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

13where ˜β t is the actual value, β t is the nominal value, and ɛ t represents the error.Referring to our general polynomial h(x) from equation (2.12), let us assume the firstT u coefficients are uncertain. We can therefore write(∑h(x, δ, ɛ) = T ∏ naβ t (x i + δ i ) α t,it=1∑+ Tut=1i=1(∏ naɛ t (x i + δ i ) α t,ii=1n a+n ∏ mi=n a+1n a+n m∏i=n a+1(1 + δ i ) α t,ix α t,i(1 + δ i ) α t,ix α t,iiin∏i=n a+n m+1n∏i=n a+n m+1The expression for h(x, δ, ɛ) is polynomial in x, δ, and ɛ. We can define δ h to be[δ h = δand the dimension of δ h is (n a + n m + T u ).ɛ] Tx α t,iix α t,iiIn this section we showed that givena polynomial, it is trivial to add uncertainty to the design variables and constantswhile maintaining the polynomial structure. This allows us to consider problems ofthe form posed in (2.1), and then later add uncertainty to the objective and maintainthe rational polynomial structure presented in equation (2.2).))