Fault Reconstruction and Fault-Tolerant Control via ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 6, JUNE 2015

3885

Fault Reconstruction and Fault-Tolerant Control via Learning Observers in Takagi–Sugeno Fuzzy Descriptor Systems With Time Delays Qingxian Jia, Wen Chen, Member, IEEE, Yingchun Zhang, and Huayi Li

Abstract—This paper addresses the problems of observer-based fault reconstruction and fault-tolerant control for Takagi–Sugeno fuzzy descriptor systems subject to time delays and external disturbances. A novel fuzzy descriptor learning observer is constructed to achieve simultaneous reconstruction of system states and actuator faults. Sufficient conditions for the existence of the proposed observer are explicitly provided. Utilizing the reconstructed fault information, a reconfigurable fuzzy fault-tolerant controller based on the separation property is designed to compensate for the impact of actuator faults on system performance by stabilizing the closed-loop system. In addition, the design of the fault reconstruction observer and the fault-tolerant controller is formulated in terms of linear matrix inequalities that can be conveniently solved using convex optimization techniques. Finally, simulation results on a truck–trailer system are presented to verify the effectiveness of the proposed approaches. Index Terms—Fault reconstruction, fault-tolerant control (FTC), learning observers (LOs), linear matrix inequalities (LMIs), Takagi–Sugeno (T–S) fuzzy descriptor systems.

I. I NTRODUCTION

T

HE increasing complexity of modern engineering systems will correspondingly increase the possibility of system faults and/or failures. The occurrence of sensor, actuator, or component failures may dramatically degrade system performance and even result in catastrophic system collapse. As a response to high requirement for system safety, reliability, and survivability, fault diagnosis and fault-tolerant control (FTC) for dynamic systems have been attractive subjects of many

Manuscript received March 30, 2014; revised April 7, 2014, August 29, 2014, October 15, 2014, and December 24, 2014; accepted January 5, 2015. Date of publication February 19, 2015; date of current version May 8, 2015. Q. Jia and H. Li are with the Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150001, China (e-mail: jqxhit@ gmail.com; [email protected]). W. Chen is with the Division of Engineering Technology, Wayne State University, Detroit, MI 48202 USA (e-mail: [email protected]). Y. Zhang is with the Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150001, China, and also with Shenzen Aerospace Dongfanghong HIT Satellite Company Ltd., Shenzhen 518057, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2015.2404784

investigations in control community and have received considerable attention during the past few decades, and a great deal of research progress has been made (see [1]–[8] and the references therein). Descriptor systems are also referred to as singular systems, differential-algebraic systems, generalized state-space systems, and implicit systems [9]. They arise from a natural modeling process in characterizing a wide class of practical systems, including electric and electronic systems, aircraft systems, and biological systems. Due to the universal existence of inherent nonlinearities in industrial and commercial processes, nonlinear descriptor systems and their applications should be specifically considered. On the other hand, it is well recognized that the Takagi–Sugeno (T–S) fuzzy model can approximate a highly complicated nonlinear system using a family of fuzzy IF-THEN rules that represent the local linear state-space models [10]– [12]. Therefore, taking advantages of both descriptor and T–S formalisms in the field of modeling, it is significantly efficient for T–S descriptor systems (TSDSs) to describe many practical processes. During the past decade, TSDSs have attracted considerable attention in both research and practical applications. A few efforts have been reported in stabilization [13], output-feedback control [14], H∞ control [15], and sliding mode control [16], just to name a few. Nevertheless, to the best of our knowledge, few research activities have been aimed at fault diagnosis and FTC in TSDSs. Reference [17] is an existing work for designing an unknown input observer with disturbance-decoupling ability to perform actuator fault detection in discrete-time TSDSs. An observer-based reliable control method is developed in [18] for continuous-time TSDSs with time delays, where a fuzzy Luenberger observer is utilized to estimate system states. Literature [19] addresses the issue of robust dissipative FTC against actuator faults and parametric uncertainties in the linear matrix inequality (LMI) optimization framework. An observer-based active FTC strategy is investigated in [20], where a proportional multi-integral observer is proposed for actuator fault reconstruction with an assumption that the faults and their derivatives are norm bounded. Additionally, observer-based fault diagnosis and FTC in T–S fuzzy systems using descriptor techniques have been also studied in [21]–[23]. Although several fault diagnosis and FTC schemes for TSDSs have been proposed, they are still challenging issues and are under investigation.

