Synthesis using approximately bisimilar abstractions: time-optimal control problems

In this paper, we present a hierarchical approach to time-optimal control using approximately bisimilar abstractions. Given a time-optimal controller for an abstraction, we present a specific procedure that allows us to compute a suboptimal controller for the original system. While the usual controller refinement procedure produces dynamic controllers that may have limitations in terms of implementation cost and robustness, the static controllers we synthesize do not suffer from these issues. Moreover, we provide guarantees by bounding below and above the performances of the synthesized controller between performances of two time-optimal controllers for problems that can be made arbitrarily close by choosing sufficiently precise abstractions. Finally, we show the effectiveness of our approach by solving time-optimal control problems for switched systems.