Necessary and Sufficient Conditions for Stability of Discrete-Time Switched Linear Systems with Ranged Dwell Time

TL;DR

This paper introduces a new stability analysis method for discrete-time switched linear systems using L-switching-cycle, generalizing existing Lyapunov approaches and reducing conservativeness with longer cycles.

Contribution

It proposes the L-switching-cycle concept and two equivalent sufficient conditions for stability, extending Lyapunov methods and improving analysis accuracy.

Findings

The conditions guarantee global uniform asymptotic stability.

L-switching-cycle can reduce conservativeness with longer cycles.

Numerical example confirms theoretical results.

Abstract

This paper deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called L-switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on L-switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same $L$-switching-cycle. These two sufficient conditions can be viewed as generalizations of the clock-dependent Lyapunov and multiple Lyapunov function methods, respectively. Furthermore, it has been proven that the proposed L-switching-cycle can eventually achieve the nonconservativeness in stability analysis as long as a sufficiently long L-switching-cycle is adopted. A numerical example is provided to illustrate our theoretical results.