Fault Detection for T–S Fuzzy Discrete Systems in Finite ... - IEEE Xplore

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 4, AUGUST 2011

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Fault Detection for T–S Fuzzy Discrete Systems in Finite-Frequency Domain Hongjiu Yang, Yuanqing Xia, and Bo Liu

Abstract—This paper investigates the problem of fault detection for Takagi–Sugeno (T–S) fuzzy discrete systems in finite-frequency domain. By means of the T–S fuzzy model, both a fuzzy fault detection filter system and the dynamics of filtering error generator are constructed. Two finite-frequency performance indices are introduced to measure fault sensitivity and disturbance robustness. Faults are considered in a middle frequency domain, while disturbances are considered in another certain finite-frequency domain interval. By using the generalized Kalman–Yakubovi˘ c–Popov Lemma in a local linear system model, the design methods are presented in terms of solutions to a set of linear matrix inequalities, which can be readily solved via standard numerical software. The design problem is formulated as a two-objective optimization algorithm. A numerical example is given to illustrate the effectiveness and potential of the developed techniques. Index Terms—Fault detection, finite-frequency domain, Kalman–Yakubovi˘ c–Popov (KYP) Lemma, linear matrix inequality (LMI), Takagi–Sugeno (T–S) fuzzy model.

I. I NTRODUCTION

I

N RECENT years, fault detection in dynamic systems has attracted considerable attention from many researchers due to the increasing demand for reliability and safety in industrial processes [1]–[3]. Some results on fault detection and isolation by using a frequency domain approach have been reported in literature, such as in [4]–[7]. There are also some recently published papers on robust H∞ filtering, for example, [8]–[10]. For the purpose of fault detection, the H− index, defined as the smallest singular value of a transfer function matrix, was proposed in [11]. A matrix factorization approach to develop the optimal fault detection scheme using H− index was given in [12] and [13]. The approach was also used to H− /H∞ fault detection problem in [14]. The optimal solutions of robust fault detection problems for time domain were given in [15], which also extends the results to frequency domain. Note that all performance indices in these literatures are defined in the whole

Manuscript received May 25, 2010; revised September 10, 2010 and November 20, 2010; accepted December 5, 2010. Date of publication January 20, 2011; date of current version July 20, 2011. This work was supported in part by the National Natural Science Foundation of China under Grant 60974011, in part by the Program for New Century Excellent Talents in University of the Peoples Republic of China (NCET-08-0047), in part by the Ph.D. Programs Foundation of the Ministry of Education of China (20091101110023), in part by the Program for Changjiang Scholars and Innovative Research Team in University, and in part by the Beijing Municipal Natural Science Foundation (4102053). This paper was recommended by Associate Editor P. Shi. The authors are with the Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, China (e-mail: [email protected]; [email protected]; [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/TSMCB.2010.2099653

