Scalable Synthesis of Safety Barrier Certificates for Networks of Stochastic Switched Systems

In this paper, we propose a compositional scheme for the safety controller synthesis of stochastic switched networks with dwell-time conditions. The proposed framework is based on a notion of so-called transition sub-barrier certificates constructed for each switched subsystem, by employing which one can compositionally synthesize safety controllers for interconnected networks over (in)finite time horizons. In our proposed scheme, we leverage dissipativity-type compositional conditions to compositionally construct transition barrier certificates for interconnected networks based on their corresponding sub-barrier certificates of individual subsystems. We show that the provided compositionality conditions can utilize the structure of the interconnection topology and be potentially satisfied independently of the number or gains of subsystems. We then utilize the constructed transition barrier certificates and quantify upper bounds on the probability that the interconnected network reaches certain unsafe regions in (in)finite time horizons. For nonlinear stochastic systems with polynomial dynamics, we employ sum-of-squares (SOS) optimization programs to search for stochastic storage certificates of each switching mode with its independent supply rate . We then focus on a particular class of nonlinear stochastic switched systems whose nonlinearities satisfy some linear-growth restriction and propose a constructive approach to search for storage certificates of each mode via satisfying some matrix inequalities. We demonstrate our proposed results by applying them to a fully-interconnected network of 400 nonlinear switched subsystems (totally 800 dimensions) accepting multiple supply rates and multiple storage certificates with dwell-time conditions.