Stability Analysis And Synthesis For Takagi-Sugeno Fuzzy Systems With Time-Delays

With development of science, technology and productivity, the controlled plants are becoming more and more complex. For lack of accurate mathematical model, and there exist various nonlinear sections, uncertainties and time delay in practical systems, it is important and significant to investigate the stability synthesis and control problem of nonlinear time delay systems. However, there is no efficient method to solve this class of problems. As we know T-S fuzzy model can approximate nonlinear system significantly, and it is easy for us to analyze. T-S fuzzy control method has been the hotspots in the investigation of nonlinear systems. In this paper, the stability analyse and control problems of nonlinear time delay systems have been investigated based on T-S fuzzy model.First, a class of continuous T-S fuzzy uncertain systems with time delays are studied. Based on Lyapunov-Krasovskii(L-K) approach, the sufficient stability conditions are proposed, and then fuzzy controller is designed via parallel distribute compensation technique to stabilize the plant. The closed-loop system is asymptotically stable with the guaranteed cost performance. A piecewise L-K approach is considered to study a class of discrete-time T-S fuzzy systems with delays. Based on the model transformation, the system stability is analyzed. The designed state observer and controller can make the closed-loop system stable with satisfying performance.Second, discrete-time T-S fuzzy system with single delay and one with multiple delays are considered, respectively. They are transformed to different uncertain systems with time-delays. The systems stability is analyzed. The output feedback controller are presented based on the PDC method and Piecewise L-K method. The generalized output feedback controller with dual index rules is designed to satisfy the high requirement in performance and control precision. Accordingly, the PDC method is extended to generalized PDC method.Third, this paper investigates the delay-dependent stability of the T-S fuzzy system with time-delays. One important lemma is advanced. By constructing aspecial L-K functional, we can proposed sufficient conditions of delay-dependent stability with less conservatism. The state feedback control methods are respectively used to stabilize the systems so that the closed-loop systems is delay-dependent stable with satisfying performance. The estimating upper bound of time-delay can also be obtained. For any time-delay less than this upper bound, the fuzzy systems are stable or stabilizable. The results demonstrate that decreasing the conservatism of delay-dependent stability conditions depends on the robust performance of fuzzy time-delay systems to some extent.Next, a class of time-delay systems with convex polytopic uncertainties are presented in accordance with the global form of T-S fuzzy systems. The stability analysis method is introduced based on polyhedral L-K functional. By increasing free variables, we structure a class of analyzing matrix dependent affinely on system parameters and these of L-K functional. The stability conditions are given based on single polyhedral L-K functional and double polyhedral L-K functional, respectively. The comparison study shows that the presented stability conditions own larger stability region and less conservatism than those of quadratic stability and simultaneous stability. Moreover, the double polyhedral L-K functional is more effective than the single polyhedral one.In the end, we investigate the application of T-S fuzzy system in chaotic systems. For the stabilization at the fixed points of chaotic systems, one uniformed controller design method is proposed for the time-delayed feedback control of chaotic systems without time-delay, for the memoryless state feedback control of time-delay chaotic systems and for the time-delayed feedback control of time-delay chaotic systems. The method not only illuminates a new approach for chaos control, but also provides a new idea for time-delayed feedback control of chaotic systems.