Interpretability of Path-Complete Techniques and Memory-based Lyapunov functions

TL;DR

This paper explores the interpretability of path-complete Lyapunov functions for switched systems, introducing memory-based Lyapunov functions that generalize existing techniques and establishing their equivalence for improved numerical efficiency.

Contribution

It introduces memory-based Lyapunov functions, demonstrating their equivalence to path-complete functions and enhancing understanding and computational efficiency.

Findings

Memory-based Lyapunov functions generalize existing techniques.

Equivalence between path-complete and memory-based Lyapunov functions.

Improved numerical efficiency demonstrated through an academic example.

Abstract

We study path-complete Lyapunov functions, which are stability criteria for switched systems, described by a combinatorial component (namely, an automaton), and a functional component (a set of candidate Lyapunov functions, called the template). We introduce a class of criteria based on what we call memory-based Lyapunov functions, which generalize several techniques in the literature. Our main result is an equivalence result: any path-complete Lyapunov function is equivalent to a memory-based Lyapunov function, however defined on another template. We show the usefulness of our result in terms of numerical efficiency via an academic example.