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

Abstract

This article is concerned with a compositional approach for the construction of control barrier certificates for large-scale interconnected stochastic systems while synthesizing hybrid controllers against high-level logic properties. Our proposed methodology involves decomposition of interconnected systems into smaller subsystems and leverages the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">control</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-barrier certificates</i> of subsystems, enabling one to construct control barrier certificates of interconnected systems by employing some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\max$</tex-math></inline-formula> -type small-gain conditions. The main goal is to synthesize hybrid controllers enforcing complex logic properties, including the ones represented by the accepting language of deterministic finite automata, while providing probabilistic guarantees on the satisfaction of given specifications in bounded-time horizons. To do so, we propose a systematic approach to first decompose high-level specifications into simple reachability tasks by utilizing automata corresponding to the complement of specifications. We then construct control sub-barrier certificates and synthesize local controllers for those simpler tasks and combine them to obtain a hybrid controller that ensures satisfaction of the complex specification with some lower bound on the probability of satisfaction. We finally apply our proposed techniques to a fully-interconnected Kuramoto network composed of 100 nonlinear oscillators.