Deciding the boundedness and dead-beat stability of constrained switching systems

TL;DR

This paper investigates the stability of constrained discrete-time linear switching systems, providing a decidable condition for boundedness and a polynomial-time algorithm for dead-beat stability, with practical applications.

Contribution

It introduces a generalized boundedness condition for constrained systems and a polynomial-time method for dead-beat stability, extending previous results to automaton-constrained switching.

Findings

Decidable sufficient condition for boundedness when maximal growth rate is one

Polynomial-time algorithm for dead-beat stability

Application demonstrated through a real-world case study

Abstract

We study computational questions related with the stability of discrete-time linear switching systems with switching sequences constrained by an automaton. We first present a decidable sufficient condition for their boundedness when the maximal exponential growth rate equals one. The condition generalizes the notion of the irreducibility of a matrix set, which is a well known sufficient condition for boundedness in the arbitrary switching (i.e. unconstrained) case. Second, we provide a polynomial time algorithm for deciding the dead-beat stability of a system, i.e. that all trajectories vanish to the origin in finite time. The algorithm generalizes one proposed by Gurvits for arbitrary switching systems, and is illustrated with a real-world case study.