Interpretability of Path-Complete Techniques and Memory-Based Lyapunov Functions

We study path-complete Lyapunov functions, which are stability criteria for switched systems, described by a combinatorial component (namely, an automaton), and a functional component (a set of candidate Lyapunov functions, called the template). We introduce a class of criteria based on what we call memory-based Lyapunov functions, which generalize several techniques in the literature. Our main result is an equivalence result: any path-complete Lyapunov function is equivalent to a memory-based Lyapunov function, however defined on another template. We show the usefulness of our result in terms of numerical efficiency via an academic example.

Automated summary

We study path-complete Lyapunov functions as stability criteria for switched systems, combining a combinatorial component (automaton) and a functional component (template of candidate Lyapunov functions). We propose a class of memory-based Lyapunov function criteria that generalize existing techniques. Our main result shows the equivalence between any path-complete Lyapunov function and a memory-based Lyapunov function defined on a different template. We demonstrate the numerical efficiency of our approach through an academic example.