Comparison of Path-Complete Lyapunov Functions via Template-Dependent Lifts

TL;DR

This paper develops a new framework for comparing path-complete Lyapunov functions in discrete-time switching systems using lifts, linking analytical properties of Lyapunov templates to stability criteria, and applies it to positive systems.

Contribution

Introduces a general methodology using lifts to compare path-complete Lyapunov functions, connecting template properties to stability certificate admissibility.

Findings

Provides new stability certificates for positive switching systems.

Establishes relations between Lyapunov templates and admissibility of lifts.

Illustrates the approach with numerical examples.

Abstract

This paper investigates, in the context of discrete-time switching systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the order relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switching systems. These tools are finally illustrated with the aim of numerical examples.