Fault-tolerant control for delayed interval type-2 fuzzy systems with nonlinear fault input

This paper addresses the stabilization of Takagi-Sugeno (T-S) fuzzy systems with time delay. In particular, the control system with parameter uncertainties are modeled through an interval type-2 (IT2) T-S fuzzy model, in which the uncertainties are handled via lower and upper membership functions. By developing some new techniques, a fault-tolerant controller is designed to ensure that the closed-loop fuzzy time-delay system is asymptotically stable with nonlinear fault input. Finally, a numerical example that demonstrate the effectiveness of the proposed conditions are provided.


Introduction
Over the past few decades many Takagi-Sugeno (T-S) fuzzy model [1,2] has became an growing interest in fuzzy control of nonlinear systems. A number of stability analysis and control design for T-S fuzzy model based on the Lyapunov function method has been reported in literature [3].
To mention a few, a stability criteria for fuzzy system with time-varying delays has been derived in [4,5]. In recent years, the stability analysis and control or filtering problems for systems with time delays or missing measurements have become an active research area.
Besides, the type-2 fuzzy set is a extension of type-1 fuzzy set [6]. In [7][8][9], it is pointed out that the type-2 fuzzy systems have the potential to provide better performance than type-1 fuzzy systems. The authors in [10,11] used IT2 membership functions to capture the nonlinear plants and design state feedback controllers for the IT2 T-S fuzzy systems. It should be pointed out that the controller design results are obtained for continuous-time IT2 T-S fuzzy systems and there lack some modeling, stability analysis, and control design results for discrete-time IT2 T-S fuzzy systems. Furthermore, the stability conditions for IT2 T-S fuzzy control systems are derived by state feedback approach in [12,13].
Nowadays, time delay is an important source of instability and poor performance for a control system, and is frequently encountered in many practical control systems. Less attention has been paid on IT2 T-S fuzzy systems with constant time delays because they can be transformed into fuzzy time delay systems via state augmentation approach in [14,15]. Recently, [16] addressed the stability analysis and controller design for interval type-2 fuzzy systems with time delay.
On the other hand, due to the growing demands of system reliability in practical systems, the study of reliable control which can guarantee the system stability [17,18]. It should be mentioned 2 that reliable control results under the assumption that the actuator fault is not always a single multiplicative fault, sometimes it couples with a certain nonlinearity, for example, the dead zone or relay is a classic actuator fault [19,20]. Obviously we cannot express these feature as a multiplicative fault. Therefore, for those complicated and practical fault cases, we need to have a further study. Moreover, to the best of our knowledge, the nonlinear fault based problem for IT2 fuzzy system has not been investigated yet in the existing literature which motivates our present study. The main contributions of this paper can be highlighted as follows: (i) This is the first attempt to investigate the fault-tolerant control design together with nonlinear for IT2 time-delay fuzzy systems. (ii) By the implementation of Lyapunov stability theory together with fuzzy Lyapunov functional approach, a set of sufficient conditions is proposed in terms of LMIs to assure the asymptotical stability of the IT2 fuzzy time-delay system.

