Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach

Abstract This paper addresses the problem of robust fault-tolerant control design scheme for a class of Takagi-Sugeno fuzzy systems subject to interval time-varying delay and external disturbances. First, by using improved delay partitioning approach, a novel n-steps iterative learning fault estimation observer under H ∞ constraint is constructed to achieve estimation of actuator fault. Then, based on the online estimation information, a fuzzy dynamic output feedback fault-tolerant controller considered interval time delay is designed to compensate for the impact of actuator faults, while guaranteing that the closed-loop system is asymptotically stable with the prescribed H ∞ performance. Moreover, all the obtained less conservative sufficient conditions for the existence of fault estimation observer and fault-tolerant controller are formulated in terms of linear matrix inequalities. Finally, the numerical examples and simulation results are presented to show the effectiveness and merits of the proposed methods.


Introduction
In real world, most physical systems are nonlinear and many researchers have been seeking the effective approaches to control nonlinear systems. Among these, there are growing interests in Takagi-Sugeno (T-S) fuzzy model based control [1,2]. It has been proved that T-S fuzzy models can be used to approximate a wider class of nonlinear system, which are realized by piecewise smoothly connecting a family of local linear models with fuzzy membership functions. This "blending" makes the subsystems of T-S fuzzy model similar to linear systems, and the fruitful results of linear system theories can be directly applied for the stability analysis and synthesis of nonlinear systems [3][4][5][6][7]. A great repercussion of T-S fuzzy models can be verified in many practical fuzzy model based control systems (see, for instance, [8][9][10][11][12][13][14] and references therein).
1. By using improved delay partitioning approach, a novel n-steps iterative learning fuzzy fault estimation observer under H 1 performance constraint is constructed to achieve the estimation of actuator faults, and the less conservative sufficient conditions for the existence of observer are explicitly provided. 2. A new type of fuzzy dynamic output feedback fault-tolerant controller considered interval time delay is designed to guarantee that the closed-loop system is asymptotically stable with the prescribed H 1 performance. 3. All obtained sufficient conditions for the existence of observer and controller are formulated in terms of strictly LMIs. Compared with the existing results, the proposed design schemes are with less conservative and wider application range, simulation examples demonstrate the effectiveness of the proposed approaches.
The rest of this paper is organized as follows. The system description and problem formulations are presented in Section 2. Section 3 presents the main results on robust fault estimation observer and fault-tolerant controller design scheme. In Section 4, simulation results of numerical example are presented to demonstrate the effectiveness and merits of the proposed methods. Finally, Section 5 concludes the paper. Notations: Throughout the paper, R n denotes the n-dimensional real Euclidean space; I denotes the identity matrix; the superscripts "T " and "-1" stand for the matrix transpose and inverse, respectively; notation X > 0.X 0/ means that matrix X is real symmetric positive definite (positive semi-definite); k k is the spectral norm. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. The symbol " " stands for matrix block induced by symmetry.

Problem formulation
Consider a nonlinear system which can be represented by the following extended T-S fuzzy time-delay model with exogenous disturbance and actuator faults simultaneously. Plant rule i : IF 1 .t / is M i1 and : : : x.t / D i .t /; 8t 2 OE 2 ; 0; i D 1; 2; : : : ; r: (1) where x.t / 2 R n is the state vector, u.t / 2 R q denotes the input vector, y.t / 2 R l stands for the system output vector. d.t / 2 R m is the exogenous disturbance input that belongs to L 2 OE0; 1/, f .t / 2 R q represents the possible actuator fault. A i , A i , B i , B d i , C i , C i and D d i are constant real matrices of appropriate dimensions. It is assumed that the pairs .A i ; B i / are controllable, and the pairs .A i ; C i / are observable, where i D 1; 2; : : : ; r.
where 1 and 2 are lower and upper bounds of state delay .t /, respectively. Through the use of fuzzy blending, the fuzzy system (1) can be inferred as follows: where Before proceeding further, we will introduce some lemmas to be needed in the development of main results throughout this paper.  29]). For any constant matrix X 2 R n n , X D X T > 0, scalar r > 0, and vector function P x W OE r; 0 ! R n such that the following integration is well defined, then For T-S nonlinear system description (3), we can see that a general system is considered in this paper, including possible state time delay, actuator fault and exogenous disturbance input simultaneously. If there is no state delay, then (3) reduces to the existing one in [30]. Moreover, the lower bound of delay is not restricted to 0 as [23], which is even more applicable to networked control systems and other practical systems.

