A complete characterization of the ordering of path-complete methods

Philippe, Matthew;Jungers, Raphaël
(2019) the 22nd ACM International Conference — Location: Montreal, Quebec, Canada (16.April.2019)

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We study criteria allowing to compare the conservativeness of stability certificates for switching systems. The stability certificates under consideration are Path-Complete Lyapunov functions (PCLFs), which are multiple Lyapunov functions with an underlying combinatorial structure. Several criteria exist to compare these certificates and we focus here on the recently introduced concept of simulation. This criterion, which is borrowed from automata theory, relies only on the combinatorial structures that underlie the PCLFs. We show that the concept of simulation provides a complete characterization of the ordering relation between PCLFs, independently of the algebraic properties of the dynamical system studied and the template of Lyapunov functions used. We summarize the state of the art on the comparison of PCLFs and raise future research directions.
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Philippe, M., & Jungers, R. (2019). A complete characterization of the ordering of path-complete methods. Proceedings of the 22nd ACM International Conference on Hybrid Systems Computation and Control - HSCC ’19, p. 138-146. https://doi.org/10.1145/3302504.3311803