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

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

“…□ Now, thanks to the convex duality theory summarized in Appendix, we can straightforwardly obtain similar results for the template P of primal copositive norms. The same statements (but with direct proofs) can be found in our preliminary paper [23].…”

“…Indeed, given a path-complete graph G, we can observe that the sum-,min-, and max-lifted graphs admit a path-complete and strongly connected component isomorphic to the initial graph. See [23] for more details. △…”

“…i.e. the componentwise minimum between v and w. In our preliminary paper [23], we focused on properties of primal copositive norms only. In what follows, we provide the statements and proofs for dual copositive norms, and moreover we show how, thanks to the convex-duality theory summarized in Appendix, the corresponding properties are re-obtained, as simple corollaries, for primal norms.…”

Necessary and Sufficient Conditions for Template-Dependent Ordering of Path-Complete Lyapunov Methods

“…In this framework, the inequalities relating the positive definite functions composing a given multiple-Lyapunov stability criterion are encoded in a directed and labeled path-complete graph G, as we will clarify. Possibly increasing the number of nodes/edges of this underlying graph G, it is possible to "asymptotically" reach necessary and sufficient criteria, even starting from a very structured set of candidate Lyapunov functions, as for example quadratics or polyhedral functions [1], [3], [8]. While this framework is now mature enough to provide effective criteria to approximate the decay rate (a.k.a.…”

“…A partial result is provided in [19], without any hypothesis on the candidate Lyapunov functions template V, relying on the notion of simulation of graphs. On the other hand, in [8], it is underlined how the comparison/ordering relations between graphs is strongly affected by the analytical properties of the chosen template V. Although it may be counter-intuitive, there are indeed examples for which increasing the size of the graph does not improve the stability certificate if we consider a particular family of Lyapunov functions. In [8], we introduced formal transformations of graphs, called template-dependent lifts, in order to improve the performance of a path-complete criterion (by enlarging the underlying graph) following particular rules which rely on closure properties of V.…”