Symbolic control of stochastic systems via approximately bisimilar finite abstractions

TL;DR

This paper extends symbolic control methods to stochastic systems by constructing finite abstractions that are approximately bisimilar, enabling automated controller synthesis for complex probabilistic models with temporal logic specifications.

Contribution

It introduces a method to create finite-state models for stochastic control systems satisfying incremental stability, facilitating correct-by-construction controller synthesis.

Findings

Finite-state models can be approximately bisimilar to stochastic systems.

Controllers for stochastic systems can be synthesized using linear temporal logic.

The approach is effective for safety-critical cyber-physical systems.

Abstract

Symbolic approaches to the control design over complex systems employ the construction of finite-state models that are related to the original control systems, then use techniques from finite-state synthesis to compute controllers satisfying specifications given in a temporal logic, and finally translate the synthesized schemes back as controllers for the concrete complex systems. Such approaches have been successfully developed and implemented for the synthesis of controllers over non-probabilistic control systems. In this paper, we extend the technique to probabilistic control systems modeled by controlled stochastic differential equations. We show that for every stochastic control system satisfying a probabilistic variant of incremental input-to-state stability, and for every given precision $\varepsilon>0$, a finite-state transition system can be constructed, which is $\varepsilon$-approximately bisimilar (in the sense of moments) to the original stochastic control system. Moreover, we provide results relating stochastic control systems to their corresponding finite-state transition systems in terms of probabilistic bisimulation relations known in the literature. We demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic. The discussed technique enables a new, automated, correct-by-construction controller synthesis approach for stochastic control systems, which are common mathematical models employed in many safety critical systems subject to structured uncertainty and are thus relevant for cyber-physical applications.