Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

TL;DR

This paper introduces path-complete graph Lyapunov functions for approximating the joint spectral radius, unifying and extending existing stability analysis techniques through semidefinite programming hierarchies.

Contribution

It defines a new class of graphs called path-complete graphs and demonstrates their effectiveness in stability analysis and approximation guarantees for switched systems.

Findings

Provides asymptotically tight hierarchies of relaxations

Unifies existing Lyapunov techniques under a common framework

Establishes approximation guarantees for specific graph classes

Abstract

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.