Continuous-time switched systems with switching frequency constraints: Path-complete stability criteria

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

“…Indeed, this equivalence implies (15), by the general result presented in Lemma 6 in Appendix. □ 4 For the formal definition of the infimal convolution ( □ ) operation between convex functions, we refer to Property 2 in Appendix.…”

“…} is a common Lyapunov function for (11). From Theorem 8 we can obtain its dual result: applying again the duality relation in (15) we have that, for any l ∈ N,…”

“…Formally, the PCLF framework involves both combinatorial and algebraic components: first, a graph that describes the set of Lyapunov inequalities and that must be path-complete in the sense that it captures every finite switching sequence, and then a set of candidate Lyapunov functions, called a template, among which a solution is sought. More recently, path-complete techniques have been proven to be effective not only for the stability problem for (1), but also in different settings, as for example constrained switched systems in [13], stabilization of switched systems in [14], and continuous-time switched systems in [15].…”

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

“…We follow the outline of the proof of Lemma 2. We consider M block-diagonal {0, 1}-matrices {A i | i ∈ ⟨M⟩} of dimension n = 2 E ⊕T for which each 2×2 block A j [ ẽ] associated to a non-existing edge ẽ ∈ E ⊕T is defined by (10). We use the template of weighted L 1 norms for which satisfying a Lyapunov inequality (s, d, i) amounts to satisfying (11).…”

“…For the purpose of understanding the relations between different multiple Lyapunov functions structures, path-complete Lyapunov functions have been proposed as a unifying and flexible approach, see [1], [18] and [10] for a partial extension in the continuous-time setting. 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.…”