New Results on Stability of Singular Stochastic Markov Jump Systems with State-Dependent Noise

TL;DR

This paper develops a stability theory for singular stochastic Markov jump systems with state-dependent noise, introducing new concepts and providing conditions for system stability in both continuous and discrete time.

Contribution

It introduces new concepts like non-impulsiveness and mean square admissibility, and extends stability conditions using $ ext{H}$-representation and pseudo-inverse techniques.

Findings

Established sufficient conditions for mean square admissibility.

Provided practical examples demonstrating effectiveness.

Extended deterministic and normal stochastic system results.

Abstract

This paper aims to develop the stability theory for singular stochastic Markov jump systems with state-dependent noise, including both continuous- and discrete-time cases. The sufficient conditions for the existence and uniqueness of a solution to the system equation are provided. Some new and fundamental concepts such as non-impulsiveness and mean square admissibility are introduced, which are different from those of other existing works. By making use of the $\mathcal{H}$-representation technique and the pseudo inverse $E^+$ of a singular matrix $E$, sufficient conditions ensuring the system to be mean square admissible are established in terms of strict linear matrix inequalities, which can be regarded as extensions of the corresponding results of deterministic singular systems and normal stochastic systems. Practical examples are given to demonstrate the effectiveness of the proposed approaches.