Actuator fault estimation
In order to estimate system faults, the n-steps iterative learning fault estimation observer is constructed as follows: where O x n .t/ 2 R n is the n-steps observer state, O y n .t / 2 R l is the observer output, and O f n .t / 2 R q is the nth step estimate of fault f .t /, n D 1; 2; :::; N is the number of fault estimation steps. Then, the objective of estimating actuator fault by observer (4) is to design the appropriate dimension gain matrices L n .t / 2 R n l , F n .t / 2 R q l in the presence of disturbance and state time delay, where L n .t / D then the error dynamic systems is deduced from (3) and (4) as follows: For simplicity, we introduce the following vectors: where h D 1 =n.n D 1; 2; :::; N / is the length of each division, N is the number (a positive integer) of divisions of the interval OE 1 ; 0 and is also the number of fault estimation steps in (4). The delay interval OE 1 ; 2 is divided into two subintervals with an unequal width as OE 1 ; 1 C ı and OE 1 C ı; 2 , where ı D 2 1 , 0 < < 1. Then, the state of error dynamics (5) can be rewritten as P e n .t / D 1 .t / 1 .t /. Therefore, the H 1 fault estimation observer design problem to be addressed in this paper can be formulated as follows: (i) The error dynamic system (5) with ! n .t / D 0 is asymptotically stable for any time-delay satisfying (2) when n D 1; 2; :::; N ; (ii) For a given scalar n , the following H 1 performance is satisfied: k! n .t /k 2 dt; n D 1; 2; :::; N for all L > 0 and ! n .t / 2 L 2 OE0; 1/ under zero initial conditions. Theorem 3.1. For the given scalars 1 , 2 , Á, n and 0 < < 1, the error dynamic system (5) is asymptotically stable with ! n .t / D 0 while satisfying a prescribed H 1 performance (6), if there exist matrices P > 0, Q n > 0, W n > 0.n D 1; 2; ::: 5 0 and Y ni .i D 1; 2; : : : ; r/, such that the following inequalities hold … i i < 0 i D 1; 2; : : : ; r (7) and nn D Q n 1 W n 1 C Q n W n ; n D 2; 3; ::: Case 1: Then the observer gain matrices can be obtained as follows: Proof. The following novel Lyapunov-Krasovskii functional candidate is constructed to prove system (5) is asymptotically stable with H 1 performance. where where the unknown matrices P > 0, S 1 > 0, S 2 > 0, S 3 > 0, R 1 > 0, R 2 > 0, Q n > 0 and W n > 0.n D 1; 2; :::; N / are to be determined.
Then, the time derivatives of V .x t ; t / along the trajectories of the argument systems (5) satisfy e T n .s/Z 33 P e n .s/ds (15) Case 1. When 1 Ä .t / Ä 1 C ı, the following equations are true: Using Lemma 2.1 and the Leibniz-Newton formula, we have Similarly, we obtain Substituting (16)-(18) into (15), a straightforward computation gives When R 1 Y 33 0, R 2 Z 33 0, and 1 Ä .t / Ä 1 C ı, the last two terms in (19) are all less than 0. Then, for any scalar Á > 0, it follows from the fact PR 1 P Ä 2ÁP C Á 2 R and Schur complement theorem, we can see if the following inequalities hold By noticing V .L/ 0 and V .0/ D 0 under zero initial conditions, we can conclude that (6) holds for all L > 0 and any nonzero ! n .t / 2 L 2 OE0; 1/. Hence, with the changes of variables as Y n .t / D P N L n .t /, we have which imply that the error dynamics (5) satisfies the prescribed H 1 performance (6). In addition, by choosing the same Lyapunov function as (14) and following the similar line in the earlier deduction under conditions (7)-(9), we can easily obtain that the time derivative of V .x t ; t / along the solution of error dynamics (5) with ! n .t / D 0 satisfies P V .t / < 0, which indicates the asymptotic stability of systems (5).

