Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach

TLDR

This paper addresses the stability analysis and control synthesis of discrete‑time switched systems. The authors employ a switched quadratic Lyapunov function and propose two linear matrix inequality conditions—one classical and one new with a slack variable—to test asymptotic stability. The two conditions are proved equivalent for stability analysis, and the slack‑variable condition is shown to be less conservative for static output‑feedback control, as illustrated by numerical results.

Abstract

This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.