Hybrid control synthesis for eventuality specifications using level set methods

This paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications. Given a hybrid system modelled by a hybrid automaton and a target set of states, the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time. Due to the existence of set-valued disturbance inputs, the problem is studied in a game-theoretic framework. Having shown that a least restrictive solution does not exist, we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property. To implement the algorithm, reachable sets of pursuit-evasion differential games need to be computed. For that reason level set methods are employed, where the boundary of the reachable set is characterized as the zero level set of a Hamilton–Jacobi equation. The procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology. Finally, to demonstrate the practical character of our results, a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology.