Approximate finite-state feedback controller of hybrid systems

From the practical point of view, finite abstraction is a useful tool for the analysis and design of hybrid systems. In general, however, it does not always exist. In this paper, we propose a design method of an approximate finite-state feedback controller for a specified predicate in a hybrid system. We partition an infinite state set of the hybrid system into a finite set and construct a transition system, called a quotient transition system, which is an approximate abstraction of the hybrid system. Then, we construct a controller by which a reachability set of the closed system satisfies approximately the predicate. We introduce a concept of a tolerate degree of the approximation to evaluate how close the reachability set is to the legal set satisfying the predicate. Moreover, we propose design procedures of an approximate finite-state feedback controller which satisfies the predicate with a specified tolerate degree.