Robust stability theory for hybrid systems

In this work, we focus on developing Lyapunov stability analysis tools for hybrid dynamical systems, which are combinations of a differential equation/inclusion on a constraint set and a difference equation/inclusion on another constraint set. This dissertation is a collection of equivalent characterizations of pre-asymptotic stability (a natural generalization of asymptotic stability), input-to-state stability (ISS), and output-to-state stability (OSS) for hybrid dynamical systems. Most of these results unify and extend the existing theory for continuous-time and discrete-time systems.; For a hybrid system satisfying mild regularity assumptions, we show that (pre-)asymptotic stability is equivalent to the existence of a smooth Lyapunov function. This result is achieved with two intermediate results, that stability with respect to a single measure (in particular, a proper indicator for a compact set on its open basin of attraction) for a hybrid system is generically robust to small state-dependent perturbations, and that the robustness of stability with respect to two measures is equivalent to a characterization of such stability in terms of a smooth Lyapunov function. As a special case, we state a converse Lyapunov theorem for systems with logic variables and use this result to establish input-to-state stabilization using hybrid feedback control. The Lyapunov characterizations of (pre-)asymptotic stability not only can be used to establish semi-global practical robustness to various types of perturbations, but also help us to generate the equivalence of OSS, nonuniform OSS, and the existence of a smooth OSS-Lyapunov function for hybrid dynamical systems that are associated with outputs.; Relying on Lyapunov characterizations of (pre-)asymptotic stability and assuming that the right-hand side of the differential equation has a convexity property with respect to inputs, we establish the equivalence of ISS, nonuniform ISS, and the existence of a smooth ISS-Lyapunov function for hybrid dynamical systems that are affected by inputs (or disturbances). We demonstrate by examples that the equivalence may fail when such a convexity property is not assumed. We also consider the case of no convexity assumption but Lipschitz continuity; in particular, we propose a strengthened ISS condition (i.e. robust ISS) as a sufficient condition for the existence of a smooth ISS-Lyapunov function, and we derive finite-horizon relaxation theorems to identify when the existence of a smooth ISS-Lyapunov function is necessary for ISS. Finally, we use the ISS results for hybrid systems to recover and generalize Lyapunov characterizations of input-output-to-state stability for continuous-time systems.