Analysis And Synthesis For A Kind Of T-S Fuzzy Descriptor Systems

Currently, although much effort has been made in the exploration of analysis and synthesis for the T-S fuzzy descriptor system, the results on this topic are far fewer than those on T-S fuzzy normal systems. The research on T-S fuzzy descriptor system is still in preliminary stage. Many unsolved problems need to do. In this paper, by using the advanced matrix theory and the linear matrix inequality (LMI) technique, the problem of analysis and synthesis for a kind of T-S fuzzy descriptor system is studied under the framework of Lyapunov stability theory, which is motivated by the linear descriptor system theory and advanced robust control theory. The main contributions are as follows:(1) The admissibility conditions of continuous T-S fuzzy descriptor systems are discussed. Based on a fuzzy Lyapunov function, the sufficient conditions of the admissibility for T-S fuzzy descriptor systems are obtained. The derivative of membership function is dealt with the form of fuzzy sum. Furthermore, the obtained admissibility conditions are relaxed in accordance with the fact that the sum of derivative of membership equals zero. It should be noted that all the conditions for the theories obtained can be formulated into LMI. The solutions of non-strict LMI are discussed by means of the YALMIP Toolbox.(2) The admissibility conditions and robust admissibility conditions of discrete T-S fuzzy descriptor systems are studied. Some new admissibility conditions of the open-loop systems without uncertainties are obtained by investigating its dual system. By introducing an assistant matrix variable, further investigations are made to design the robust controller with the form of strict LMIs, which make the closed-loop systems are admissible for all uncertain parameters.(3) The admissibility conditions and H∞control problems of continuous T-S descriptor fuzzy system are discussed. The original system can be generalized as the augmented system, and then the admissibility conditions are obtained based on a new fuzzy Lyapunov function and some controllers. Furthermore, the admissibility of the subsystems is not required to get the admissibility conditions for the open-loop system; on the other hand, the multiplied term of input control matrix by control gain matrix is not required to get the admissibility conditions for the closed-loop system, which lessens the effect on the solution of control gain from the original system. By virtue of the property of membership function, the H∞control conditions are generalized, and then such conditions in the form of strict LMI are also obtained. (4) The H∞control problem of discrete T-S fuzzy descriptor system is investigated. By introducing some assistant matrix variables, new admissibility conditions are obtained. Due to indefiniteness of matrix P, Schur complement cannot be applied to solve the nonlinear Lyapunov inequality. The difficulty is overcome by the approach proposed in this paper. Furthermore, the H∞control conditions in the form of strict inequality are discussed. By using the fuzzy Lyapunov function, the feedback state controller, static output feedback controller and dynamic output feedback controller are investigated. The results gained in this paper can be generalized in the case of systems with norm-bounded uncertainties.(5) The delay-dependent stability condition of T-S fuzzy descriptor systems with time-varying delay is discussed. By defining a new Lyapunov function and the term consisted of state in the integral multiplied by the derivative of state is dealt with the Moon inequality. Delay-dependent stability criterion is investigated. With the effective application of weights, a less conservative fuzzy controller design method is given. The relaxed design method includes the interactions of the different subsystems into one matrix. The explicit expression for the desired stabilizing controller is given by using the LMIs and the cone complementarity linearization iterative algorithm.The stabilization problem for T-S fuzzy descriptor systems with uncertainty and time-delay are investigated. The conditions for the closed-loop systems to be asymptotically stable are proposed in the cases of state feedback controller and static output feedback controller. Departing from traditional approaches, which aim to find a common invertible matrix for all rules, sufficient conditions to guarantee the robust stability for such fuzzy systems are derived in term of matrix measures of system matrices. These conditions are further transformed into LMI. By solving LMI, a state feedback controller and static output feedback controller that can stabilize the considered uncertain time-delay system can easily be obtained.(6) The problem of reliable control for T-S fuzzy descriptor systems with time-delay is investigated. A more practical model of actuator failures than outage is considered. Based on strict LMI, a method which designs a less conservative reliable controller is proposed. The resultant control system is admissible when all control components are operational. In fact, these properties of the proposed control systems still hold even when some control components experience failures. Moreover, theses results extend the observer-based reliable fuzzy controller, which is further designed with the help of separate principle.(7) The derivative feedback control problem of T-S fuzzy descriptor system is discussed. By using the rank describing the matrix expression, the conditions that the system can be normalized and regularized based on the proportional and derivative state feedback (PDSF) controller and proportional and derivative output feedback (PDOF) controller are investigated, respectively. However, it needs further attempt to find a more practical criterion. For the continuous T-S fuzzy descriptor system, the admissibility conditions are derived in the form of matrix inequality.