Problem Formulation and Preliminaries
A nonlinear plant with time-delay and nonlinear actuator fault subjected to parameter uncertainties is considered and spoken to by the following IT2 fuzzy model with lower and upper bound membership functions.
whereM iα is an IT2 fuzzy set of ith rule corresponding to the function ζ α (x(t)), α = 1, 2, · · · , p, i = 1, 2, · · · , r, p is a positive integer, x(t) ∈ R n is the system state vector, u F (t) ∈ R m is the control input vector with fault, h(t) is the time-varying delay and satisfies h(t) ∈ (0, h],ḣ(t) ≤ µ, h and µ are known positive numbers, φ(t) is the initial condition. A i ∈ R n×n , A di ∈ R n×n and B i ∈ R n×m are known input matrices, respectively. Further, ∆A i (t) denotes the parametric uncertainties satisfying ∆A i (t) = E i ∆ i (t)F i , where E i and F i are known constant matrices with appropriate dimensions, and ∆ i (t) is an unknown time-varying matrix, which is Lebesque measurable in t and satisfies ∆ T i (t)∆ i (t) ≤ I. The firing strength of the i th rule is the following interval sets: where θ i (x(t)) = p α=1 µM iα (ζ α (x(t))) ≥ 0, θ i (x(t)) = p α=1 µM iα (ζ α (x(t))) ≥ 0 in which µM ia (ζ α (x(t))) and µM iα (ζ α (x(t))) denote the lower and upper membership functions respectively satisfying the property µM ia (ζ α (x(t))) ≥ µM ia (ζ α (x(t))) ≥ 0, and θ i (x(t)) and θ i (x(t)) indicate the lower and upper grades of membership respectively. The inferred IT2 fuzzy model can be defined as follows: whereθ i (x(t)) denotes the grades of membership of the embedded membership functions and with r i=1θ i (x(t)) = 1 in which α i (x(t)) ∈ [0, 1], α i (x(t)) ∈ [0, 1] are nonlinear functions with the property that α i (x(t)) + α i (x(t)) = 1. In practical situation, some actuators of the control system often work in an abnormal status aroused from an integrated factor, such as the linear gain missing, the nonlinearity of dead zone and so on. The fault control input for the actuator deviates the true value in a complicated way including linear and nonlinear part. Here, we consider the nonlinear actuator fault model as where 0 < E 1 = diag{e 1 , e 2 , ..., e m } ≤ I and the vector function g(u(t)) = [g 1 (u(t)), g 2 (u(t)), ..., g m (u(t))] T which satisfies |g i (u(t))| ≤ √ k i |u i (t)|, i ∈ I = {i|i = 1, 2, ..., m}. Further, the inequality (5) can be rewritten as where E 2 = diag{α 1 , α 2 , ..., α m }. An IT2 fuzzy controller with l rules of the following format is proposed to stabilize the nonlinear plant spoke to by the IT2 T-S fuzzy model (3).
We need to visit a fundamental lemma to be used in the following proof.
, and a vector functioṅ x : [−h, 0) → R n such that the integration in the following inequality is well defined, then it holds that where

