From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems

This article is concerned with a compositional approach for the construction of control barrier certificates large-scale interconnected stochastic systems while synthesizing hybrid controllers against high-level logic properties. Our proposed methodology involves decomposition into smaller subsystems and leverages notion control xmlns:xlink="http://www.w3.org/1999/xlink">sub-barrier certificates subsystems, enabling one to construct by employing some $\max$ -type small-gain conditions. The main goal synthesize enforcing complex properties, including ones represented accepting language deterministic finite automata, providing probabilistic guarantees on satisfaction given specifications in bounded-time horizons. To do so, we propose systematic first decompose simple reachability tasks utilizing automata corresponding complement specifications. We then sub-barrier local those simpler combine them obtain controller that ensures specification lower bound probability satisfaction. finally apply our techniques fully-interconnected Kuramoto network composed 100 nonlinear oscillators.