Analysis And Synthesis Of Delayed IT2 Systems Based On The Bessel-Legendre Inequality

In terms of type-1 T-S model,complex nonlinear systems can be approximately represented as a weighted sum of local linear subsystems.However,when the parameter uncertainties exist in the system,the type-2 fuzzy models have better performance than the type-1 ones.As a special case of the type 2 fuzzy set,the interval type 2 fuzzy set not only maintains the advantages of the type 2 fuzzy set,but also reduces computational burdens.Based on the T-S model,this paper studies the problems of stability analysis and controller design for three classes of delayed interval type-2 fuzzy systems.The research content of this paper mainly includes:Part I: The problems of stability analysis and combined memory and memoryless state feedback controller design for the interval type-2 fuzzy systems with constant delay are investigated.Time delay exists in both the state and the control input of the system.By constructing an augmented Lyapunov-Krasovskii functional with triple-integral terms and using the freematrix-based integral inequality and Jensen double integral inequality as well as Finsler's lemma,a state feedback controller including memoryless state feedback and delayed state feedback is given in the form of linear matrix inequality.The effectiveness of the proposed method is verified via a simulation example.Part II: The problems of stability analysis and controller design for interval type-2 fuzzy systems with time-varying delays are studied.By the features of the Bessel-Legendre integral inequality,we construct an augmented Lyapunov-Krasovskii functional with triple-integral terms.By using the Lyapunov-Krasovskii functional method and the second-order BesselLegendre integral inequality as well as the reciprocally convex combination,a tight bound of the derivative for the Lyapunov-Krasovskii functional is obtained.Thus,a less conservative asymptotic stability condition of the delayed interval type-2 fuzzy system is established.Based on the obtained stability condition and Finsler's lemma,a state feedback controller for the delayed interval type-2 fuzzy system is developed.Three numerical examples are given to illustrate the feasibility of the proposed method.Part III: The problem of extended dissipative controller design for the interval type-2fuzzy systems with time-varying delay is investigated.By constructing an augmented Lyapunov-Krasovskii functional with triple-integral terms and employing the second-order Bessel-Legendre integral inequality and the reciprocally convex combination as well as the extended Jensen double integral inequality,a sufficient asymptotic stability condition for the delayed interval type-2 fuzzy systems with extended dissipative performance is established.By matrix variable transformation,a sufficient condition for the existence of extended dissipativecontroller is given in the form of linear matrix inequality.The effectiveness of the derived method is verified by a simulation example.