0278-0046 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

3886


Recently, researchers have been paying much attention to learning observer (LO)-based fault reconstruction [24]–[29]. Its main advantage is that it could accurately reconstruct timevarying faults, particularly fast-varying faults. At present, few studies have been focused on LO design in T–S fuzzy systems. Therefore, there is a strong incentive for us to investigate fault reconstruction and FTC for TSDSs via LOs. In view of these, we will address the problems of fuzzy LO-based fault reconstruction and reconfigurable FTC for continuoustime TSDSs subject to time delays and external disturbances. In this paper, the main contributions that are worth emphasizing are summarized in the following three aspects. 1) A novel fuzzy descriptor LO (FDLO) is constructed to achieve simultaneous reconstruction of system states and actuator faults, and sufficient conditions for the existence of the FDLO are explicitly provided. 2) The FDLO is designed to completely decouple external disturbances from fault reconstruction based on disturbance-decoupling techniques. 3) The FDLO and the fuzzy fault-tolerant controller (F2 TC) are independently designed based on the separation property [30], [31]. It is worth noting that the separation property has not been developed in any LOs and LObased FTC. In addition, a systematic design method for the FDLO and the F2 TC is suggested using LMI optimization techniques; thus, it is convenient to calculate their gain matrices. Throughout this paper, A > 0 (A < 0) denotes that A is positive (negative) definite; λmin (A) and λmax (A) are the minimum and maximum eigenvalues of A, respectively; A† denotes a generalized inverse of A; · and · ∞ represent the Euclidean and infinity norms of a vector or a matrix, respectively; ∗ means symmetric term; In is an identity matrix of size n; C+ denotes the set of all complex numbers with nonnegative real part; r = {1, . . . , r}; and ∀ means “for all”. II. S YSTEM F ORMULATION The considered continuous-time TSDS is composed of a set of fuzzy IF-THEN rules, each of which represents a local linear input–output relationship of the nonlinear system. The ith rule of this T–S fuzzy descriptor model is in the following form. Plant Rule i: IF z1 (t) is Mi1 and . . . zq (t) is Miq , THEN ⎧ E x(t) ˙ = Ai x(t) + Ahi x(t − h) + Bi u(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ + Dd(t) + Fi fa (t) (1) ⎪ y(t) = Cx(t) ⎪ ⎪ ⎪ ⎪ ⎩ x(t) = φ(t), t ∈ [−h, 0] where x(t) ∈ Rn , u(t) ∈ Rm , and y(t) ∈ Rp ; they represent system states, control inputs, and measurement outputs, respectively. External disturbances d(t) ∈ Rν , and actuator faults fa (t) ∈ Rm and could be constant or time varying. Herein, z(t) = [z1 (t), . . . , zq (t)]T , representing the premise variable vector that is supposed to be measurable. Fuzzy sets Mij (i = 1, . . . , r; j = 1, . . . , q) are characterized by the membership

function; and r and q are the numbers of IF-THEN rules and of premise variables, respectively. Constant matrix E is singular, i.e., rank(E) = s < n. Initial vector φ(t) is a continuous function when t ∈ [−h, 0], where time delay h is a positive constant scalar, and φ(t) ∈ Rn . Without loss of generality, it is supposed that matrices Fi and C are of full column rank and of full row rank, respectively. Hence, the overall fuzzy model achieved by fuzzy blending of each individual plant rule is described by ⎧ r ⎪ ⎪ ⎪ E x(t) ˙ = μi (z(t)) {Ai x(t) + Ahi x(t − h) ⎪ ⎪ ⎨ i=1 (2) ⎪ + Bi u(t) + Dd(t) + Fi fa (t)} ⎪ ⎪ ⎪ ⎪ ⎩ y(t) =Cx(t) where μi (z(t)) is defined as ωi (z(t)) μi (z(t)) = r i=1 ωi (z(t))

ωi (z(t)) =

q

μij (zj (t)) (3)

j=1

function of where μij (·) is the grade of the membership μij . It is assumed in this paper that ri=1 ωi (z(t)) > 0 with ωi (z(t)) ≥ 0, ∀ t. Therefore, we have ri=1 μi (z(t)) = 1 with 0 ≤ μi (z(t)) ≤ 1, ∀ t. In order to address the main results, the following assumptions are made. Assumption 1:

E rank = n. (4) C Assumption 2: The triple matrix (E, Ai , C) is R-detectable [32], ∀ i ∈ r, that is,

sE − Ai rank ∀ i ∈ r. (5) = n, s ∈ C+ C Assumption 3:

E rank C

D 0

= n + rank(D).

