here265.64 22.84 14.09 2.00 68.71 392.60[w n = 199.950 kHzF (x ∗ C , Σ C) = 5.700 × 10 −4] Tc 2 (x ∗ C ) = 3.0 × 10−7s(x ∗ C , Σ C) = 5.697 × 10[−4∇ x F (x ∗ C , Σ C) = −.003 −.066 −.902 1.34 .002 −.001]× 10 −4Constraints 2, 8, and 9 from table (5.2) are active at the solution. Aside from thefact that there are active constraints, the main difference between the uncorrelatedand correlated designs are in the variables L 1 , h m , and b m . The size of the proof massfor the correlated design is about 70% of the size for the uncorrelated design and thelegs are slightly longer. Figure (5.9) illustrates the robust design x ∗ C , and figure (5.12)shows a distribution of resonant frequencies when this design is subjected to correlated
Page 1 and 2: Robust Optimization:Design in MEMSb
Page 4 and 5: 36 Conclusion 46Bibliography 48A Al
Page 6 and 7: 5List of Tables3.1 Probabilities wi
Page 8 and 9: 7Chapter 1IntroductionMicro-electro
Page 11 and 12: 10performance, f(x, 0), is equal to
Page 13 and 14: 12function f(x), there are several
Page 15 and 16: 14Chapter 3Optimization AlgorithmIn
Page 17 and 18: 16paribus. This can be very effecti
Page 19 and 20: 18to the active linear constraints
Page 21 and 22: 20can all the constraints be satisf
Page 23 and 24: 22Chapter 4Optimization ExamplesWe
Page 25 and 26: 2475007300220200t 1t2t 37100180Cost
Page 27 and 28: 26design variables, other than the
Page 29 and 30: 28Chapter 5Robust Design Examples5.
Page 31 and 32: 30where ¯f is the w 2 design , Σ
Page 33 and 34: 32design along the line c 2 (x) = 0
Page 35 and 36: 34550500450(5)global solutionPath 1
Page 37 and 38: 36Figure 5.6: Diagram of Six Variab
Page 39 and 40: 38problem is well-defined. Before a
Page 41 and 42: 405.2.3 Robust DesignWe will use th
Page 43: 42evident by noting that the c 2 (x
Page 47 and 48: 46Chapter 6ConclusionIn this report
Page 49 and 50: 48Bibliography[1] What is MEMS Tech
Page 51: 50Appendix AAlkylation Constantsi c

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