Stability Analysis Of T-S Fuzzy Models

Fuzzy control has been becoming one of the hot topics in control theory in recent years and has been successfully used in many domains such as engineering and household appli-ances.As one of the several models approximating nonlinear systems,Takagi-Sugeno(T-S)fuzzy models have attracted considerable attention.T-S fuzzy models are constructed by blending several linear subsystems with the membership functions.Such a good construc-tion makes it possible to apply the mature linear control theory to analyze nonlinear systems.Therefore,the studies of T-S fuzzy models,especially of the stability of T-S fuzzy models,are very meaningful in both theory and practice.On the basis of the existing works,the dissertation further investigates the stability of the T-S fuzzy models and achieves some meaningful results by improving some of the existing methods.The contributions of this dissertation are summarized as follows:1.The problem of the stability of discrete-time T-S fuzzy systems is investigated by ap-plying the Kronecker method.In these days,much effort has been devoted into the improvement of the Lyapunov matrix in the Lyapunov function while the state vector is always unchanged.According to this situation,this dissertation presents a new Lya-punov function,i.e.,homogeneous polynomial parameter-dependent and Kronecker-product(HPPD-KP)Lyapunov function.Based on the HPPD-KP Lyapunov function,more relaxed conditions are obtained,Moreover,the larger the Kronecker power,the more relaxed the obtained conditions.2.The problem of the stability of discrete-time T-S fuzzy models is further investigated by applying the double-homogeneous-polynomial method.The HPPD-KP Lyapunov function usually brings a lot of variables as it is shown in the Kronecker form.According to this situation,the double homogeneous polynomial parameter-dependent(DHPPD)Lyapunov function is constructed,which includes almost all of the existing ones.Based on the DHPPD Lyapunov function,more relaxed conditions are obtained by applying the complete square matricial representation(CSMR)of the homogeneous polynomials.3.The problem of the stability of continuous-time T-S fuzzy models is investigated by ap-plying the linear-integral method.The merit of the linear-integral Lyapunov function is that its time derivative does not contain time derivatives of the membership functions.By fully considering the structural information of T-S fuzzy systems,this dissertation presents the q-dimensional linear-integral Lyapunov function by extending the original one.Based on the q-dimensional linear-integral Lyapunov function,more relaxed condi-tions are obtained for both continuous-time T-S fuzzy models and continuous-time T-S fuzzy time-delay models.4.The membership functions contain much important information about the T-S fuzzy models.Therefore,based on the quadratic and non-quadratic Lyapunov functions,this dissertation further investigates the stability and stabilization of the discrete-time T-S fuzzy models by applying the in.formation of the order relationship among the member-ship functions.In order to further reduce the conservatism,a new method of constructing the transformation matrix is proposed.5.The free-weighting matrix technique and the method of inequalities are two widely-used methods to study the stability of T-S fuzzy time-delay models.As the discrete coun-terpart of the free-matrix-based integral inequality,a new summation inequality,i.e.,the free-matrix-based summation inequality,is presented in this dissertation,which in-cludes the Jensen summation inequality and the Wirtinger-based summation inequality.By constructing appropriate Lyapunov-Krasovskii functionals and applying the free-matrix-based summation inequality,this dissertation investigates the stability of both linear time-delay models and T-S fuzzy time-delay models.As a result,more relaxed conditions are obtained.