Using Symbolic Dynamics to Compare Path-Complete Lyapunov Functions

In this paper, we study discrete-time linear switched systems by leveraging tools from symbolic dynamics and language theory. More specifically, viewing path-complete Lyapunov functions (PCLF) associated with the switched system as finite automata, we investigate the coverings of bi-infinite words generated via a PCLF and develop a framework for comparing two different PCLFs via these coverings. However, in most of the cases PCLFs are not comparable with regards to one corresponding to a better stability criterion than the other. For this purpose, we utilize the notion of support sets, which is a subset of paths in a PCLF that is sufficient to obtain the same performance index as that of the entire set of bi-infinite words generated by the PCLF, to obtain partial relations between two coverings. We also illustrate a numerical example to justify the study of support sets via the covering framework.