Invariance-like results for Nonautonomous Switched Systems

TL;DR

This paper extends the LaSalle-Yoshizawa Theorem to nonsmooth switched systems, providing conditions under which stability results hold using common Lyapunov functions and regularization techniques.

Contribution

It introduces a generalized LaSalle-Yoshizawa framework for nonsmooth switched systems, including regular and non-regular Lyapunov functions, with applications to adaptive control.

Findings

Filippov and Krasovskii regularizations are within convex hulls of subsystem regularizations.

A common Lyapunov function with negative semidefinite derivative ensures stability.

Generalized derivative concepts enable stability analysis for nonsmooth, switched systems.

Abstract

This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A candidate common Lyapunov function that has a negative semidefinite derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle-Yoshizawa results for the switched system. Results for regular and non-regular candidate Lyapunov functions are presented using an appropriate generalization of the time derivative. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.