The Path-Complete Formalism for Switched Systems: Stability and Beyond

Abstract

Switched systems play a crucial role in modern engineering thanks to their ability to capture complex behaviours involving transitions between different operational modes. However, analyzing their stability remains a challenging task due to the intricate interplay between discrete switching and dynamics. This complexity calls for sophisticated mathematical tools. While Lyapunov theory remains a cornerstone of stability analysis, traditional methods often fall short when applied to switched systems, prompting ongoing efforts to extend the theory to better accommodate their unique characteristics.
This talk presents the framework of path-complete Lyapunov functions, which offers a fresh perspective by integrating combinatorial structures to represent switching behaviour. Specifically, a path-complete Lyapunov function comprises two components: a combinatorial element, represented by an automaton (a directed graph) that encodes admissible switching sequences, and an algebraic element, consisting of a collection of Lyapunov functions—one for each node in the graph. The graph edges govern how these Lyapunov pieces interact and decrease across transitions. This framework is particularly appealing for the analysis of switched systems because it allows for the construction of tailored, nonstandard, and less conservative stability criteria, all while mitigating the combinatorial complexity that often burdens classical optimization techniques.
While originally introduced for stability analysis, the path-complete Lyapunov framework has been recently extended to encompass constrained switching systems, stabilization via switching sequence and control Lyapunov function design, and safety through path-complete barrier functions. This talk will highlight these recent developments and their unifying role within the framework.