Multiple Lyapunov Functions and Memory: A Symbolic Dynamics Approach to Systems and Control

TL;DR

This paper introduces a symbolic dynamics-based framework for Lyapunov stability analysis of hybrid systems, connecting multiple Lyapunov functions with memory and prediction techniques, leading to improved stability conditions and numerical methods.

Contribution

It develops a novel language-theoretic approach linking multiple Lyapunov functions with memory and prediction, offering new stability criteria and numerical schemes for hybrid systems.

Findings

Unified Lyapunov function equivalent to multiple Lyapunov functions

New stability conditions incorporating memory and future states

Numerical schemes outperform existing techniques

Abstract

We propose a novel framework for the Lyapunov analysis of an important class of hybrid systems, inspired by the theory of symbolic dynamics and earlier results on the restricted class of switched systems. This new framework allows us to leverage language theory tools in order to provide a universal characterization of Lyapunov stability for this class of systems. We establish, in particular, a formal connection between multiple Lyapunov functions and techniques based on memorization and/or prediction of the discrete part of the state. This allows us to provide an equivalent (single) Lyapunov function, for any given multiple-Lyapunov criterion. By leveraging our language-theoretic formalism, a new class of stability conditions is then obtained when considering both memory and future values of the state in a joint fashion, providing new numerical schemes that outperform existing technique. Our techniques are then illustrated on numerical examples.