On stability of randomly switched nonlinear systems

TL;DR

This paper analyzes the stability of randomly switched nonlinear systems, providing Lyapunov-based conditions for almost sure stability under slow switching, with applications to Markov jump systems and feedback stabilization.

Contribution

It introduces new stability criteria for randomly switched nonlinear systems under slow switching conditions, including Markov jump systems, and offers explicit feedback control schemes.

Findings

Sufficient conditions for almost sure global asymptotic stability

Applicability to Markov jump systems

Explicit feedback stabilization methods

Abstract

This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure global asymptotic stability using Lyapunov-based methods when individual subsystems are stable and a certain ``slow switching'' condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results therefore hold for Markov jump systems in particular. For systems with control inputs we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems.