Stabilizing Randomly Switched Systems

TL;DR

This paper develops conditions for the stability and stabilization of randomly switched systems with jump stochastic switching signals, using Lyapunov functions to design feedback controllers even when subsystems are unstable.

Contribution

It introduces new stability criteria for non-Markovian switching signals and provides a systematic approach for feedback stabilization of nonlinear systems.

Findings

Established sufficient conditions for stochastic stability in various senses.

Designed stabilizing feedback controllers for affine control subsystems.

Provided universal formulas for feedback stabilization of nonlinear systems.

Abstract

This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design