Control Of Nonlinear Systems Based On T-S Fuzzy Hyperbolic Model And T-S Fuzzy PDE Model

As is well known, fuzzy logic system utilizing fuzzy sets and fuzzy inference toprocess uncertain information is an effective way of processing complex nonlinearsystems. So far, the resulting fuzzy control is an important method for studying thecontrol design of nonlinear systems. In particular, the fuzzy control technique based onthe so-called Takagi-Sugeno (T-S) fuzzy model has been widely employed for thecontrol design of nonlinear systems, since the classical linear system theory can beapplied to the analysis and controller synthesis of nonlinear systems. As a commonbelief, this technique is conceptually simple and effective for controlling complexnonlinear systems. Recently, some novel T-S models were proposed, such as, T-S fuzzybilinear model, T-S fuzzy nonlinear model, T-S fuzzy hyperbolic partial differentialequation (PDE) model and so on.According to the Lyapunov stability theorem, parallel distribution compensation(PDC) method, robust control theory, H control theory and non-fragile guaranteedcost control theory, combining linear matrix inequality (LMI) techniques, this thesisconcerns the stabilization of nonlinear ordinary differential equation (ODE) systems andnonlinear PDE systems based on T-S fuzzy hyperbolic model and T-S fuzzy hyperbolicPDE model, respectively.The main research works can be summarized as follows:Firstly, a T-S fuzzy hyperbolic model is proposed for the stabilization of a class ofnonlinear ODE systems. The consequence of the proposed model is a hyperbolictangent dynamic model, and it is employed to represent the nonlinear system. Byconstructing a new Lyapunov function and according to the Lyapunov stability theorem,the stability conditions of the open-loop system are derived via linear matrixinequalities (LMIs). Then, the PDC method is used to design a fuzzy hyperboliccontroller, and the asymptotic stability conditions of the closed-loop system areformulated via LMIs. Comparing with the control approach based on T-S fuzzy model,our design approach can achieve much smaller control amplitude in the case of the statestabilization time is almost the same, and it can be referred to as “soft” constraintcontrol approach. Finally, the effectiveness and advantage of the proposed schemes areillustrated by a mathematical constructive example and the Van de Vusse example.Secondly, the “soft” constraint control approach based on T-S fuzzy hyperbolicmodel is employed to the robust Hfuzzy control of nonlinear ODE systems. According to the Lyapunov theorem theory, the robust H fuzzy controller designprocedure is developed in terms of LMIs. Comparing with the H fuzzy controlapproach based on T-S fuzzy model, our proposed approach can achieve much smallercontrol amplitude.Thirdly, the “soft” constraint control approach based on T-S fuzzy hyperbolicmodel is used to the non-fragile guaranteed cost control of nonlinear ODE systems. Anuncertain T-S fuzzy hyperbolic model is employed to represent the nonlinear system,and the non-fragile controller is designed by PDC method, and sufficient conditions areformulated via LMIs such that the system is asymptotically stable and the cost functionsatisfies an upper bound. In contrast to the non-fragile guaranteed cost control approachbased on T-S fuzzy model, our proposed method can achieve much smaller controlamplitude. Finally, the Van de Vusse example is given to illustrate the effectiveness andadvantage of the proposed approach.Fourthly, the non-fragile guaranteed cost control for nonlinear first-order hyperbolicpartial differential equations (PDEs) is concerned. An uncertain T-S fuzzy hyperbolicPDE model is presented to exactly represent the nonlinear hyperbolic PDE system.Then, the state-feedback non-fragile controller distributed in space is designed by PDCmethod, and some sufficient conditions are derived in terms of spatial differential linearmatrix inequalities (SDLMIs) such that the T-S fuzzy hyperbolic PDE system isasymptotically stable and the cost function keeps an upper bound. Compared with theexisting approaches, our proposed approach can tolerate larger controller gain variations.Finally, a nonlinear hyperbolic PDE system is presented to illustrate the effectivenessand advantage of the developed design methodology.Finally, some conclusions and future research works are given.