Viable Control of Hybrid Systems

Hybrid systems are continuous variable, continuous time systems with a phased operation. Within each phase the system evolves continuously according to the dynamical law of that phase; when an event occurs, the system makes a transition from one phase to the next. In this paper, we study hybrid systems with nondeterministic discrete and continuous behaviors. We use nondeterministic nite automata to model the discrete behavior and state dependent diierential inclusions to model the continuous behavior. By viability we mean the system's ability to take an innnite number of discrete transitions. Viability can be used to express safety and fairness properties over the system's state trajectories. To ensure viability, the system's evolution must be restricted so that discrete transitions occur within spe-ciic subsets of their enabling conditions. Following Aubin 3], we call these subsets the system's viability kernel. We give results pertaining to certain continuity properties of the viability kernel, we give conditions under which the viability kernel can be computed in a nite number of steps, and we synthesize a hybrid controller that yields all viable trajectories.