Compositional Synthesis of Control Barrier Certificates for Networks of Stochastic Systems against $\omega$-Regular Specifications

TL;DR

This paper presents a compositional method for synthesizing control barrier certificates for interconnected stochastic systems to ensure $$-regular specifications with probabilistic guarantees, using automata-based decomposition and SOS optimization.

Contribution

It introduces a systematic, compositional framework leveraging automata and SOS techniques to synthesize control policies for complex stochastic systems against $$-regular specifications.

Findings

Effective control barrier certificates synthesized for interconnected systems.

Probabilistic guarantees achieved for $$-regular specifications.

Validated approach on a physical case study.

Abstract

This paper is concerned with a compositional scheme for the construction of control barrier certificates for interconnected discrete-time stochastic systems. The main objective is to synthesize switching control policies against $\omega$-regular properties that can be described by accepting languages of deterministic Streett automata (DSA) along with providing probabilistic guarantees for the satisfaction of such specifications. The proposed framework leverages the interconnection topology and a notion of so-called \emph{control sub-barrier certificates} of subsystems, which are used to compositionally construct control barrier certificates of interconnected systems by imposing some dissipativity-type compositionality conditions. We propose a systematic approach to decompose high-level $\omega$-regular specifications into simpler tasks by utilizing the automata corresponding to the specifications. In addition, we formulate an alternating direction method of multipliers (ADMM) optimization problem in order to obtain suitable control sub-barrier certificates of subsystems while satisfying compositionality conditions. For systems with polynomial dynamics, we provide a sum-of-squares (SOS) optimization problem for the computation of control sub-barrier certificates and local control policies of subsystems. Finally, we demonstrate the effectiveness of our proposed approaches by applying them to a physical case study.