frequency domain range. However, the effects of modeling disturbances and faults occupy in different frequency ranges according to the actual situation. Therefore, designing an observer to detect the fault by satisfying certain performances in finitefrequency domain is a very good alternative to consider. In [16], the smallest nonzero singular value of the transfer function from fault to residual over a finite-frequency band was used to evaluate the worst case fault sensitivity. However, the worst case H− index calculated is time consuming to search for good weighting functions and makes the design more complex. With the aid of the generalized Kalman–Yakubovi˘ c–Popov (GKYP) Lemma, new linear matrix inequality (LMI) conditions were given for the finite-frequency performance index; please refer to [17]–[19]. By using the GKYP Lemma, the fault detection problems in finite-frequency domain both for discrete and continuous systems were investigated in [20] and [21], respectively. Furthermore, a fault detection problem for the low-frequency domain of linear time-delay continuous systems was converted into a detection observer design problem in [22]. It is known that all the results considered linear invariant model; however, many processes are not linear based on the actual situation. It is hard to use the GKYP Lemma in the fault detection problem for finite-frequency domain in the nonlinear case, which motivates us to make an effort in this paper. The Takagi–Sugeno (T–S) fuzzy model described by a family of fuzzy IF – THEN rules was first introduced in [23]. In other words, it formulates the complex nonlinear systems into a framework that interpolates some affine local models by a set of fuzzy membership functions. Based on this framework, a systematic analysis and design procedure for complex nonlinear systems can be possibly developed in view of the powerful control theories and techniques in linear systems. Therefore, many important results on T–S fuzzy systems have been reported, such as in [24]–[27] and the references therein. An adaptive fuzzy neural network control problem was considered via a T–S fuzzy model for a robot manipulator, including the actuator dynamics in [28]. An adaptive fuzzy sliding control method was used for a double-pendulum-and-cart system in [29]. A modeling and control problem of fuzzy discrete event systems was investigated in [30]. The stability analysis and controller design of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions were considered in [31]. Relaxed stabilization criteria for discrete-time T–S fuzzy control systems based on a switching fuzzy model and a piecewise Lyapunov function were investigated in [32]. The control synthesis of continuous-time T–S fuzzy systems with local nonlinear models was considered in [33]. State-based control of fuzzy discrete-event systems was given in [34].

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 4, AUGUST 2011

Furthermore, a feasible solution of the fault detection problem for nonlinear systems can be converted to that of fault detection for T–S fuzzy systems. In [35], the robust H∞ fault estimation problem was investigated for uncertain time-delay T–S fuzzy models. Considering intermittent measurements in T–S fuzzy systems, the problem of fault detection was also investigated in [36]. For the T–S fuzzy Itô stochastic system, a robust H∞ sensor fault detection problem was dealt with in [37]. An LMI approach was proposed to solve the H∞ /H− fault detection problem for uncertain fuzzy systems in [38]. A fuzzy fault detection problem for nonlinear stochastic time-delay Markov jump systems was studied in [39]. The problem of H∞ fuzzy controller synthesis for a class of discrete nonlinear active faulttolerant control systems in a stochastic setting was concerned with in [40]. However, most of the aforementioned results are concerned with infinite-frequency domain. In reality, most physical systems are not only complex nonlinear but also in finite-frequency domain. Thus, the combination of fuzzy models and fault detection methods in finite-frequency domain is an important and practical problem, which will be investigated in this paper. This paper is organized as follows. In Section II, the modeling of a T–S fuzzy system in the logical dynamical framework is described. Section III gives the main results for designing a fault detection observer in finite-frequency domain for the fuzzy system. In Section IV, we present numerical simulation results. Conclusions are given in Section V. Notations: In the following, if not explicitly stated, matrices are to have compatible dimensions. The scalar j = √ assumed −1, Rn denotes the n-dimensional Euclidean space, Z + denotes the set of nonnegative integers, the notation X > Y (X ≥ Y ) means that the matrix X−Y is positive definite (X−Y is semipositive definite, respectively), and I is the identity matrix of appropriate dimension. For any matrix A, AT denotes the transpose of matrix A, A−1 denotes the inverse of matrix A, A† = AT (AAT )−1 denotes its Moore–Penrose inverse, {Xi }Si=1 denotes the set {X1 , X2 , . . . , XS }. The shorthand diag{M1 , M2 , . . . , Mr } denotes a block diagonal matrix with diagonal blocks being the matrices M1 , M2 , . . . , Mr , and  ·  denotes the Euclidean norm for vectors or the spectral norms of matrices. The symmetric terms in a symmetric matrix are denoted by ∗.

tor; and f (k) ∈ Rnf denotes the fault input vector. Ai , Bdi , Bf i , Ci , Ddi , and Df i are known constant real matrices with appropriate dimensions. Through the use of fuzzy “blending,” the resulting fuzzy system model is inferred as the weighted average of the local models of the form x(k + 1) =

