Compositional Construction of Control Barrier Certificates for Large-Scale Stochastic Switched Systems

In this paper, we propose a compositional framework for the construction of control barrier certificates for large-scale stochastic switched systems accepting multiple control barrier certificates with some dwell-time conditions. The proposed scheme is based on a notion of so-called augmented pseudo-barrier certificates computed for each switched subsystem, using which one can compositionally synthesize state-feedback controllers for interconnected systems enforcing safety specifications over a finite-time horizon. In particular, we first leverage sufficient max-type small-gain conditions to compositionally construct augmented control barrier certificates for interconnected systems based on the corresponding augmented pseudo-barrier certificates of subsystems. Then we quantify upper bounds on exit probabilities

• the probability that an interconnected system reaches certain unsafe regions
• in a finite-time horizon using the constructed augmented barrier certificates. We employ a technique based on a counter-example guided inductive synthesis (CEGIS) approach to search for control barrier certificates of each mode while synthesizing safety controllers providing switching signals. We demonstrate our proposed results by applying them first to a room temperature network containing 1000 rooms. Finally, we apply our techniques to a network of 500 switched subsystems (totally 1000 dimensions) accepting multiple barrier certificates with a dwell-time condition, and provide upper bounds on the probability that the interconnected system reaches some unsafe region in a finite-time horizon.