Uniform Stability of Switched Linear Systems: Extensions of LaSalle's Invariance Principle

Abstract

This paper addresses the uniform stability of switched linear systems, where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set. We provide a collection of results that can be viewed as extensions of LaSalle's Invariance Principle to certain classes of switched linear systems. Using these results one can deduce asymptotic stability using multiple Lyapunov functions whose Lie derivatives are only negative semidefinite. Depending on the regularity assumptions placed on the switching signals, one may be able to conclude just asymptotic stability or (uniform) exponential stability. We show by counter-example that the results obtained are tight.