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

39We were able to find Φ 1000 by minimizing the squared error between w 2 n and w 2 targetfrom different initial conditions. This can be written as the following optimizationminx( )w2n − wtarget2 2s.t. gi (x) ≤ 0 (5.10)This problem is equivalent to minimizing the c 2 (x) term in the robust design problemof equation (2.9). The optimization problem posed in (5.10) is rational polynomialwith polynomial constraints, and thus we were able to use our optimization algorithmto find unique designs. Figure (5.7) illustrates four of the more extreme designs fromΦ 1000 . Each of these structures has a resonant frequency of 200 kHz and satisfiesthe constraints. They demonstrate that there is significant freedom in how the designvariables are chosen. The question we will now ask is “which design is the most robustto geometric uncertainties, and how can we find that design?”600600500500400400y (µm)300y (µm)30020020010010000 200 400 600x (µm)00 200 400 600x (µm)600600500500400400y (µm)300y (µm)30020020010010000 200 400 600x (µm)00 200 400 600x (µm)Figure 5.7: Four unique crab-leg resonator designs from Φ 1000 .