Main Results
In this section, we are in a position to discuss the stability of the IT2 fuzzy system with timevarying delay and nonlinear actuator fault. If we set ∆A i (t) = 0 in (11), then we obtain the resulting nominal fuzzy system aṡ Specifically, we need to introduce suitable Lyapunov-Krasovskii functionals to establish new sufficient conditions that can be expressed in terms of LMIs for stability of system (12).
Theorem 3.1 For given positive scalars µ, h and known actuator fault matrix E 1 , IT2 fuzzy system (12) is asymptotically stable, if there exist symmetric matrices P > 0, Q > 0, R > 0, S > 0 and N ij > 0 of appropriate dimensions such that the following LMIs are satisfied for i = 1, . . . , r, j = 1, . . . , l: and other parameter positions are zero. Moreover, if the above conditions are satisfied, then the reliable feedback controller gain matrices are given by K j = Y j X −1 .
Proof: Consider the candidate of Lyapunov-Krasovskii functional (LKF) for fuzzy system (12) in the following form Taking the time derivative of V (x(t)) with respect to t along the trajectory of fuzzy system (12), Applying Lemma 2.1 in (17), the integral part ofV (x(t)) can be expressed as Then, it follows from inequality (6) that and k is any positive scalar. Applying Schur complement to (19) and combine with inequality (17), we can obtain the inequality as follows:V . By using Remark 3 in [12], the above inequality can be obtain thaṫ with N ij ≥ 0. The stability condition for the closed-loop system (12) can be written as Define X = P −1 ,Q = XQX,R = XRX,Ŝ = XSX, −XR −1 X ≤ R − 2P , Y j = K j X and then pre-and post-multiplying the above matrix U ij by diag{X, X, X, I, I} andN ij =XN ijX ,X = diag{X, X, X}, the resulting inequality (20) is equivalent to (15). Thus, the closed-loop fuzzy system (12) is asymptotically stable, which completes the proof. Now, we further extend the result in Theorem 3.1 to discuss reliable control design for uncertain closed-loop fuzzy system (11).
Theorem 3.2 For given positive scalars µ, h and known actuator fault matrix E 1 , the uncertain IT2 fuzzy system (11) is asymptotically stable, if there exist symmetric matrices P > 0, Q > 0, R > 0, S > 0 and N ij > 0 of appropriate dimensions and positive scalars i such that the following LMIs are satisfied for i = 1, . . . , r, j = 1, . . . , l together with (13) holds:  (14) and (15), then by following the similar steps of Theorem 3.1, we can easily get By Lemma 2.6 [20], a sufficient condition guaranteeing (23) for system (11) is that there exists a positive number > 0 such that By the virtue of Schur complement Lemma in [20], it can be concluded that (24) is equivalent to LMIs (21) and (22) which completes the proof of this theorem. Based on the method proposed in (5) with E 1 = I (fault is absence) and g(u(t)) = 0 (nonlinear is absence), the following corollaries can be easily obtained. Corollary 3.3 For given positive scalars µ and h, IT2 fuzzy system (12) is asymptotically stable, if there exist symmetric matrices P > 0, Q > 0, R > 0, S > 0 and N ij > 0 of appropriate dimensions such that the following LMIs are satisfied for i = 1, . . . , r, j = 1, . . . , l together with (13) holds: for all i, j, k, i 1 , i 2 , . . . , i n and the parameters are defined as in Theorem 3.1. Moreover, the feedback controller gain matrices are given by K j = Y j X −1 .
Proof: Substituting the actuator fault matrix E 1 = I and nonlinear g(u(t)) = 0 in (12), then by following the similar steps of Theorem 3.1. Therefore, it omits here. Corollary 3.4 For given positive scalars µ and h, the uncertain IT2 fuzzy system (11) is asymptotically stable, if there exist symmetric matrices P > 0, Q > 0, R > 0, S > 0 and N ij > 0 of appropriate dimensions and positive scalars i such that the following LMIs are satisfied for i = 1, . . . , r, j = 1, . . . , l together with (13) holds: hF i X 0] T and the other parameters are defined as in Theorem 3.1. Moreover, the desired robust feedback controller gain matrices are given by K j = Y j X −1 .
Proof: Substituting the actuator fault matrix E 1 = I and nonlinear g(u(t)) = 0 in (11), then by following the similar steps of Theorem 3.2. Therefore, it is omitted here.

Numerical Example
In this section, we give a numerical results with simulation for the closed-loop systems for both, the Type-1 and the interval Type-2 fuzzy time delay systems with nonlinear actuator fault.
Consider a three-rule IT2 fuzzy model with time-varying delay and nonlinear actuator fault as followṡ where 22 ≤ a ≤ 30 and 20 ≤ b ≤ 25. In addition, we choose

cos(t) are uncertainties and time delay
h(t) = 0.1(1 + 9 sin 2 t). Moreover, the nonlinear assume that g(u(t)) = E 2 u(t) sin(u(t)). On stability of interval Type-2 fuzzy systems compare with Type-1 fuzzy systems. There are two simulation tasks performed for fuzzy system (29).

Conclusion
This paper focuses on the challenges of theoretical underpinning including notation, learning type-2 sets and systems. The problem of state-feedback controller design for a class of IT2 fuzzy time delay system with nonlinear actuator fault has been investigated. By using the Lyapunov-Krasovskii functional approach and LMIs techniques, a sufficient condition is derived such that the closed loop IT2 T-S fuzzy systems asymptotically stable under the nonlinear actuator fault. From the given example, one can see that the IT2 T-S fuzzy system with nonlinear actuator failure can still maintain a certain control performance by using the designed fault-tolerant controller. Another objective of this paper is to derive a novel uncertainty measure for IT2 membership functions with clearer presentation. In the future, the practical application and standard measures of type-2 fuzzy sets will be an in-depth study in order to solve more and more practical problems.