Robust Stability For Discrete-time Uncertain Singular Markov Jump Systems

In practice, there are many dynamical systems with jump in their structures and parameters because of abrupt random changes phenomenon, such as breakdowns of components, changes in the interconnections of subsystems, abrupt changes disturbance of outside, environment condition changes, abrupt change moves of nonlinear systems working points after linearization, the different impaction of the environment on systems in different control stages and so on. These abrupt changes can make stochastic fluctuations of systems in different stages. Researchers find that this kind of random change rule usually follows the change rule of Markov process through a lot of researches. The system matrices of the linear Markov jump systems jump randomly in a series of discrete moments while keeping linear between jump. Linear Markov jump systems are the extension of linear system models and many properties of linear Markov jump systems are similar to general linear systems, so they become the extensive jumping models researched by researchers in the current international.The phenomenon of time-delay is extremely common for the factors of the observation, the signal transmission and the insensitivity of the physical devices in all kinds of industrial systems. In addition, every system has constraints in their nature. In special, actuator saturation is the most common phenomena in real world. However, most of the controller design methods in the modern control theory often assume that the control input of the system is infinite and ignores the presence of actuator saturation. At present, some research results about typical control systems with actuators saturation such as singular systems, switched systems, and large-scale decentralized systems, etc, are very little, especially articles about time-delay are even less. This dissertation will dedicate our efforts on robust stability for discrete-time uncertain singular Markov jump systems.Firstly, the state space model of discrete-time random uncertain singular Markov jump systems is established, then the sufficient and necessary condition of regular, causal and stochastically stable about the nominal system of discrete-time random uncertain singular Markov jump systems is given and the detailed proof of theorem is given. Further, the robust stability for discrete-time uncertain singular Markov jump systems with actuator saturation and time delay is researched. A sufficient condition for the systems to be regular, causal and stochastically stable is given. In order to apply the matlab LMI toolbox, the condition is changed to be the form of linear matrix inequalities. The design problem of robust state feedback controller is solved based on the method of linear matrix inequality under the condition and full knowledge of transition probabilities and the state feedback controller is achieved by solving the optimization problem of the corresponding linear matrix inequalities. Finally, a numerical example is presented to verify the research.