Optimal control with regular objectives using an abstraction-refinement approach

This paper presents a method to synthesize a sequence of control inputs for a discrete-time piecewise linear system equipped with a cost function, such that the controlled system behavior satisfies a finite-word linear-time temporal objective with minimal cost. An abstract finite state weighted transition system is constructed, such that the cost of the optimal control on the abstract system provides an upper bound on the cost of the optimal control for the original system. Specifically, the abstract system is constructed from finite partitions of the state and input spaces by solving optimization problems, and a sequence of suboptimal controllers is obtained by considering a sequence of uniformly refined partitions. Furthermore, the costs achieved by the sequence of suboptimal controllers converge to the optimal cost for the piecewise linear system. The abstraction refinement algorithm is implemented in the tool OptCAR. The feasibility of this approach is illustrated on an example, by constructing automatically, sub-optimal controllers with improving optimal costs.