(6)

Remark 1: Assumption 1 implies that there exists a full-row † E rank matrix [ T1 T2 ] = such that C

E [ T1 T2 ] = In . (7) C It also guarantees the impulse observability of the triple matrix (E, Ai , C), ∀ i ∈ r [32]. Both Assumption 1 and Assumption 2 are necessary conditions for the existence of the designed observer in the latter section. Assumption 3 is presented for disturbance-decoupling design of the proposed observer. It is worth noting that such an assumption is common in the design of disturbance-decoupling observer for descriptor systems. Similar assumptions can be also found in [17] and [32] and the references therein.

JIA et al.: FAULT RECONSTRUCTION AND FTC VIA LOs IN T–S FUZZY DESCRIPTOR SYSTEMS

III. FAULT R ECONSTRUCTION B ASED ON AN FDLO A. Design of an FDLO In order to reconstruct actuator faults in the TSDS (2), a novel FDLO is constructed as follows: ω(t) ˙ =

r

μi (z(t))

× TAi x ˆ(t)+TAhi x ˆ(t−h)+TBi u(t)+L1i (y(t)− yˆ(t)) ˆ (8a) + L2i (y(t − h) − yˆ(t − h)) + T Fi fa (t) i=1

x ˆ(t) = ω(t) + Hy(t) yˆ(t) = C x ˆ(t)

(8b) (8c)

fâ (t) = K1 fâ (t − τ ) +

r

μi (z(t)) K2i (y(t) − yˆ(t))

(8d)

i=1

where ω(t) ∈ Rn , and x ˆ(t) ∈ Rn ; they are observer states and estimated states, respectively. Signals yˆ(t) and yˆ(t − h) are estimated system outputs at the sampling time t and t − h, respectively; yˆ(t) ∈ Rp , and yˆ(t − h) ∈ Rp . Fault reconstruction signal fâ (t) ∈ Rm , which is updated by both its previous information at t − τ and current output estimation error. Parameter τ is the learning interval [29]. Diagonal matrix K1 = diag{σ1 , . . . , σm } with σi ∈ (0, 1]; and T , H, K1 , L1i , L2i , and K2i are gain matrices with appropriate dimensions to be determined later. Assumption 1 implies that there exist gain matrices T and H such that T E + HC = In .

(9)

State estimation errors, output estimation errors, and fault reˆ(t), construction errors can be represented by ex (t) = x(t) − x ˆ ey (t) = y(t) − yˆ(t), and ef (t) = fa (t) − fa (t), respectively. By using (9), the following estimation error dynamics can be obtained:

3887

Remark 2: In order to design gain matrices T and H such that constraints (9) and (11) are simultaneously satisfied, an augmented matrix equation composing of (9) and (11) is organized as

E D (13) [T H ] = [ In 0 ] . C 0 According to [34], a solution of (13) exists if and only if ⎤ ⎡

E D E D . (14) rank ⎣ C 0 ⎦ = rank C 0 In 0 Under Assumption 3, it is easily verified that (14) holds. Therefore, a general solution for [T H] [34] can be obtained as

† E D [ T H ] = [ In 0 ] C 0

† E D E D −N In+p − (15) C 0 C 0 where N ∈ Rn×(n+p) , representing an arbitrary matrix. In addition, to guarantee fault detectability by the proposed FDLO, matrix T should be selected such that rank(T Fi ) = rank(Fi ), ∀ i ∈ r. To establish stability and convergence of the presented FDLO such that accuracy of fault reconstruction can be guaranteed, the following assumption is needed. Assumption 4: Assume that fã (t)∞ ≤ kf , where the vector fã (t) := fa (t) − K1 fa (t − τ ), and kf is a small positive constant. Using Assumption 4, one obtains ef (t) = K1 ef (t−τ )−

r

μi (z(t)) K2i Cex (t)+ fã (t). (16)

