Convergence Rate of Nonlinear Switched Systems

TL;DR

This paper investigates the convergence rates of nonlinear switched systems, showing solutions can converge arbitrarily slowly or exponentially fast depending on system properties and input conditions.

Contribution

It provides explicit convergence rate bounds for analytic switched systems with common Lyapunov functions under specific assumptions.

Findings

Solutions can converge arbitrarily slowly for certain inputs.

Explicit exponential convergence rates are derived for systems with fixed dwell-time.

Convergence behavior depends on system stability and Lyapunov function properties.

Abstract

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems. We first consider a switched system which is asymptotically stable for a class of inputs but not for all inputs. We show that solutions corresponding to that class of inputs converge arbitrarily slowly to the origin. Then we consider analytic switched systems for which a common weak quadratic Lyapunov function exists. Under two different sets of assumptions we provide explicit exponential convergence rates for inputs with a fixed dwell-time.