Finite-step alternating approximately bi-similar symbolic model for Markov jump systems

Abstract

Establishing a bi-similar abstraction (symbolic model) of control systems has been demonstrated as a viable approach for implementing temporal logic tasks on control systems. Previous research primarily focused on logic tasks over an infinite time horizon. However, practical scenarios often involve control tasks defined within finite time intervals. Hence, this paper addresses the finite-time bi-similar problem specifically for a significant class of stochastic switched systems known as Markov jump systems (MJSs). To achieve this, two novel concepts are introduced: finite-time stochastic incremental stability (FSIS) and finite-step alternating approximate bi-simulation relation. A sufficient condition is firstly established for FSIS of MJSs by leveraging the incremental weak infinitesimal operator. Subsequently, a symbolic model is developed for MJSs by employing gridding techniques. Furthermore, a sufficient condition is derived to ensure that the developed symbolic model exhibits finite-step alternating approximate bi-similarity with the original MJSs. Finally, two numerical examples illustrate the application of the proposed results.