Stability of Markov regenerative switched linear systems

TL;DR

This paper establishes a necessary and sufficient condition for the mean stability of Markov regenerative switched linear systems, encompassing Markov jump and semi-Markov jump systems, based on Schur stability of a specific matrix.

Contribution

It introduces a unified stability criterion for Markov regenerative switched linear systems, generalizing existing models like Markov jump and semi-Markov systems.

Findings

Mean stability is characterized by Schur stability of a specific matrix.

The criterion applies when the stability order is even or the system is positive.

Provides a complete stability condition for a broad class of switched systems.

Abstract

In this paper, we give a necessary and sufficient condition for mean stability of switched linear systems having a Markov regenerative process as its switching signal. This class of switched linear systems, which we call Markov regenerative switched linear systems, contains Markov jump linear systems and semi-Markov jump linear systems as special cases. We show that a Markov regenerative switched linear system is $m$th mean stable if and only if a particular matrix is Schur stable, under the assumption that either $m$ is even or the system is positive.