Control Synthesis For Continuous-time Takagi-Sugeno Fuzzy System Based On Non-quadratic Lyapunov Functions

In recent years, the Takagi-Sugeno(T-S) fuzzy systems were studied mainly by using the quadratic Lyapunov functions. However, the quadratic Lyapunov function contains only a positive definite matrix, and in most cases, a single positive definite matrix can not satisfy all of the subsystems, which leads to much conservativeness.Based on the predecessor’s foundation, in this paper, the problem of local stabilization,fuzzy filter design and fuzzy observer design for continuous-time T-S fuzzy system are studied by using the non-quadratic Lyapunov function. The content are as follows:1.The local stabilization for continuous-time T-S fuzzy system is studied. First of all, the shortcomings of the existing results are pointed out by an example and a new method is proposed to overcome the disadvantage. Then, the non-quadratic Lyapunov function are introduced and the time derivatives of the membership function are discussed in detail to obtain the new local stabilization conditions. Further, larger stabilization regions are obtained by using convex optimization constraint conditions that are expressed linear matrix inequalities. Finally, the acquired results are applied to the simulation example to demonstrate the effectiveness of the method.2.The filtering design with immeasurable premise variables for continuous-time T-S fuzzy system is studied. First of all, the time derivatives of the membership function are expressed as linear matrix inequality by using the Schur complement lemma. Then,the non-quadratic Lyapunov function, which has diagonal form, is designed and the function contains the same number of rules as the positive definite matrix has. Further,the error system is asymptotically stable by using the Lyapunov stability theory. Finally,the simulation examples are given to verify the effectiveness of the method.3.The observer design for continuous-time T-S fuzzy system is studied. First of all,the new bound conditions of the time derivatives of the membership function are obtained by using convex optimization and can be applied to any the fuzzy systems no matter whether the rules are even or odd. The new conditions overcome the disadvantages of the existing conditions that can only applied to an even number of fuzzy rules of the system. Then, considering the disturbance bounded by energy, the observer is designed and two bounded domains are obtained, which ensures that the trajectory starting from one region will remain in another region. Further, the H?observer and the controller are designed to guarantee the system have stronger ability to resist the external disturbances. Finally, the simulation examples are given to verify the effectiveness of the method.