S 

y(k) =

S 

x(k + 1) = Ai x(k) + Bdi d(k) + Bf i f (k)

(1)

y(k) = Ci x(k) + Ddi d(k) + Df i f (k)

(2)

where i = 1, 2, . . . , S; S is the number of IF – THEN rules; μ1 (k), μ2 (k), . . . , μg (k) are the premise variables; Fli (l = 1, 2, . . . , g) denotes the fuzzy sets; x(k) ∈ Rn is the state variable; u(k) ∈ Rnu is the control input; y(k) ∈ Rny is the measured output; d(k) ∈ Rnd represents the disturbance vec-

hi (μ(k)) [Ci x(k) + Ddi d(k) + Df i f (k)] (4)

i=1

with ωi (μ(k))

hi (μ(k)) = S

i=1 ωi (μ(k))

,

ωi (μ(k)) =

g 

Fli (μl (k))

l=1

in which Fli (μl (k)) is the grade of membership of μl (k) in the fuzzy set Fli . Here, we assume that ωi (μ(k)) ≥ 0,  i = 1, 2, . . . , S, and Si=1 ωi (μ(k)) > 0 for all k ≥ 0. There fore, hi (μ(k)) ≥ 0, and Si=1 hi (μ(k)) = 1 for all k ≥ 0. Let ρ be a set of basis functions satisfying hi (μ(k)) ≥ 0 and S i=1 hi (μ(k)) = 1. In order to detect the system fault, a fuzzy fault detection observer is designed as follows. Plant Rule i: If μ1 (k) is F1i , μ2 (k) is F2i , . . . , μg (k) is Fgi , then x ˆ(k + 1) = Ai x ˆ(k) + Li (y(k) − yˆ(k)) yˆ(k) = Ci x ˆ(k) r(k) = y(k) − yˆ(k)

(5) (6) (7)

where r(k) is the residual signal which carries information on the time and location of the occurrence of the faults and x ˆ(k) ∈ Rn and yˆ(k) ∈ Rny denote the state and output estimation vectors, respectively. The design parameter is the observer gain matrices Li . Then, the dynamic global fault detection filter models can be constructed as follows: x ˆ(k + 1) =

II. P ROBLEM F ORMULATION The nonlinear discrete-time system whose faults are to be detected is represented by the following T–S fuzzy model. Plant Rule i: If μ1 (k) is F1i , μ2 (k) is F2i , . . . , μg (k) is Fgi , then

hi (μ(k)) [Ai x(k) + Bdi d(k) + Bf i f (k)] (3)

i=1

yˆ(k) = r(k) =

S  i=1 S  i=1 S 

hi (μ(k)) [Ai x ˆ(k) + Li (y(k) − yˆ(k))]

(8)

hi (μ(k)) [Ci x ˆ(k)]

(9)

hi (μ(k)) [Ci x(k) − Ci x ˆ(k)

i=1

+ Ddi d(k) + Df i f (k)] .

(10)

Defining the state estimate error e(k) = x(k) − x ˆ(k) and x ¯(k) = [xT (k) eT (k)]T , we have the following state error dynamic equation at node i: x ¯(k + 1) = Ail x ¯(k) + Bdil d(k) + Bf il f (k) r(k) = Cil x ¯(k) + Ddi d(k) + Df i f (k)

(11) (12)

YANG et al.: FAULT DETECTION FOR T–S FUZZY DISCRETE SYSTEMS

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where 

 0 Ail = , Ai − Li Cl   Bf i , Bf il = Bf i − Li Df l Ai 0



Bdil

 Bdi = , Bdi − Li Ddl

Cil = [ Ci − Cl

Cl ] .

