Lyapunov Characterization for ISS of Impulsive Switched Systems

TL;DR

This paper establishes necessary and sufficient conditions for input-to-state stability (ISS) of impulsive switched systems using time-varying Lyapunov functions, broadening applicability with less restrictive switching conditions.

Contribution

It introduces two types of time-varying ISS-Lyapunov functions and proves their equivalence to ISS, extending previous work by relaxing dwell time constraints.

Findings

Necessary and sufficient ISS conditions established

Construction method for decreasing Lyapunov functions from non-decreasing ones

Broader applicability due to relaxed dwell and leave time constraints

Abstract

In this study, we investigate the ISS of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is non-decreasing and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a technique for constructing a decreasing ISS-Lyapunov function from a non-decreasing one, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee the ISS of a particular class of impulsive switched systems when the switching signal is unknown.