i=1

Remark 3: In the FDLO (8), learning interval τ can be adjusted to guarantee fault-reconstructing accuracy. It could be selected large if a fault is constant or slowly varying; otherwise, r it should be chosen small. In particular, in sampled-data control e˙ x (t) = μi (z(t)) systems, it can be taken as the sampling interval or as an integer i=1 multiple of the sampling interval. Assumption 4 and (16) imply × {(T Ai − L1i C)ex (t) + (T Ahi − L2i C) ex (t − h) that, for each i ∈ m, σi is selected close to 1; in other words, gain matrix K1 is designed close to Im , and learning interval + T Dd(t) + T Fi ef (t)} τ is selected to be sufficiently small. Then, fã (t)∞ can be ey (t) = Cex (t). (10) guaranteed to be sufficiently small for both constant and timeOn the basis of the disturbance-decoupling techniques proposed varying faults. Therefore, under Assumption 4, the proposed FDLO-based fault-reconstructing scheme can be applied to in [17] and [33], matrix T is selected such that reconstruct time-varying faults, particularly fast-varying faults. Remark 4: The fault-constraining condition described in TD = 0 (11) Assumption 4 in this paper is less restrictive compared with hence, estimation error dynamics (10) can be simplified as those in [20] and [35] because the norm-bounded conditions on the derivatives of faults are not required in this paper. r Remark 5: Compared with the fuzzy unknown input obμi (z(t)) {(T Ai − L1i C)ex (t) e˙ x (t) = servers proposed in [17] and [33], only two gain matrices T i=1 and H in (8a) and (8b) and two equality constraints (9) and (11) + (T Ahi −L2i C)ex (t−h)+T Fi ef (t)} are involved for disturbance-decoupling purpose. It is therefore ey (t) =Cex (t). (12) easier to design the FDLO.

3888


× P T Fi − α0 (K2i C)T fã (t)

B. Stability Analysis of the FDLO Here, the stability and convergence of the FDLO (8) are proved, and detailed design of observer gain matrices is discussed. Theorem 1: Consider the TSDS shown in (2). If there exist positive definite symmetric matrices P ∈ Rn×n , Q ∈ Rn×n , Q1 ∈ Rm×m , and K1 ∈ Rm×m , as well as matrices K2i ∈ Rm×p , Y1i ∈ Rn×p , and Y2i ∈ Rn×p , such that, ∀ i ∈ r

Ξ1 P T Ahi − Y2i C 0, then the presented FDLO (8) can realize that state estimation error and fault reconstruction error are uniformly ultimately bounded. Observer gain matrices can be accordingly determined by L1i = P −1 Y1i and L2i = P −1 Y2i , ∀ i ∈ r. Proof: Consider the following Lyapunov–Krasovskii functional candidate: T

Y1iT

t V (t) =

eTx (t)P ex (t)

eTx (s)Qex (s)ds

+

t eTf (s)Q1 ef (s)ds.

(20)

t−τ

r

× eTx (t) P (T Ai −L1i C)+(T Ai −L1i C)T P +Q ex (t) + 2eTx (t)P T Fi ef (t)+2eTx (t)P (T Ahi −L2i C)ex (t−h) − eTx (t−h)Qex (t−h)−0 eTf (t)Q1 ef (t) + α0 eTf (t)ef (t)−eTf (t−τ )Q1 ef (t−τ )

(21)

where α0 = (1 + 0 )λmax (Q1 ) with 0 ≥ 0. Substituting (16) into (21) leads to r

= −α0 eTx (t)(K2i C)T K2i Cex (t) ≤ 0.

(23)

Furthermore, with the help of (18) and (23), (22) can be simplified as ˙ ≤ V(t)

r

μi (z(t))

i=1

T ex (t) P (T Ai −L1i C)+(T Ai −L1i C)T P + Q ex (t) + 2eTx (t)P (T Ahi − L2i C)ex (t − h) − eTx (t − h)Qex (t − h) + 2α0 eTf (t − τ )K1T fã (t) + α0 eTf (t − τ )K1T K1 ef (t − τ ) + α0 fãT (t)fã (t) − 0 eTf (t)Q1 ef (t) − eTf (t − τ )Q1 ef (t − τ ) . (24)

V˙ (t) ≤

r

μi (z(t))

⎧ ⎨ × [ eTx (t) ⎩

eTx (t − h)

eTf (t) ]

⎤ 0 Ξ1 P T Ahi − Y2i C −Q 0 ⎦ ×⎣ ∗ ∗ ∗ −0 Q1 ⎡ ⎤ ex (t) α0 α1 ˜T × ⎣ ex (t − h) ⎦ + f (t)fã (t) α1 − 1 a ef (t) ⎫ ⎬ + eTf (t − τ ) α0 α1 K1T K1 − Q1 ef (t − τ ) ⎭ ⎡

μi (z(t))

i=1

˙ ≤ V(t)

If the condition (18) holds, one obtains −eTx (t) 2P T Fi − α0 (K2i C)T K2i Cex (t)

i=1

The derivative of the Laypunov candidate with respect to time can be derived as ˙ ≤ V(t)

+ 2α0 eTf (t − τ )K1T fã (t) − 0 eTf (t)Q1 ef (t) (22) − eTf (t − τ )Q1 ef (t − τ ) .

