Viability in Hybrid Systems

Hybrid systems are interacting systems of digital automata and continuous plants subject to disturbances. The digital automata are used to force the state trajectory of the continuous plant to obey a performance specification. For the basic concepts and notation for hybrid systems, see Kohn and Nerode (1993), and other papers in the same volume. Here we introduce tools for analyzing enforcing viability of all possible plant state trajectories of a hybrid system by suitable choices of finite state control automata. Thus, the performance specification considered here is that the state of the plant remain in a prescribed viability set of states at all times (Aubin, 1991). The tools introduced are local viability graphs and viability graphs for hybrid systems. We construct control automata which guarantee viability as the fixpoints of certain operators on graphs. When control and state spaces are compact, the viability set is closed, and a non-empty closed subset of a viability graph is given with a sturdiness property, one can extract finite state automata guaranteeing viable trajectories. This paper is a sequel to Kohn and Nerode (1993), especially Appendix II.