Switching systems with dwell time: computation of the maximal Lyapunov exponent

TL;DR

This paper investigates the stability of continuous-time switching systems with dwell time constraints by reformulating them as mixed discrete-continuous systems on graphs, and develops methods to compute their maximal Lyapunov exponent.

Contribution

It introduces a stability theory for mixed systems on graphs, including invariant Lyapunov norms, and adapts spectral radius algorithms for Lyapunov exponent computation.

Findings

Proves existence of invariant Lyapunov norms for mixed systems.

Adapts spectral radius algorithms to compute Lyapunov exponents.

Analyzes stability conditions for systems with mode-dependent dwell times.

Abstract

We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete-continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its structure in various cases, including discrete-time systems for which discrete actions have inhomogeneous time durations. This allows us to adapt recent methods for the joint spectral radius computation (Gripenberg's algorithm and the Invariant Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on graphs.