Based on the following global nominal residual fuzzy system:

x ¯(k + 1) =

S 

hi (μ(k))

i=1

S 

hl (μ(k))

l=1

¯(k) + Bdil d(k) + Bf il f (k)] × [Ail x r(k) =

S 

hi (μ(k))

i=1

S 

(13)

hl (μ(k))

l=1

¯(k) + Ddi d(k) + Df i f (k)] . × [Cil x

(14)

We let

A(h) =

S 

hi (μ(k))

i=1

Bd (h) =

Bf (h) =

C(h) =

S 

hi (μ(k))

S 

i=1

l=1

S 

S 

hi (μ(k))

i=1

l=1

S 

S 

hi (μ(k))

S 

hl (μ(k)) Cil

Df (h) =

S 

hi (μ(k)) Df i

i=1

S 

hi (μ(k))

S 

hl (μ(k)) [Cil x ¯(k) + Ddi d(k)]

l=1

S  i=1

hl (μ(k)) Bf il

hi (μ(k)) Ddi ,

i=1

rf (k) =

hl (μ(k)) Bdil

l=1

i=1

rd (k) =

hl (μ(k)) Ail

l=1

i=1

Dd (h) =

S 

hi (μ(k))

S 

hl (μ(k)) [Cil x ¯(k) + Df i f (k)]

l=1

with h := (h1 , h2 , . . . , hS ) ∈ ρ. In the next, we give the following two definitions on H∞ and H− performance indices for the global nominal residual fuzzy system (13) and (14). Definition 1: Considering the global nominal residual fuzzy system (13) and (14), if the following condition is satisfied: H∞ := rd (k)2 / d(k)2 < γ

where γ is a given real positive scalar which denotes the worst case criterion for the effect of disturbances on the residual r(k). Then, the parameter γ is called the H∞ index bound of the control. The smaller the γ is, the more robust the generator becomes and the more unknown the input is restrained. Definition 2: For the global nominal residual fuzzy system (13) and (14), if the following condition is satisfied: H− := rf (k)2 / f (k)2 > β where β is a given real positive scalar that is a measurement of the fault sensitivity in the worst case from fault f (k) to residual r(k). Then, the parameter β is called the H− index bound of the control. The larger the β is, the more sensitive the generator becomes and the more detectable the fault information residual generator can capture. The objective of fuzzy fault detection is to design a suitable filter Li and to determine the residual evaluation function and an appropriate threshold to realize the following three conditions: Aa The state estimate error approaches asymptotically zero. Ab The effects of disturbances to the residual r(k) are reduced. Ac The effects of faults to the residual r(k) are increased. Based on practical situations, the effects of modeling disturbances and faults occupy in two different frequency ranges. Therefore, we consider the following two middle frequency ranges for z = ejθ both in disturbances and faults in this paper: Ωd = {ejθ |θ ∈ R, ϑd1 ≤ θ ≤ ϑd2 }

(15)

Ωf = {ejθ |θ ∈ R, ϑf 1 ≤ θ ≤ ϑf 2 }

(16)

where ϑd1 , ϑd2 , ϑf 1 , and ϑf 2 are given positive scalars which reflect the individual frequency ranges of disturbances and faults. By considering the disturbance and fault frequency ranges of (15) and (16), the conditions Aa–Ac can be rewritten as follows. A1 The fuzzy system (13) is asymptotically stable. A2 H∞ := r(k)2 /d(k)2 < γ ∀θ ∈ Ωd . A3 H− := r(k)2 /f (k)2 > β ∀θ ∈ Ωf . In this paper, we will design a fuzzy fault detection filter for the global fuzzy system (3) and (4) that satisfies the former conditions A1–A3 simultaneously. Remark 1: The global fuzzy system (13) and (14) is a timevarying system; however, the subsystem (11) and (12) is a linear time-invariant system. We will use the methods in finite frequency for the linear time-invariant system (11) and (12), which means that we realize H∞ and H− performance indices using finite-frequency GKYP Lemma for system (11) and (12). After some mathematical operations, it will be proven that the global fuzzy system (13) and (14) satisfies the H∞ and H− performance indices. Before ending this section, the following lemmas will be used to prove our main results.

