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

41Figure (5.8) illustrates the two uniform etch scenarios on a beam with an end mass.The vector ξ = [−1 −1 1 1 1 1] T represents a uniform etch for the crab-leg structure.If we take λ to be a gaussian random variable, with standard deviation σ, then we canwrite the correlated uncertainty vector as δ = λξ. When λ is positive, this correspondsto the design, x + λξ, having been underetched, and conversely, negative λ indicatesthe structure was overetched. We can therefore write the correlated covariance matrix,Σ C , asΣ C = σ 2 ξξ T (5.14)We will proceed by looking at the results for these two uncertainty models.Figure 5.8: Illustration depicting under and overetch of a beam with an end-mass.Uncorrelated ResultsUsing our rational polynomial optimization algorithm to solve the problem posedin (5.12), we found the following robust design for the uncorrelated uncertainty casex ∗ U = 241.68 20.56 14.49 2.03 87.84 442.71[w n = 199.960 kHzF (x ∗ U , Σ U) = 4.2362 × 10 −4] Tc 2 (x ∗ U ) = 1.6 × 10−7s(x ∗ U , Σ U) = 4.2346 × 10[−4∇ x F (x ∗ U , Σ U) = − .057 .025 4.780 .408 .056 .011]× 10 −5The robust design, x ∗ U , for the uncorrelated uncertainty case is shown in figure (5.9).The functions c 2 (x) and s(x, Σ) were defined in equation (2.10). Clearly, the nominal(δ = 0) resonant frequency, w n , of this design almost nails the target. This is also