s/ds
The proof can be completed in a similar formulation to Case 1 and is omitted here for simplification. This completes the proof of Theorem 3.1.
In order to compare our results with the existing ones, based on the improved delay-decomposing approach of Theorem 3.1, we suggest to develop a delay-dependent stability condition for the nominal unforced fuzzy system of (3), which can be written as Corollary 3.2. For the given scalars 1 , 2 , d and 0 < < 1, the system (21) with d.t / D 0 is asymptotically stable for any time-varying delay .t / satisfying (2), if there exist matrices P > 0, Q n > 0, W n > 0.n D 1; 2; :::; N /, 5 0 , such that the following set of inequalities hold: and other elements of the matrix ‚ i in Case 1 and 2 are defined in (10)-(13).

Remark 3.3.
To constrain the effect of input disturbance from P f .t /, different from the existing fault estimation observer design result in [14,21,26,27], [30,31], the information of f n 1 .t / is considered in the design scheme of fault estimation observer (4), in which P f n 1 .t / can increasingly weaken the effect intensity of input disturbance from P f .t / in the error dynamics. Therefore, when the delay partitioning procedure change from one to N , we can see that FE observer based on the N-steps time delay partitioning approach not only reduces the conservativeness of result, but also better depicts the size and shape of faults with the increase of steps number N . The simulation examples illustrate the effectiveness and merits of the design method.
Remark 3.5. Based on such variable decomposition method, the improved FE observer (4) may increase the maximum allowable upper bounds on 2 for fixed lower bound 1 while giving fault estimation, if one can set a suitable dividing point with relation to . For seeking an appropriate , an algorithm is given as follows: Step 1: For given d , choose upper bound on ı satisfying (7)- (9), select this upper bound as initial value ı 0 of ı.
Step 2: Set appropriate step lengths, ı st ep and st ep for ı and , respectively. Set k as a counter and choose k D 1. Meanwhile, let ı D ı 0 C ı st ep and the initial value 0 of equals step .
Step 5: Let k D k C 1, if k st ep < 1, then go to Step 3. otherwise, stop.

Fault tolerant controller design
On the basis of the obtained online fault estimation information, we design a fault tolerant controller considering interval time delay to guarantee stability in the presence of system faults. Since the state x.t / is unmeasurable, we use the fuzzy dynamical output feedback controller scheme [38] as follows: R q n , D c 2 R q l are the designed controller matrices, and A c .

t //, then one can obtain the closed-loop systems
For simplicity, we introduce the following vectors: Then, the closed-loop argument systems (23) can be rewritten as So far, the problem of robust dynamic output feedback control for the closed-loop fuzzy system is to design the gain matrices of (22) such that: (i) The closed-loop fuzzy system (23) with Q !.t / D 0 is asymptotically stable for any time-delay satisfying (2); (ii) For a given scalar n > 0, the following H 1 performance is satisfied: for all L > 0 and Q !.t / 2 L 2 OE0; 1/ under zero initial conditions. In what follows, a useful lemma is needed, which is given here for completeness of the next theorem. : : : : : : : : : : : : : : : : : : : : : where " ij D 2 6 6 6 4 : : : : : : : : : : : : : : : : : : : : : Then, the gain matrices of the dynamic output feedback fault tolerant controller are given by where M , N satisfy MN T D I X Y .
Proof. For any scalar Á > 0, by the Schur complement theorem and the fact .ÁR P /R 1 .ÁR P / that PR 1 P Ä 2ÁP C Á 2 R, we can conclude that (26) holds if the following inequality hold: Then, we partition the symmetric positive definite matrix P and its inverse matrix P 1 into components …" ƒ F T 1 ; F T 1 ; :::; F T 1 ; F T 1 ; F T 1 ; F T 1 ; I g, then, pre and post multiplying (29) by d i agf ‡ T 1 ; F T 1 ; F T 1 ; I g.
Remark 3.8. As in [30], from D F T 1 PF 1 > 0, we can obtain Y > 0 and X Y 1 < 0 which imply that I XY is nonsingular. Therefore, we can always find nonsingular matrices M and N satisfying MN T D I X Y , and they can be calculated by the QR function of Matlab toolbox. Remark 3.9. Note that (27)- (28) are LMIs. This indicates that it can be included as an optimization variable, which can be exploited to reduce the attenuation level bound. Then, the minimum attenuation level of H 1 performance can be obtained by solving a convex optimization problem P: min # subject to (27)- (28) with # D 2 . Different from the existing results, the main advantage of the proposed design method is the reduction of conservatism by presenting a delay-dependent result. Also, an interval time-varying delay has been considered in the design scheme.

