Construction of approximations of stochastic control systems: A compositional approach

In this paper, we provide a compositional framework for the construction of infinite approximations of interconnected stochastic control systems. Our approach is based on a notion of so-called stochastic simulation functions that are associated with interfaces. The stochastic simulation functions are used to quantify the approximation error while the interfaces are used to lift the controllers synthesized for the approximation to the controllers for the original stochastic system. In the first part of the paper, we analyze interconnected stochastic control systems which consist of several stochastic control subsystems. We derive sufficient conditions that facilitate the compositional construction of stochastic simulation functions together with the associated interfaces. Specifically, we show how to construct a stochastic simulation function with the corresponding interface for the interconnected stochastic control system from the simulation functions and interfaces of the individual stochastic control subsystems. In the second part of the paper, we focus on linear stochastic control systems. We extend a methodology, which is known for the non-probabilistic case, to construct infinite approximations of linear stochastic control systems together with their stochastic simulation functions and the corresponding interfaces. Finally, we illustrate the effectiveness of the proposed results on the interconnection of four linear stochastic control subsystems.