Symbolic Models for Retarded Jump-Diffusion Systems

In this paper, we provide for the first time an automated, correct-by-construction, controller synthesis scheme for a class of infinite dimensional stochastic hybrid systems, namely, hybrid stochastic retarded systems. First, we construct finite dimensional abstractions approximately bisimilar to original infinite dimensional stochastic systems having some stability property, namely, incremental input-to-state stability. Second, we construct finite abstractions approximately bisimilar to constructed finite dimensional abstractions. Both types of abstractions are derived without any state-space discretization. By using the so-called transitivity property of approximate bisimulation relations, we establish that the constructed finite abstractions are also approximately bisimilar to original infinite dimensional stochastic hybrid systems with a precision that can be chosen a-priori. Given those finite abstractions, one can synthesize controllers for infinite dimensional systems satisfying some high-level logic requirements in a systematic way. Moreover, we provide sufficient conditions for the proposed notion of incremental stability in terms of the existence of so-called incremental Lyapunov functions which reduce to some linear matrix inequalities (LMI) in the linear case. Finally, the effectiveness of the results are illustrated by synthesizing a controller regulating the temperatures in a ten-room building modeled as a hybrid stochastic delayed system.