Robust Optimization: Design in MEMS - University of California ...

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29AhLFigure 5.1: Diagram of Two DOF Resonator.we define an uncertain design vector to bewhere δ = [δ h˜x = x + δ (5.1)δ L ]. Since our algorithm is designed for rational polynomial functions,we choose the uncertain objective function, f(x, δ), to be w 2 n. The expression for w 2 n,in terms of x and δ, is given byf(x, δ) = w 2 n =( ) ( )2E (h + δh ) 3ρA (L + δ L ) 3We will assume that the design is subject to the following constraints(5.2)h min ≤ h ≤ h max10h ≤ L ≤ L max(5.3)These constraints have been imposed for illustrative purposes, however they could bethe result of MEMS design rules, die size requirements, fabrication limitations or anumber of other design factors.Now that we have a rational polynomial objective function and a set of constraints,we can pose the robust design problem that we would like to solve. This is essentially)a restatement of equation (2.9)( (f(x, ) 2 0) − ¯fmin+ 1¯f (∇x ¯f2 2 f(x, 0)Σ∇ T 2 f(x, 0))s.t. g i (x) ≤ 0 (5.4)
30where ¯f is the w 2 design , Σ is the uncertainty covariance matrix, and g i(x) are theconstraints. The constraints given in (5.3) can be written in a vector g(x) as⎡ ⎤h min − hh − h maxg(x) =⎢⎣ 10h − L ⎥⎦L − L max(5.5)For simplicity we will assume that only the beam width, h, is uncertain. Thecovariance matrix Σ is thereforeΣ =[σ2h 00 0](5.6)where σ h is the standard deviation of the uncertainty in h. The gradient of f(x, δ)with respect to δ is given by[∇ 2 f(x, 0) =6Eh 2ρAL 3]− 6Eh3ρAL 4(5.7)For this example we choose a target resonant frequency of 20kHz, approximately125,000 rad/s. The relevant design parameters are listed in table (5.1). For notationalconvenience, we once again refer to the two terms that make up the cost function inequation (5.4) asc 2 (x) =s(x, Σ) = 1¯f 2( ) 2f(x,0)− ¯f¯f(∇f(x)Σ∇ T f(x) )The term c 2 (x) is a measure of the deviation of the nominal design, f(x, 0), fromthe target, ¯f, while s(x, Σ) is a measure of the sensitivity at the nominal design.Minimizing c 2 (x) is equivalent to reducing the spread between the nominal and targetfrequency. By choosing the design variables such that f(x, 0) = ¯f, the c 2 (x) cost canbe eliminated.We will begin the analysis of this problem by looking at the feasible design space.Figure (5.2) illustrates the four linear constraints given in equation (5.5). The feasibleregion is inside the quadrilateral and is clearly convex. The dashed line in figure (5.2)represents the set of designs that have a resonant frequency of 20 kHz. Mathematically,we can say that it is the set {x | f(x, 0) = ¯f}, and therefore the associated
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: 28Chapter 5Robust Design Examples5.
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 and 44: 42evident by noting that the c 2 (x
Page 45 and 46: 44Robust Uncorrelated DesignAverage
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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