Structured Feedback Synthesis for Stability and Performance of Switched Systems

This article addresses the synthesis of fixed-order output feedback controllers for stability and performance of continuous-time switched linear systems with dwell time constraints or arbitrary switching. Specifically, this article starts by considering the stabilization problem, which is addressed by searching for a family of homogeneous polynomial Lyapunov functions parameterized polynomially by the sought controller. In order to conduct this search, polynomials are introduced for approximating the matrix exponential and for quantifying the feasibility of the Lyapunov inequalities. It is shown that a stabilizing controller exists if and only if a condition built solving three convex optimization problems with linear matrix inequalities holds for polynomials of degree sufficiently large. Analogous conditions for the existence of a controller ensuring desired upper bounds on the H
₂
norm and on the rms gain of the closed-loop system are derived by searching for a family of homogeneous rational Lyapunov functions parameterized rationally by the sought controller.