A Necessary and Sufficient Condition for Robust Asymptotic Stabilizability of Continuous-Time Uncertain Switched Linear Systems

Abstract — The main contribution of this paper is a necessary and sufficient rank condition derived for the robust asymptotic stabilizability of switched linear systems with timevariant parametric uncertainties, thus improving the sufficient only conditions found in the literature. The method is based on polyhedral Lyapunov-like functions for each unstable subsystems, which represent generalizations of the polytopic Lyapunov functions in the classical sense. Under certain rank condition, which is related to these Lyapunov-like functions, the stabilizing switching law is constructed and a global Lyapunov function is composed, which guarantees asymptotic stability for the closed-loop switched linear system. The rank condition is also proved to be necessary for the existence of an asymptotically stabilizing switching control law for the switched linear systems. I