Framework for Studying Stability of Switching Max-Plus Linear Systems

TL;DR

This paper introduces a framework for analyzing the stability of switching max-plus linear systems, addressing challenges due to non-monotonicity by relaxing assumptions to positive invariance of max-plus cones.

Contribution

It develops new stability notions for switching max-plus systems and generalizes stability analysis using max-plus cone invariance, advancing the understanding of such systems.

Findings

Identified loss of equivalence of stability notions in switching systems.

Proposed relaxation to positive invariance of max-plus cones.

Established a foundational approach for stability analysis of switching max-plus systems.

Abstract

We propose a framework for studying the stability of discrete-event systems modelled as switching max-plus linear systems. In this framework, we propose a set of notions of stability for generic discrete-event systems in the max-plus algebra. Then we show the loss of equivalence of these notions for switching max-plus linear systems due to the lack of global monotonicity and the accompanying difficulty in rigorous analysis. This serves as a motivation to relax the assumption on monotonicity of the dynamics to positive invariance of max-plus cones. Then we proceed to generalise the notions of stability when the dynamics is restricted to such cones. The stability analysis approach presented in this paper serves as a first step to study the stability of a general class of switching max-plus linear systems.