Strong Implication-Form ISS-Lyapunov Functions for Discontinuous Discrete-Time Systems

Input-to-State Stability (ISS) and the ISSLyapunov function have proved to be useful tools for the analysis and design of nonlinear systems in a variety of contexts. Motivated by the fact that many feedback control laws, such as model predictive control or event-based control, lead to discontinuous discrete-time dynamics, we investigate ISS-Lyapunov functions for such systems. ISS-Lyapunov functions were originally introduced in a so-called implication-form and, in many cases, this has been shown to be equivalent to an ISS-Lyapunov function of dissipation-form. However, for discontinuous dynamics, we demonstrate via an example that this equivalence no longer holds. We therefore propose a stronger implication-form ISS-Lyapunov which re-establishes the equivalence to dissipation-form ISS-Lyapunov functions and to the ISS property for discontinuous systems.