Study On Some Control Problems Of T-S Fuzzy Systems

With the developments of the science, we have more requirements for control sys-tems and the control systems are more complex. The controlled object or process often is nonlinear, subject to some random disturbances and uncertainties, so that its mathe-matical model, which can satisfy our demands, is not available. For the control system, the conventional control theory is not applicable. However, the fuzzy control techniques with a property of human language can overcome the difficulties. In particular, stability analysis and control synthesis of nonlinear systems described by T-S fuzzy models appeal many researchers’interest and many important progresses have been achieved. Due to the inherent nonlinearities of T-S fuzzy systems, it is still a very important task to exploit more effective conditions for stability analysis and control synthesis.Based on the previous work of other researchers, this thesis further studies some control problems of T-S fuzzy systems, which are given as follows:By using the prop-erties of fuzzy membership functions, a new LMI technique, which can introduce more LMI variables than the existing ones, is proposed. Further, by the new technique, the conditions with less conservatism are given for stability analysis and control synthesis for continuous-time T-S fuzzy systems; By using the properties of the null space of sys-tem output matrix, a new LMI-based condition for designing H^ static output feedback controllers for discrete-time T-S fuzzy systems is given. In contrast to the existing ap-proaches, the new one removes the constraint on the structure of Lyapunov matrices, then it can gives less conservative results; A new control scheme for T-S fuzzy systems, i.e., switching PDC control scheme, is proposed for the first time. The corresponding LMI-based control design approaches are given. The proposed approaches can be reduced to the PDC controller design methods or switching linear controller design methods under some additional constraints, then they can give less conservative results; A new type of T-S fuzzy models, which consist of local nonlinear subsystems, is constructed for a class of nonlinear systems with strong nonlinearities, and the corresponding conditions for design-ing controllers are given. The proposed approach can overcome the difficulty of designing controllers based on the conventional T-S fuzzy models due to too many fuzzy rules in the model. It can reduce the computational burden and give less conservative results; A controller design method for singularly perturbed fuzzy systems, which can improve the upper bound of the singular perturbation parameter by designing controllers, is given. The proposed method overcomes the difficulty that the existing approaches cannot improve the upper bound of the singular perturbation parameter by designing controllers. Moreover, a new condition for designing mode-independent controllers for Markovian jump fuzzy systems is given. More slack variables are introduced in the proposed condition, then the proposed method is more flexible in the freedom and can given less conservative results than the existing ones.The considered main issues are given as follows:In Chapter 1, the background and the development of fuzzy control are summarized and analyzed.In Chapter 2, a new LMI technique, which can introduce more variables, is ex-ploited, then an LMI-based method with a linear search for designing fuzzy controllers for continuous-time T-S fuzzy systems is obtained. In contrast to the existing approaches, the new one is more flexible in the freedom due to more slack variables, therefore, it can give less conservative results. Numerical examples are further given for showing the effectiveness of the proposed approach. Moreover, by using the properties of the null space of output matrices, a parameter-dependent slack variable with lower-triangular structure is introduced, then a new method for designing static output feedback controllers is obtained. In contrast to the existing approaches, the proposed method can remove the constraint on Lyapunov matrices and give less conservative results. Numerical examples validate the effectiveness of the proposed approachIn Chapter 3. by analyzing the properties of fuzzy systems, a new control scheme, i.e., switching PDC control scheme, is proposed for the first time. LMI-based condi-tions for designing switching PDC controllers are given respectively for continuous and discrete-time T-S fuzzy systems. The existing PDC controller design methods or switch-ing linear controller design methods can be considered as a special case of the new control scheme. Therefore, the proposed method can give less conservative results than the ex-isting ones. Numerical examples are given to show the effectiveness of the proposed approaches.In Chapter 4, fuzzy controller design problems of a class of nonlinear systems with complex nonlinearities are considered. The T-S fuzzy models of the class of nonlinear systems are of very many fuzzy rules, so that the methods for designing controller have heavy computational burden or cannot get a feasible solution. Moreover, the obtained controllers are not convenient for implementing in engineering. In order to overcome the difficulty, a new type of T-S fuzzy models with local nonlinear subsystems are proposed and the corresponding control synthesis techniques are given. In contrast to the existing approaches, the proposed one can obtain a fuzzy controller with less fuzzy rules and has less computational burden. In particular, some important properties of nonlinear systems can be reserved due to the use of the nonlinear subsystems, then it also can give less conservative results. The effectiveness of the proposed method is validated by numerical examples.In Chapter 5, the problems of the controller design, and the estimation of the up-per bound of singular perturbation parameter of singularly perturbed fuzzy systems are considered. A controller design method with improving the upper bound of the singular perturbation parameter is given for the first time for continuous-time singularly perturbed fuzzy systems. Moreover, two LMI-based conditions for designing Hx controllers only by slow state information for discrete-time singularly perturbed fuzzy systems are pro-posed, where one of them can improve the upper of singular perturbation parameter by designing controllers. Both of them remove the constraint on Riccati equation or inequal-ity approaches, where the systems have to satisfy the regularity. The effectiveness of the proposed methods are validated by numerical examples.In Chapter 6, based on stochastic Lyapunov functions and by introducing slack vari-ables for separating the system matrix and Lyapunov matrix, an LMI-based condition for designing mode-independent controllers is given for Markovian jump fuzzy systems. In contrast to the existing approach, the proposed method is of more LMI variables, then can give less conservative results. Numerical examples further show the effectiveness of the proposed method.Finally, the results of the dissertation are summarized and further research topics are pointed out.