Low order multicriteria H∞ design via bilinear matrix inequalities

A practical and systematic procedure for optimization of low order H_∞ controllers is presented. Separate H_∞ measures for low, mid and high frequency closed loop properties are formulated and used to construct a constrained optimization problem with the controller parameters as unknowns. By virtue of the bounded real lemma, the optimization problem can be rewritten into bilinear matrix inequality (BMI) form with one Lyapunov function for each performance measure as additional unknowns. As is well known, by forcing these Lyapunov functions to be identical, the problem can be transformed to a linear matrix inequality (LMI) problem. Closed loop robustness and performance is compared in three examples for controllers designed based on the BMI and the LMI formulations as well as H_∞ loop-shaping and PID controllers with a second order filter on the derivative part. Importantly, the LMI controller is used as an initial guess to the BMI algorithm, which is solved with a P-K iteration. General non-convex design problems are solved with algorithms in the TOMLAB optimization environment. Results show that closed loop robustness and performance can be improved by allowing the Lyapunov functions to be different, even when the controller order is significantly reduced compared to the H_∞-LMI controller.