For any scalar α > 0, 2X T Y ≤ (1/α)X T X + αY T Y ; thus, (24) can be further transformed into

t−h

+

+ α0 fãT (t)fã (t) − eTx (t − h)Qex (t − h)

(18)

and

T

+ α0 eTf (t−τ )K1T K1 ef (t−τ )

μi (z(t))

i=1

× eTx (t)[P (T Ai −L1i C)+(T Ai −L1i C)T P +Q ex (t) + 2eTx(t)P (T Ahi −L2i C)ex (t−h)+2eTx(t) × P T Fi −α0 (K2i C)T K1 ef (t−τ )−eTx (t) × 2P T Fi − α0 (K2i C)T K2i Cex (t)+2eTx (t)

(25)

where Y1i = P L1i , Y2i = P L2i , and α1 = 1 + 1 with 1 > 0. According to (17), we have ⎡ ⎤ Ξ1 P T Ahi − Y2i C 0 Ξ=⎣ ∗ (26) −Q 0 ⎦ < 0. ∗ ∗ −0 Q1 If conditions (19) and (26) hold, it follows from (25) that V˙ (t) ≤ −ξζ(t)2 + δ

(27)

JIA et al.: FAULT RECONSTRUCTION AND FTC VIA LOs IN T–S FUZZY DESCRIPTOR SYSTEMS

where ξ = λmin (−Ξ), ζ(t) = [eTx (t) eTx (t − h) eTf (t)]T , and δ = (α0 α1 /α1 − 1)kf2 . It follows that V˙ (t) < 0 for ξζ(t)2 > δ, which means that the trajectory of ζ(t) that is outside of a small set Ψ = {ζ(t)|ζ(t)2 ≤ δ/ξ} will converge to this set Ψ according to the Lyapunov stability theory; thus, ζ(t) is uniformly ultimately bounded. It can be noted that the condition (17) is an LMI that can be conveniently solved using standard LMI tools, whereas the condition (18) is a matrix equality that is difficult to be directly solved using the MATLAB LMI toolbox [23], [28], [29]. To tackle this difficult problem, we rewrite linear equality condition (18) as T (T Fi )T P −α0 K2i C < γIn (28) (T Fi )T P −α0 K2i C where γ is a positive scalar. By using Schur complement lemma [36], (28) is equivalent to

−γIm (T Fi )T P − α0 K2i C < 0. (29) ∗ −In The design problem of (17) and (18) is now converted into the following minimization problem: Minimize γ, subject to (17) and (29).

3889

The preceding two conditions mean that the pair (T Ai , C) is detectable, ∀ i ∈ r, that is

sIn − T Ai rank ∀ i ∈ r. (31) = n, s ∈ C+ C It can be also understood that (31) is a necessary condition for (17) and (18) to hold ∀ i ∈ r. In what follows, we will further discuss necessary conditions for (31) to hold such that the applicable scope of the proposed FDLO can be clarified. Based on (31), the following derivation can be obtained:

sIn − T Ai s(T E + HC) − T Ai rank = rank C C

sT E − T Ai = rank C

T 0 sE − Ai = rank 0 Ip C

! sE − Ai ≤ min rank(T ) + p, rank . C (32)

(30)