Numerical example
In this section, three examples are provided to demonstrate the effectiveness of the proposed approaches. which can be exactly expressed as a nominal T-S delayed system with the following rules [4][5][6][7], [37]: The membership functions for above rules 1, 2 are 1 .
The purpose here is to find the allowable maximum time-delay value 2 under which the fuzzy system is stable.
Considering interval time-varying delay, the upper delay bounds 2 derived from [4][5][6][7], [37] and the method proposed in this paper are tabulated in Table 1 under different values of 1 . It is seen from Table 1 that the results obtained from Corollary 3.2 (d is unknown, set S 1 D 0) of this paper are significantly better than those obtained from the other methods. Example 4.2. Consider a two rule T-S fuzzy system borrowed from [4,7], [34][35][36]: To compare with the existing results, the improvement of this paper is shown in Table 2. It can be concluded that the obtained results in this paper are less conservative than those of [4,7], [34][35][36]. Moreover, if we assume timevarying delay .t/ satisfies (2), the delay-dependent fault estimation conditions proposed in [21][22][23][24][25], [27] fail to give a feasible solution. However, by using LMI toolbox in matlab, a feasible solution of Theorem 3.1 can be obtained for .t/ D 1 C 0:1si nt. 1 D 0:9; 2 D 1:1/ and other cases. In order to further illustrate the effectiveness of proposed approach, the problem of FTC for T-S fuzzy systems with interval time delay is considered in the next example. Example 4.3. We apply the above analysis technique to design robust fault estimation observer and dynamic output feedback fault-tolerant controller for a computer simulated truck-trailer system borrowed from [39]. The time delay model with actuator fault f .t / and disturbance d.t / is given by T-S fuzzy systems as follows  It is observed that, compared with CAFE, although there is initial estimation error, the n-steps estimation approach not only provides better rapidity of fault estimation but also achieves more accurate estimation of actuator fault by increasing steps N . Moreover, improved delay partitioning approach is employed to deal with interval time delay for increasing the maximum allowable delay bounds, while the approach in [21,23,27], [26] fail to give a feasible estimation. All of these make it meaningful for the approach to be implemented in practice.  Simulation results for the stability of the closed-loop systems and the systems output response are shown in Fig. 6 and Fig. 7-8. It can be seen that although the open-loop systems are unstable, the proposed design still achieves the performance under actuator faults, and the stability of closed-loop systems is guaranteed while satisfying the prescribed H 1 performance. As indicated by the simulation result graph, we can see that whether the interval time delay fuzzy systems are considered with constant fault or time-varying fault, the fuzzy n-steps fault estimation observer can almost realize accurate fault estimation, and the fuzzy dynamic output feedback control strategy can effective accommodate the effect of actuator faults on system performance.

Conclusion
In this paper, by using improved delay partitioning approach, a novel n-steps robust fault estimation observer has been constructed for a class of T-S fuzzy model with interval time-varying delay and external disturbances. Then, utilizing realtime information on estimated faults, a fuzzy dynamic output feedback fault-tolerant controller considering interval time delay is proposed to accommodate the effect of actuator faults while satisfying the prescribed H 1 performance. An advantage of the proposed approach is that with the increase of steps n, not only the approach can give a better performance of FE and FTC, but also the maximum allowable delay bounds of fuzzy systems is increased. Finally, some examples have clearly verified the effectiveness of the proposed method for FE and FTC. This paper focus on robust FTC for T-S fuzzy systems with actuator fault and does not consider sensor fault. The consideration of the system with actuator and sensor fault simultaneously will be studied in our future work.