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, and when an event occurs, the system makes a transition from one phase to the next. Thus, hybrid systems display both discrete and continuous behavior. The discrete behavior consists of abrupt transitions between phases, and the continuous behavior consists of smooth ows within phases. The state of the system can be thought of as a pair|the discrete state and the continuous state. The discrete state identiies a ow, and the continuous state identiies a position in it. For a transition to take place, the discrete state and the continuous state together must satisfy a condition that enables the transition. Once a transition occurs, the discrete state and the continuous state are changed abruptly. Then, until the next transition, the continuous state evolves according to the ow identiied by the new discrete state. Thus, the discrete dynamics and the continuous dynamics innuence each other. In our development, we are concerned with viable control of nondeterministic hybrid systems. The discrete behavior is nondeterministic if multiple transitions can be enabled at some time, and the system can choose any one of them. The continuous behavior is nondeterministic if each discrete state identiies a nonempty set of ows and at least one set is not a singleton. The continuous state can follow any ow in the set at each time in the given phase of operation. In viable control, the controller uses a control law that identiies all control trajectories that yield the desired behavior; some trajectory within the law is picked nondeterministically. We use nondeterministic nite automata to model the discrete behavior and state dependent diierential inclusions to model the continuous behavior, and we deene the maximal nonblocking causal control strategy. We give computability results and synthesize a viable state feedback controller for hybrid systems. The special case of discrete event systems is obtained when the continuous behavior is absent. We describe discrete event systems in the propositional linear temporal logic formalism. We deene the maximal nonblocking causal control strategy for discrete event systems, and develop the ideas of three-way partitioning of controls and syntactic control. It is important to note that we give a syntactic procedure for synthesizing a viable feedback controller for discrete event systems.