Windows of opportunity for the stability of jump linear systems: Almost sure versus moment convergence

Maurizio Porfiri, Russell Jeter, Igor Belykh

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we examine the role of the switching period on the stochastic stability of jump linear systems. More specifically, we consider a jump linear system in which the state matrix switches every m time steps randomly within a finite set of realizations, without a memory of past switching instances. Through the computation of the Lyapunov exponents, we study δ-moment and almost sure stability of the system. For scalar systems, we demonstrate that almost sure stability is independent of m, while δ-moment stability can be modulated through the selection of the switching period. For higher-dimensional problems, we discover a richer influence of m on stochastic stability, quantified in an almost sure and a δ-moment sense. Through the detailed analysis of an archetypical two-dimensional problem, we illustrate the existence of disconnected windows of opportunity where the system is asymptotically stable. Outside of these windows, the system is unstable, even though it switches between two Schur-stable state matrices.

Original languageEnglish (US)
Pages (from-to)323-329
Number of pages7
JournalAutomatica
Volume100
DOIs
StatePublished - Feb 2019

Keywords

  • Discrete-time
  • Linear stochastic systems
  • Lyapunov exponent
  • Mean-square
  • Stability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

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