Robust Stability and Performance via Fixed-Order Dynamic Compensation

Abstract

Two robust control-design problems are considered. The Robust Stabilization Problem involves deterministically modeled, bounded but unknown, time-varying parameter variations, while the Robust Performance Problem includes, in addition, a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations. For both problems the design goal is a fixed-order (i.e., reduced- or full-order) dynamic (strictly proper) feedback compensator. A sufficient condition for solving the Robust Stabilization Problem is given by means of a quadratic Lyapunov function parameterized by the compensator gains. For the Robust Performance Problem the Lyapunov function provides an upper bound for the closed-loop performance. This leads to consideration of the Auxiliary Minimization Problem: Minimize the performance bound over the class of fixed-order controllers subject to the Lyapunov-function constraint. Necessary conditions for optimality in the auxiliary problem thus serve as sufficient conditions for robust stability and performance in the original problem. Two particular bounds are considered for constructing the quadratic Lyapunov function. The first corresponds to a right shift/ multiplicative white noise model, while the second was suggested by recent work of Petersen and Hollot. The main result is an extended version of the optimal projection equations for fixed-order dynamic compensation whose solutions are guaranteed to provide both robust stability and robust performance.