Stability Analysis Of Interval Type-2 Fuzzy System With Time Delays Based On Linear Matrix Inequality Approach

In the past fifty years,fuzzy theory has been applied by many scholars and researchers into different fields,such as computer science and communication networks,industrial control systems,signal processing,digital communications,diagnostic medicine,decision-making and financial investment for its advantages in word modeling and dealing with uncertainty.Interval type-2 fuzzy sets are equivalent to the collection of an infinite number of type-1 fuzzy sets.Massive literatures have proved that interval type-2 fuzzy systems have better ability in dealing with uncertainty than their corresponding type-1 fuzzy systems.Fuzzy TSK models are used to represent nonlinear systems,and the mathematical models of the closedloop interval type-2 fuzzy TSK system are given in this paper.When the system encounters time delays,external disturbances,stochastic disturbances,uncertain parameters and event-triggered mechanism,the stability conditions of interval type2 fuzzy TSK systems are analyzed according to various actual environment,using appropriate methods,then corresponding interval type-2 fuzzy TSK controllers are designed.Based on Lyapunov-Krasovskii stability method and LMI technique,the stability problem of the closed-loop interval type-2 fuzzy TSK system with time delays is studied.Some theoretical results are obtained,and numerical examples have been carried out to show the effectiveness of the proposed methods.In order to effectively reduce the conservatism of the proposed methods,Jensen's inequality,Wirtinger's inequality,delay fractioning technique,and convex combination technique are used to simplify the inequalities.The main contents are listed as follows:(1)The problem of stabilization control for the interval type-2 fuzzy TSK system with time-varying delays is studied.By selecting the appropriate LyapunovKrasovskii functionals,applying Jensen's inequality and convex combination technique to reduce the conservatism,the sufficient conditions for the asymptotically stable of the interval type-2 fuzzy TSK system with bounded timevarying delays are obtained.(2)The design problem of the H? controller for the interval type-2 fuzzy TSK system with time-delay is discussed.By dividing the delay into N equal parts,the Lyapunov-Krasovskii functionals based on the variable N is constructed,then sufficient conditions for the asymptotically stable of the interval type-2 fuzzy TSK system that satisfy the requirement of H? control with the external disturbance are given.The stability condition based on the variable N is obtained in the form of LMIs.By increasing the partition of the delay,the conservatism of the system could reduce to some extend largely,while the upper bound of the delay increases.(3)The design problem of the non-fragile controller for the interval type-2 fuzzy TSK system with time delays is analyzed.By selecting the appropriate Lyapunov-Krasovskii functionals and using the Wirtinger's inequality to reduce the conservatism,the stability conditions of the conventional and nonfragile controllers for the interval type-2 fuzzy TSK system are obtained respectively.(4)Aiming at a class of nonlinear systems with time delays and stochastic disturbances,the observer-based interval type-2 fuzzy TSK control system is designed,then asymptotically stable conditions in the mean-square sense of the interval type-2 fuzzy TSK stochastic system are derived.The stochastic behavior in the system may influence the signal transmission and the result of ineffectiveness of the state variable measurement,so the observer of type-1 fuzzy TSK system is introduced to improve the reliability of the state variable measurement.Besides,by using the decoupling method,the observer gains and feedback gains can be designed and solved simultaneously.(5)A dissipative control problem for nonlinear networked control systems(NCSs)with time delays is introduced.For reducing the network bandwidth stresses,the closed-loop interval type-2 fuzzy TSK system based on event-triggered scheme is designed.Dissipativity is taken into account in the system,by selecting the appropriate Lyapunov-Krasovskii functionals,then the closedloop interval type-2 fuzzy TSK system satisfies sufficient conditions that the system is asymptotically stable and strictly(Q,S,R)-?-dissipative.The last not the least,some unsolved problems and development directions for the Interval type-2 fuzzy theory and the practical application are given.