Stability Analysis And Stabilization For T-S Fuzzy Systems With Time-delay Under Imperfect Premise Matching

Since the Takagi-Sugeno (T-S) fuzzy model was firstly proposed by Japanesescholars Takagi and Sugeno, much more extensive research areas for the study offuzzy control theory have been provided. It has been theoretically demonstrated thatT-S fuzzy model can be approximated any nonlinear systems on any degree ofaccuracy since the proposed of the principle of the universal approximation.Therefore, it is effectively to analysis and synthesis some performance of nonlinearsystem based on T-S fuzzy control method, and can be obtained some better controlresults. With the rapid development of industrial technology and wide application ofcomputer technology in the actual engineering system, more and more highlynonlinear and time lag phenomenon appear in the control systems. However, theexistence of time-delay makes the systems have a special difficult in control theoryand engineering practice, and it is the major source of leading the actual systems topoor performance, even leading the systems to remain in a unstable condition.Therefore, the stability issue for time-delay systems is very important in bothcontrol theory and practical application. However, it is very difficult to model andcontrol for nonlinear systems directly. As the T-S fuzzy model can be describednonlinear systems effectively, the stability issue for T-S fuzzy time-delay systemshas drawn a great deal of research attention both in the theory and engineering area.Under National Natural Science Foundation of China (60874084), the T-S fuzzymodel with time-delay under imperfect premise matching is proposed in this paper,in which the fuzzy time-delay model and fuzzy controller share differentmembership functions. Some less conservative stability and robust stabilityconditions are obtained for the systems. Meanwhile, a new controller design method,which different from the traditional Parallel Distributed Compensation (PDC)controller design method is proposed. The new design method makes up theshortage of the PDC design method, and enhances design flexibility.The delay-independent stability and controller design issues for continuous anddiscrete T-S fuzzy time-delay under imperfect premise matching are investigated,respectively. As the imperfect premise matching proposed, the fuzzy controller canchoose different membership functions from those of the fuzzy model. Therefore,different from the existing analysis method, we consider the information of the twomembership functions in the analysis, and present the relationship of them. So someless conservativeness delay-independent stability conditions are obtained.Meanwhile, based on the stability conditions, the controller design method underimperfect premise matching is also obtained, which breaks the former fuzzy controller design constrains and makes the selection of membership functions forthe fuzzy controller much more freeness. Then controller design flexibility can beenhanced. Especially, when the structure of the membership functions for the fuzzytime-delay model are very complexity or contain some uncertainties, the proposeddesign method can reduce the difficulty, and avoid that the fuzzy controller can notbe designed, and retain the robustness of fuzzy controller. Finally, some numericalexamples are given to illustrate the effective and advantage of the proposedmethods.The delay-dependent stability and stabilization issues for T-S fuzzy time-delaysystems are investigated. Firstly, a new Lyapunov function is introduced to analysisthe stability of T-S fuzzy system with constant time delay. With the integralinequality and free-weighting matrix methods, some less conservative stabilityconditions are obtained. Furthermore, a new controller design method is alsoproposed under imperfect premise matching. Moreover, the above fuzzy model weconsidered is extended to T-S fuzzy systems with interval time-varying delay, a newLyapunov function with the upper and lower bound of time-delay is proposed, andsome new free-weighting matrix method is to be considered instead of using integralinequality. There is no integral area reduction and contain all the useful informationof the systems in the analysis. Compare with the existing results, much larger upperof delay can be got, which implies that the robust stability condition is lessconservative. Finally, some numerical examples are further to illustrate theeffectiveness and advantage of the proposed methods.For the stability and stabilization issues of uncertain T-S fuzzy systems withconstant time delay under imperfect premise matching, a new form of theaugmented Lyapunov-Krasovskii function, in which contains a triple-integral term isfirstly introduced. Two integral inequalities and a parameterized modeltransformation method with less free-weighting matrices are used to derive adelay-dependent sufficient robust stability conditions. Compare with the existingliteratures, the Lyapunov function we introduced is more generalized than theexisting one. It can provide a more relaxed constraint conditions for the informationof each systems, which lead to more stability area for the system. Meanwhile, thereare less free matrixes variables than the existing results because of introducing theintegral inequality, which will reduce the computational demand. Therefore, therobust stability conditions are less conservative, simple form and less decisionvariable, which implies that our method is more advanced. Meanwhile, a new robuststabilization approach is also proposed under imperfect premise matching based onthe robust stability conditions, which can greatly improve design flexibility andlower the implementation cost of the fuzzy controller. Furthermore, we put theabove analysis method to analyzing the robust stability problem of the system with time-varying delay. Some less conservative delay-dependent robust stabilityconditions are also obtained. Therefore, the triple-integral term plays essentialfunction for reducing the conservative stability conditions. Finally, numericalexamples show the advantage and effectiveness of the proposed method.The robust stability and stabilization issues for T-S fuzzy systems withtime-varying state and input delay under imperfect premise matching are to beinvestigated. Here, the state delay and input delay are all time-varying delay, andthey are not equal. Different from the foremost analysis method, a new Lyapunovfunction with an additional triple-integral term and contains the information of inputdelay and state delay is introduced, and some information of both membershipfunction of fuzzy model and fuzzy controller are also considered. Therefore, a newless conservativeness robust stability conditions are obtained. Meanwhile, a solvemethod for the feedback control gains is also achieved. Finally, numerical examplesare given to illustrate the less conservativeness of the proposed results, and theeffectiveness and flexibility of the proposed design method.