Stability theory for hybrid dynamical systems

Hybrid systems which are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system are of great current interest. In the present paper we first formulate a definition of hybrid dynamical system which covers a very large number of classes of hybrid systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of invariant set (e.g., equilibrium) for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for the uniform stability and the uniform asymptotic stability of an invariant set of a hybrid dynamical system. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). To demonstrate the applicability of the developed theory, we present two specific examples of hybrid dynamical systems and we conduct a stability analysis of one of these examples (a sampled-data feedback control system with a nonlinear (continuous-time) plant and a linear (discrete-time) controller).