Synthesis of switching controllers using approximately bisimilar multiscale abstractions

When available, discrete abstractions provide an appealing approach to controller synthesis. Recently, an approach for computing discrete abstractions of incrementally stable switched systems has been proposed, using the notion of approximate bisimulation. This approach is based on sampling of time and space where the sampling parameters must satisfy some relation in order to achieve a certain precision. Particularly, the smaller the sampling period, the finer the lattice approximating the state-space and the larger the number of states in the abstraction. This renders the use of these abstractions for synthesis of fast switching controllers computationally prohibitive. In this paper, we present a novel class of multiscale discrete abstractions for switched systems that allows us to deal with fast switching while keeping the number of states in the abstraction at a reasonable level. The transitions of our abstractions have various durations: for transitions of longer duration, it is sufficient to consider abstract states on a coarse lattice; for transitions of shorter duration, it becomes necessary to use finer lattices. These finer lattices are effectively used only on a restricted area of the state-space where the fast switching occurs. We show how to use these abstractions for multiscale synthesis of self-triggered switching controllers for reachability specifications under time optimization. We illustrate the merits of our approach by applying it to the boost DC-DC converter.