The preceding minimization problem can be solved by using the Solvers minx in the LMI toolbox of MATLAB. Alternatively, a sufficient small positive scalar γ can be selected such that matrices P and K2i can be computed to make (T Fi )T P approximate to α0 K2i C with satisfactory precision by solving (17) and (29). The results proposed in Theorem 1 can be readily extended to the case of constant actuator faults, as claimed by the following corollary. Corollary 1: Assume that K1 = Im . If there exist positive definite symmetric matrices P ∈ Rn×n and Q ∈ Rn×n and matrices K2i ∈ Rm×p , Y1i ∈ Rn×p , and Y2i ∈ Rn×p to satisfy conditions (17) and (18) with α0 = 1, ∀ i ∈ r, then the presented FDLO (8) guarantees asymptotic reconstruction of constant actuator faults. Moreover, observer gain matrices can be computed as L1i = P −1 Y1i and L2i = P −1 Y2i , ∀ i ∈ r. Proof: The proof of Corollary 1 is similar to that of Theorem 1. The detailed proof is thus omitted. Remark 6: If K1 = Im , then fã (t) ≡ 0, kf = 0, and δ = 0 when constant actuator faults occur in the considered system (2). Thus, via Corollary 1, the FDLO can achieve asymptotic reconstruction of constant actuator faults without requirement for Assumption 4. In other words, the FDLO can guarantee unbiased reconstruction of constant actuator faults. Remark 7: It should be noted that necessary conditions for the solvability of the proposed FDLO design using Theorem 1 and Corollary 1 must be satisfied. In reference to [37] and [38], the following necessary conditions for (17) and (18) to be solvable ∀ i ∈ r can be provided in terms of invariant zeros as follows. 1) rank(CT Fi ) = rank(T Fi ) = m, ∀ i ∈ r. 2) No invariant zeros of (T Ai , T Fi , C) lie in C+ , ∀ i ∈ r.

Therefore, if the condition (31) holds, the triple matrix (E, Ai , C) should be R-detectable, ∀ i ∈ r. Meanwhile, it is necessary that rank(T ) ≥ n − p. In addition, according to (11) and aforementioned condition (1), the following conditions hold: rank(T ) ≤ n − ν, rank(T ) ≥ m, and p ≥ m. Therefore, rank[D Fi ] ≤ n, ∀ i ∈ r. IV. D ESIGN OF FAULT-TOLERANT C ONTROLLERS B ASED ON FDLO S Here, using reconstructed fault information provided by the FDLO, we will construct a reconfigurable F2 TC scheme such that closed-loop TSDS (2) is stable. Before presenting the main results, the following assumptions are made. Assumption 5: B1 = · · · = Br = B. Assumption 6: rank(BFi ) = rank(B), ∀ i ∈ r. Remark 8: Assumption 5 is a common assumption for system (2), and similar assumptions can be found in [16] and [20] and the references therein. Assumption 6 implies that there exists a nonzero matrix F¯i ∈ Rm×m such that Fi = B F¯i , ∀ i ∈ r [35], [39]. Furthermore, it is obtained that (In − BB † )Fi = (In − BB † )B F¯i = 0

∀ i ∈ r.

(33)

In order to compensate for the fault effect on the TSDS such that the faulty system is as close as possible to the prefault system, a state-feedback-based reconfigurable F2 TC is constructed as follows: u(t) = un (t) + uf (t)

(34)

where un is the normal state-feedback controller in fault-free condition, and uf is an additional controller, which is used to

3890


accommodate actuator faults. Specifically, controllers un and uf are expressed as follows: un (t) = −

r

μi (z(t)) K3i x ˆ(t)

(35)

r

μi (z(t)) Fi fâ (t)

(36)

i=1

where K3i ∈ Rm×n , ∀ i ∈ r, representing the controller gain matrices to be determined later. Therefore, the closed-loop TSDS without external disturbances becomes E x(t) ˙ =

r

μi (z(t))

i=1

× {Ai x(t) − BK3i x ˆ(t) + Ahi x(t − h) +(In − BB † )Fi fâ (t) − Fi fâ (t) + Fi fa (t) =

r

"

μi (z(t)){(Ai −BK3i )x(t)+Ahi x(t−h)+Δi (t)}

i=1

(37) where Δi (t) = [BK3i Fi ][ex (t) ef (t)]T . To design controller gain matrices K3i , ∀ i ∈ r, such that the closed-loop TSDS with faults is still stable, the following theorem is provided. Theorem 2: Suppose that Theorem 1 holds. If there exist positive definite symmetric matrices X ∈ Rn×n and Q2 ∈ Rn×n , a matrix Mi ∈ Rm×n , and a positive scalar ηc such that, ∀ i ∈ r, the following relations hold: X T E T = EX ≥ 0 and

⎡

Θ11 ⎣ ∗ ∗

Ahi η c I n − Q2 ∗

⎤ X ⎦