Stability And Robust Stabilization For Nonlinear Uncertain Discrete-Time Descriptor Markov Jump Systems

In this paper, the stability,stabilization,robust stability and robust state feed-back stabilization problems for a class of nonlinear uncertain discrete-time de-scriptor Markov jump systems are investigated. The nonlinear function satisfies a quadratic constraint,uncertainties are bounded parameters uncertainties.The following results are obtained:(1) Stability for nonlinear discrete-time descriptor Markov jump system is investigated. First, in the case of full knowledge transition probabilities, based on Lyapunov theory and the existence theorem of hidden function, linear matrix inequality (LMI) sufficient condition is developed which guarantees that systems are regular, causal, have unique solution in neighborhood of the origin, and are stochastically stable. With this condition, based on singular value decomposi-tion approach and some matrix inequalities, another LMI condition is developed which guarantees that the systems are regular, causal, have unique solution in neighborhood of the origin, and are stochastically stable. Then,stability condi-tion for the systems with partial knowledge transition probabilities is given. The numerical examples verify the effectiveness and accuracy of the methods.(2) State feedback stabilization for nonlinear discrete-time descriptor Markov jump systems is studied. Based on the stability obtained conditions, the state feedback stabilization controllers are obtained via LMIS for full knowledge tran-sition probabilities and partial knowledge transition probabilities. The numerical examples verify the effectiveness and accuracy of the methods.(3) Robust stability and robust state feedback stabilization for nonlinear uncertain discrete-time descriptor Markov jump systems is discussed. For full knowledge transition probabilities and partial knowledge transition probabilities, sufficient conditions of robust stability and robust state feedback stabilization are obtained via LMIS respectively. The numerical examples verify the effectiveness and accuracy of the method.