Stability and exact observability of discrete stochastic singular systems based on generalised Lyapunov equations

The stability and exact observability problems of discrete stochastic singular systems based on generalised Lyapunov equations are investigated in this study. First, some new conditions for the existence and uniqueness of the solutions to discrete stochastic singular systems are given, and then, a new type generalised Lyapunov equation is proposed. Based on these results, a sufficient condition for stochastic stability of the systems is derived via the solution of the proposed generalised Lyapunov equation. Moreover, the relationship between the solution of the generalised Lyapunov equation and structure properties of discrete stochastic singular systems is discussed as well. Especially, the relationship between the solution of the generalised Lyapunov equation and the stochastic stability of discrete stochastic singular systems with non-causal behaviour is studied for the first time. Second, another new type generalised Lyapunov equation for exactly observable discrete stochastic singular systems is proposed. Meanwhile, the relationships among the stochastic stability of the systems, exact observability and the solution of the proposed generalised Lyapunov equation are investigated. Finally, simulation examples are worked out to validate the effectiveness of the obtained theoretical results.