Logic-based design and synthesis of controllers for hybrid systems

In this paper, we formulate and solve a quite general class of hybrid controller synthesis problems. The plant consists of a finite number of continuous systems, given by differential equations over a common state space; the controller steers the plant state by determining when to discretely switch between the various differential equations; and the closed-loop trajectories correspond to those of (a subclass of) the widely accepted hybrid automaton model. In addition to the well-studied classes of safety (reachability or invariance) and liveness (non-blocking and non-Zeno) performance specifications, we deal with a general class of active behavioural specifications, requiring that trajectories traverse in prescribed sequences through the blocks of a given finite partition of the plant state space. We give an abstract algorithm which solves this controller synthesis problem for arbitrary differential equations with unique solutions, and give an analytic proof of finite termination from an assumption of compactness of the sets given in the data of the specifications. Moreover, controllers produced by the synthesis algorithm exibit a quantifiable degree of robustness. In formulating the algorithm, and proving its correctness, we make essential use of modal logic as a formalism for reasoning about sets of plant states, and various operators on sets arising from the differential equations. The control problem and our synthesis algorithm are illustrated with a 2-dimensional example with linear differential equations (with spiral flows), with graphic output generated by our prototype software implementation based on an approximate and discretized representation of sets of plant states. ∗Research partially supported by US Office of Naval Research, Grant N 00014-98-1-0535.