Max-Min Lyapunov Functions for Switched Systems and Related Differential Inclusions

TL;DR

This paper develops max-min Lyapunov functions from basic positive definite functions to analyze stability in switched systems and differential inclusions, offering less conservative and more versatile stability conditions.

Contribution

It introduces a method to construct Lyapunov functions using max-min combinations, applicable to complex nonlinear systems with state-dependent switching.

Findings

Provides stability conditions for differential inclusions and switched systems.

Demonstrates the usefulness of nonconvex Lyapunov functions.

Includes examples illustrating the effectiveness of the proposed approach.

Abstract

Starting from a finite family of continuously differentiable positive definite functions, we study conditions under which a function obtained by max-min combinations is a Lyapunov function, establishing stability for two kinds of nonlinear dynamical systems: a) Differential inclusions where the set-valued right-hand-side comprises the convex hull of a finite number of vector fields, and b) Autonomous switched systems with a state-dependent switching signal. We investigate generalized notions of directional derivatives for these max-min functions, and use them in deriving stability conditions with various degrees of conservatism, where more conservative conditions are numerically more tractable. The proposed constructions also provide nonconvex Lyapunov functions, which are shown to be useful for systems with state-dependent switching that do not admit a convex Lyapunov function. Several examples are included to illustrate the results.