Comparison of path-complete Lyapunov functions via template-dependent lifts

This paper investigates, in the context of discrete-time switched 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 ordering 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 switched systems. These tools are finally illustrated with the aim of numerical examples.