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

TL;DR

This paper develops a compositional method for constructing control barrier certificates for large stochastic systems, enabling the synthesis of hybrid controllers that satisfy complex high-level specifications with probabilistic guarantees.

Contribution

It introduces a systematic approach combining small-gain conditions, automata-based specification decomposition, and SOS/CEGIS methods for scalable control synthesis in stochastic systems.

Findings

Successfully applied to two physical case studies.

Achieved probabilistic satisfaction guarantees for complex specifications.

Demonstrated scalability through compositional decomposition.

Abstract

This paper 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 control sub-barrier certificates of subsystems, enabling one to construct control barrier certificates of interconnected systems by employing some max-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. To compute control sub-barrier certificates and corresponding local controllers, we provide two systematic approaches based on sum-of-squares (SOS) optimization program and counter-example guided inductive synthesis (CEGIS) framework. We finally apply our proposed techniques to two physical case studies.