Studies On Stability And Stabilization Of Nonlinear Systems Under Delayed Impulsive Effects

Impulsive control has wide applications in many disciplines including communication, biology and economics because of its various advantages such as lower cost, high reliability, as well as easy of realization. From the viewpoint of control theory, impulses are samples of system output at discrete moments. In the transmission of the impulse information, transmission delays are unavoidable due to the effects of network congestion and other factors. Large transmission delay can degrade the system performance and can even lead to instability. Therefore, it is of importance to analyze the effects of impulse delays on stability and stabilization design in a quantitative manner and ensure the robustness of system stability towards small impulse delays. The emphasis of this thesis is focused on the delayed impulsive synchronization of nonlinear time-delay Lur’e system, the delayed impulsive control of T-S fuzzy delay system and the stability problem of nonlinear singularly perturbed systems with delayed impulses. Some new methods for stability analysis and stabilization design for systems with the delayed impulsive effects are proposed. The main work is as follows:(1) The synchronization problem of nonlinear time-delay Lur’e system with delayed impulse is studied. The transmission delays in the system output signal are considered. By introducing a time-varying Lyapunov function associated with the impulsive time sequence, the stability property of synchronization error system is analyzed by using Razhumikhin method combined with the technique of convex combination. A new global time-delay output feedback impulsive synchronization scheme is presented. The new synchronization scheme is robust with respect to small transmission delays. Applying the synchronization scheme to the impulsive synchronization problem of two identical delayed Chua’s circuits, the output feedback control law of impulsive synchronization is derived. Moreover, an upper bound of transmission delay for global synchronization is obtained.(2) The problem of delayed impulsive stabilization of T-S fuzzy delay systems is considered. By introducing T-S fuzzy rules, a class of nonlinear delay systems is represented as T-S fuzzy delay systems. A fuzzy delayed impulsive control scheme is proposed. By means of time-varying Lyapunov function and the analysis technique based on Razhumikhin-Lyapunov function, new stabilization criteria based on fuzzy impulsive control laws are established. The new criteria reveal the effects of the length of the impulsive intervals and the size of impulsive delays on the design of the fuzzy impulsive control law. In comparison with the existing results, the upper bounds of impulsive interval that ensure the system to be impulsively stabilizable are significantly improved. Furthermore, the problem of finding impulsive controller gain matrices is converted to the solvability problem of a set of linear matrix inequalities (LMIs). The allowable upper bounds of impulse delays can be achieved by solving the LMIs.(3) The exponential stability problem for nonlinear singularly perturbed systems with delayed impulses is investigated. By using a two-time scale decomposition technique, the full-order system is decomposed into a slow subsystem with delayed impulses and a fast subsystem without impulse. By means of the Lyapunov functions of the two lower order subsystems, a vector Lyapunov function for the full-order system is built. By applying the proposed impulsive vector differential inequality technique, exponential stability criteria for sufficiently small singular perturbation parameter is presented. A stability bound on perturbation parameter is also derived. The derived criteria depend on the size of the impulse delays. When the impulse delays are absent, the stability criteria cover the existing results. Applying the obtained results to the DC-motor system with unmodeled dynamics and impulse effect, new stability conditions are derived.