Converse Lyapunov Results for Stability of Switched Systems with Average Dwell-Time

TL;DR

This paper characterizes the stability of nonlinear switched systems under average dwell-time constraints using necessary and sufficient conditions with multiple Lyapunov functions, extending previous results to average dwell-time scenarios.

Contribution

It provides a novel converse Lyapunov theorem for systems with average dwell-time, independent of individual subsystem flows, and applies to linear switched systems as a special case.

Findings

Counterexample shows lower bounds for dwell-time do not imply stability for average dwell-time.

New inequalities for stability do not depend on subsystem flows and are easier to verify.

Results extend stability analysis to average dwell-time constrained switched systems.

Abstract

This article provides a characterization of stability for switched nonlinear systems under average dwell-time constraints, in terms of necessary and sufficient conditions involving multiple Lyapunov functions. Earlier converse results focus on switched systems with dwell-time constraints only, and the resulting inequalities depend on the flow of individual subsystems. With the help of a counterexample, we show that a lower bound that guarantees stability for dwell-time switching signals may not necessarily imply stability for switching signals with same lower bound on the average dwell-time. Based on these two observations, we provide a converse result for the average dwell-time constrained systems in terms of inequalities which do not depend on the flow of individual subsystems and are easier to check. The particular case of linear switched systems is studied as a corollary to our main result.