Compositional Design of Safety Controllers for Large-scale Stochastic Hybrid Systems

Abstract: In this work, we propose a compositional scheme based on small-gain reasoning for the safety controller synthesis of interconnected stochastic hybrid systems with both continuous evolutions and instantaneous jumps. In our proposed setting, we first offer an augmented scheme to represent each stochastic hybrid subsystem with continuous and discrete evolutions in a unified framework, ensuring that the state trajectories match those of the original hybrid systems. We then introduce the concept of augmented control sub-barrier certificates (A-CSBC) for each subsystem, which allows the construction of augmented control barrier certificates (A-CBC) for interconnected systems and their safety controllers under small-gain compositional conditions. We eventually leverage the constructed A-CBC and quantify a guaranteed probabilistic bound across the safety of the interconnected system. While the computational complexity of designing a barrier certificate and its safety controller grows polynomially with network dimension using sum-of-squares (SOS) optimization program, our compositional approach significantly reduces it to a linear scale with respect to the number of subsystems. We verify the efficacy of our proposed approach over an interconnected stochastic hybrid system composed of $1000$ nonlinear subsystems.