Template-Dependent Lifts for Path-Complete Stability Criteria and Application to Positive Switching Systems

In the framework of discrete-time switching systems, we analyze and compare various stability certificates relying on graph constructions. To this aim, we define several abstract expansions of graphs (so-called lifts), which depend on the chosen family of candidate Lyapunov functions (the template). We show that the validity of a given lift is linked with the analytical properties of the
template. This allows us to generate new lifts, and as a by-product, to obtain comparison criteria that go beyond the concept of simulation recently introduced in the literature. We apply our constructions to the case of copositive linear norms for positive switching systems, leading to novel stability criteria that outperform the state of the art. We provide further results relying on convex
duality and we demonstrate via numerical examples how the comparison among different stability criteria is affected by the properties of the copositive norms template.