Stability Analysis And Controller Design For T-S Fuzzy Time-delay Systems With Matched Conditions

It is well known that time delay exists widely in various engineering fields.The existence of time delay is not only one of the important factors leading to the instability of the system,but also affects the performance of the system.At present,many complex systems involved in engineering are nonlinear,so the research on the stability of nonlinear time-delay systems is particularly important,which is one of the hot issues of modern control theory.In particular,we have obtained an effective method to study complex nonlinear time-delay systems by combining the practical experience of experts with the theoretical knowledge of linear systems,whcih has been widely concerned by scholars both at home and abroad,after the Takagi-Sugeno(T-S)fuzzy model which can approximate a nonlinear system with arbitrary precision.With the linear matrix inequality(LMI)theory,we can transform the study of the stability of complex nonlinear time-delay systems into solving a series of linear matrix inequalities,which further promotes the development of T-S fuzzy time-delay systems.Therefore,the study on the stability of T-S fuzzy time-delay systems is not only of great theoretical significance,but also of great practical significance.Based on this background,the stability and controller design of fuzzy time-delay systems are studied in this paper.At present,from the relevant literature research,we can conclude that the mainstream method still uses the time domain analysis method:It is based on the T-S fuzzy model,establishes appropriate Lyaunov-Krasovskii functionals(LKF),and obtains the derivative of LKF.By scaling integral inequality and matrix inequality,convert it into the form of linear matrix inequalities,using the method of linear matrix inequalities we obtain sufficient conditions for T-S fuzzy time-delay systems stability and controller design criteria.In the existing literatures,there are many researches on the stability of T-S fuzzy time-delay systems based on state feedback,and they have achieved great results in reducing conservatism.However,the literatures on static output feedback(SOF)and dynamic output feedback(DOF)are limited,and there are no very good ways for reducing conservatism.At the same time,there is no systematic method that can both reduce the conservatism of state feedback and output feedback.Therefore,this paper focuses on three types of problems: One is the stability and controller design problem for T-S fuzzy time-delay systems with state feedback or static output feedback;Two is the stability and controller design problem for feedback memory T-S fuzzy time-delay systems with state feedback or static output feedback;Three is the problem of dynamic output feedback stabilization and dynamic output feedback H_∞ controller design for T-S fuzzy systems with time-varying delay.In this paper,the adjustment matrix L is introduced in the theorem proving process,and the stability condition is adjusted by setting some parameters in the L matrix,so as to reduce the conservatism.For the state feedback problem,we adopt the method based on lemma 9 in the process of transforming into the form of linear matrix inequalities.For the output feedback problem,we introduce the adjustment parameter β in the process of matrix inequality transformation,and we can obtain a conservative lower stability condition through the proper selection of the value for β.Finally,for the research of stability for T-S fuzzy systems with time-delay,this paper draws the relevant theorem results through examples and compared with the results of existing literature.The simulation of some complicated instances is given,and the state response curves of the closed-loop system and the open-loop system are also depicted.For study on dynamic output feedback stability of T-S fuzzy time-varying delay systems,this article has carried on the simulation through specific cases and presents the simulation results of the chart.Moreover,the state response curves of the closed loop system and the open loop system are plotted.By comparing the results of the case studies and the simulation charts,we draw the following conclusions:(1)The time domain analysis method,which is based on T-S fuzzy model,use Lyapunov stability criterion,the contraction and transformation of inequalities,and eventually transformed into a series of LMIs for solving,is a very effective method to solve complex nonlinear system stability and controller design problems.(2)In the process of stability analysis,introducing adjustment matrix L and setting the appropriate parameter value in L can reduce conservatism in a way.While,this method is also very versatile.(3)F or time-delay T-S systems with state feedback,the method based on lemma 9 can not noly simplify the process of obtaining stability conditions of LMIs form,but also reduce conservatism.(4)When we study the problem of output feedback(including static and dynamic),introducing a proper parameter β can effectively reduce conservatism.