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Lemma 1 [41]: Assume that the following fuzzy system is asymptotically stable: x(k + 1) = A(h)x(k) + B(h)u(k)

(17)

y(k) = C(h)x(k) + D(h)u(k)

(18)

where

Lemma 4: By using Lemma 2, considering system (17) in Lemma 1 and a matrix R with appropriate dimension, the LMI (19) in Lemma 1 is satisfied if there exist matrix variables X, Y , and V of appropriate dimensions, satisfying X ∈ X(C, R) and  T

A(h) =

S 

hi (μ(k))

i=1

B(h) =

S 

S 

hi (μ(k))

D(h) =

where

S 

  X(C, R) = C† XR + (I − C† C)V | det(X) = 0     A(h)T C(h)T −C(h)T A= B= B(h)T D(h)T −D(h)T

hl (μ(k)) Bil

l=1

hi (μ(k))

i=1 S 

hl (μ(k)) Ail

l=1

i=1

C(h) =

S 

S 

   T  −X −X Ξ 0 T + T < AX + BY R AX + BY R 0 Π

hl (μ(k)) Cil

l=1

C =[I

hi (μ(k)) Di .

0 ].

Moreover, T is the permutation matrix defined as

i=1

Let γ > 0 be a given constant and two positive scalars ϑ1 and ϑ2 be given, then system (17) has a finite-frequency H∞ index with parameter γ if there exist fuzzy basis matrix P (h) = P (h)T and matrix Q = QT > 0, for any h ∈ ρ, h+ := (h1 (μk+1 ), . . . , hS (μk+1 )) ∈ ρ such that   T  A(h) B(h) A(h) B(h) Ξ I 0 I 0  T   C(h) D(h) C(h) D(h) + Π < 0 (19) 0 I 0 I where   −P (h+ ) ejϑc Q Ξ= e−jϑc Q P (h) − (2 cos ϑo )Q

 Π=

I 0

0 −γ 2 I



with ϑc = (ϑ1 + ϑ2 )/2 and ϑo = (ϑ2 − ϑ1 )/2. Lemma 2 [42] (Finsler’s Lemma): Let x ∈ Rn , Q ∈ Rn×n , and U ∈ Rn×m . Let U ⊥ be any matrix such that U ⊥ U = 0. The following statements are equivalent. 1) xT Qx < 0 ∀U T x = 0, x = 0. T 2) U ⊥ QU ⊥ < 0. 3) ∃μ ∈ R such that Q − μU U T < 0. 4) ∃Y ∈ Rm×n such that

[ M1

M2

M3

M4 ]T = [ M1

M3

M2

M4 ].

The lemma can be easily obtained based on the results in [20]. III. M AIN R ESULTS A. Stability Condition Theorem 1: The global nominal residual fuzzy system (13) with d(k) = 0 and f (k) = 0 is asymptotically stable if there exist symmetric positive definite matrices {Ri > 0}Si=1 and {Pi > 0}Si=1 , and real matrices Z, X, and {Yi }Si=1 , such that for all i, l,  ∈ {1, 2, . . . , S}, the following LMIs hold: 

 R −qZ T −qZ pZ +qZ T Ai 0 and Qd > 0, and real matrices

⎡

−Rdl ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

0 −Pdl ∗ ∗ ∗ ∗

T V1d T V2d Φi (3, 3) ∗ ∗ ∗

Φi (1, 4) T T −V2d Ci + V2d Cl T T V4d − V3d Ci + V3d Cl Φi (4, 4) ∗ ∗

  X (C, Ed ) = C† X Ed + (I − C† C)Vd | det(X ) = 0     A(h)T C(h)T −C(h)T A(h) = B(h) = Bd (h)T Dd (h)T −Dd (h)T C =[I

X

0]

T −V1d Cl T − V2d Cl + ejϑdc Qd T V5d − V3d Cl

Φi (4, 5) Φi (5, 5) ∗

Y = [ 0 LT X ] .

