Almost sure Stability of Stochastic Switched Systems: Graph lifts-based Approach

TL;DR

This paper introduces a graph lifts-based approach to analyze the almost sure stability of stochastic switched systems with automaton-constrained switching signals, extending existing methods with hierarchical techniques and Lyapunov functions.

Contribution

It generalizes stability analysis tools for stochastic switched systems using graph lifts and multiple Lyapunov functions, providing hierarchical bounds on decay rates.

Findings

Established almost sure stability criteria using graph lifts.

Provided hierarchical methods for tight decay rate bounds.

Illustrated techniques with a numerical example.

Abstract

In this paper, we develop tools to establish almost sure stability of stochastic switched systems whose switching signal is constrained by an automaton. After having provided the necessary generalizations of existing results in the setting of stochastic graphs, we provide a characterization of almost sure stability in terms of multiple Lyapunov functions. We introduce the concept of lifts, providing formal expansions of stochastic graphs, together with the guarantee of conserving the underlying probability framework. We show how these techniques, firstly introduced in the deterministic setting, provide hierarchical methods in order to compute tight upper bounds for the almost sure decay rate. The theoretical developments are finally illustrated via a numerical example.