Safety Controller Synthesis for Incrementally Stable Switched Systems Using Multiscale Symbolic Models

We propose an approach to the synthesis of safety controllers for a class of switched systems, based on the use of multiscale symbolic models that describe transitions of various durations and whose sets of states are given by a sequence of embedded lattices approximating the state-space, the finer lattices being accessible only by transitions of shorter duration. We prove that these multiscale symbolic models are approximately bisimilar to the original switched system provided it enjoys an incremental stability property attested by the existence of a common Lyapunov function or of multiple Lyapunov functions with a minimal dwell-time. Then, for specifications given by a safety automaton, we present a controller synthesis algorithm that exploits the specificities of multiscale symbolic models. We formalize the notion of maximal lazy safety controller which gives priority to transitions of longer durations; the shorter transitions and thus the finer scales of the symbolic model are effectively explored only when safety cannot be ensured at the coarser level and fast switching is needed. We propose a synthesis algorithm where symbolic models can be computed on the fly, this allows us to keep the number of symbolic states as low as possible. We provide computational evidence that shows drastic improvements of the complexity of controller synthesis using multiscale symbolic models instead of uniform ones.