⎤ T −V1d Ddi T −V2d Ddi ⎥ ⎥ T V6d − V3d Ddi ⎥ ⎥ 0}Si=1 and {Qf > 0}Si=1 , and real matrices {Rf i }Si=1 , {Pf i }Si=1 , {Yi }Si=1 , Z, X and {Vιf }, (ι = 1, 2, . . . , 6), such that, for all i, l,  ∈ {1, 2, . . . , S}, the LMI (28), shown at the bottom of the page, holds, with Ψi (1, 1) = −Rf l + Z T + Z, Φi (2, 2) = −Pf l + X T + X, T T + V3f , Ψi (1, 4) = Z − Z T Ai − V1f Ci + Φi (3, 3) = I + V3f T jϑf c T T V1f Cl + e Hf , Ψi (1, 6) = −ZBf i − Z Bf i − V1f Df i , T T T T Ψi (2, 5) = Ψi (2, 4) = −V2f Ci + V2f Cl − Y Ci + Yi Cl , T X −X T Ai −V2f Cl − YiT Cl +ejϑf c Qf , Ψi (2, 6) = −XBf i − T T T Df i +YlT Df i , Ψi (3, 4) = V4f −V3f Ci +V3f Cl , X T Bf i −V2f T T T T T Ψi (4, 4) = −Ai Z − Z Ai − Ci V4f + V4f Ci + Cl V4f + T Cl +Rf i −(2 cos ϑf o )Hf , Ψi (4, 5) = −CiT V5f +ClT V5f + V4f



−X A(h)X + B(h)YEd

⎡



⎡

0 ⎢ −V1d =⎣ C(h)T V1d Dd (h)T V1d

Ψi (1, 1) ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

0 Ψi (2, 2) ∗ ∗ ∗ ∗

0 −V2d C(h)T V2d Dd (h)T V2d

T V1f T V2f Ψi (3, 3) ∗ ∗ ∗

Ψi (1, 4) Ψi (2, 4) Ψi (3, 4) Ψi (4, 4) ∗ ∗

T T Cl , Ψi (4, 6) = AT CiT Yi − Cl Yi − V4f i ZBf  − Ci V6f + T T T T T Cl V6f −Ci Y Bf l +Cl Yi Bf  −Z Bf i −V4f Df i , Ψi (5, 5) = T T T T T −AT i X − X Ai − Cl V5f + Cl Yi −V5f Cl + Yi Cl + Pf i − T T (2 cos ϑf o )Qf , Ψi (5, 6) = Ai XBf  − Cl V6f − ClT Yi Bf  − T Df i , and Ψi (6, 6) = β 2 I +BfTi (Z T + X T Bf i + YT Df i − V5f T Df i , X T +Z +X)Bf  +BfTi YiT Df  +DfTi Yl Bf  −DfTi V6f −V6f T where Yi = Li X. Proof: In Lemma 1, choosing Π = diag{−I, β 2 I} with β as a given constant, we have H− index for system (17). Following the proof of Theorem 2 and changing the subscript d to f , choosing Ef = [I 0 I − Bf ], we get LMI (28). From Definition 2, the global nominal residual fuzzy system (13) with d(k) = 0 satisfies the H− index bound β in finite  middle frequency range Ωf if LMI condition (28) holds.

D. Fuzzy Fault Detection Filter In summary, conditions Aa–Ac hold if Theorems 1–3 are satisfied simultaneously. Then, we have the following theorem. Theorem 4: The global nominal residual fuzzy system (13) is asymptotically stable and satisfies the H∞ index bound γ and H− index bound β in finite middle frequency range Ωd and Ωf , respectively, if there exist symmetric positive definite matrices {Ri > 0}Si=1 , {Pi > 0}Si=1 , Hdi > 0, Qd > 0, Hf > 0, and Qf > 0, and real matrices H, Q, {Rdi }Si=1 , {Pdi }Si=1 , {Rf i }Si=1 , {Pf i }Si=1 , Z, X, {Vιd }, and {Vιf }(ι = 1, 2, . . . , 6), such that, for all i, l,  ∈ {1, 2, . . . , S}, the following LMIs hold: LMIs (21), (20), (25), and (28)

(29)

where Yi = LT i X. According to the method in [47], the combined constraints A1–A3 such as in the way of H− index β > >α H∞ index γ can, of course, be chosen, and this combination also makes sense. However, this is not recommended. The reasons are given as follows.

−X −V3d A(h)T X + C(h)T V3d − C(h)T Y Bd (h)T X + Dd (h)T V3d − Dd (h)T Y

T −V1f Cl Ψi (2, 5) T V5f − V3f Cl Ψi (4, 5) Ψi (5, 5) ∗

⎤ Ψi (1, 6) ⎥ Ψi (2, 6) ⎥ T V6f i − V3f Df i ⎥ ⎥ Jth , Remark 2: Note that most of the performance indices in the literature, such as [11], [14], and [15], are defined in the whole frequency domain range. In practice, the effects of modeling uncertainties and faults may occupy in different frequency ranges. Therefore, designing a filter to detect the fault by satisfying rigorous performances in finite-frequency domain is a key task in this topic. Remark 3: Although weighting functions [45] and frequency gridding [46] could be utilized to deal with the finite-frequency requirements, both methods have a hard time avoiding the computational burden and guaranteeing gain property performances

where



     0.3 0.2 0.7 0.4 0.8 , A2 = , Bd1 = A1 = 0.3 0.7 0.2 0.5 1.6       0.9 1.4 1.6 , Bf 1 = , Bf 2 = Bd2 = 1.5 1.2 1.1 C1 = [ 1.5 1.6 ] , C2 = [ 1.1 1.3 ] , Dd1 = [0.8] Dd2 = [0.7], Df 1 = [1.1], Df 2 = [0.9]

and the membership functions for rules 1 and 2 are M1 (x1 (k)) =

1 , M2 (x1 (k)) = 1−M1 (x1 (k)) . 1+exp(−2x1 (k))

To analyze the effects of fault and disturbance on the residual of the detection observer, consider the stuck fault, e.g.,  0.5, k > 100 f (k) = 0, elsewhere. Let the disturbance be d(k) = 1.6e−0.05k cos(k) + 0.16 sin(k). Assume that the frequency range of disturbances is 0.6 ≤ θd ≤ 1.4 and the frequency range of faults is 1.2 ≤ θf ≤ 2.8. Let p = 1 and q = 2, given β = 9.27041; solving the optimization problem (30), we have γ = 2.3241 × 10−007 and the fault detection observer gain matrices     −0.1188 −0.0510 L2 = . L1 = 0.0971 −0.0883 Considering the fact that a real state vector in the fuzzy system can be replaced by an estimated state vector using the fault detection observer obtained in Theorem 4, we first give the simulation results of the state estimate responses of the fuzzy system in this example for the initial conditions x ˆ1 (0) = ˆ1 (k) and x ˆ2 (k) are denoted x ˆ2 (0) = 0, shown in Fig. 1, where x by xo1 (k) and xo2 (k), respectively. For the initial condition y(0) = 0, the simulation result of the estimated output of fuzzy system in this example is shown in Fig. 2, where yˆ(k) is denoted by yo(k).

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 4, AUGUST 2011

Fig. 1. State estimate response of x ˆ(k). Fig. 4.

Output of y(k).

Fig. 5.

Residual output r(k).

Fig. 6.

Residual evaluation Jr (n).

Fig. 2. Estimated output of yˆ(k).

Fig. 3. State responses of x(k).

With the initial conditions x1 (0) = x2 (0) = y(0) = 0, the graphs of the real states and output are shown in Figs. 3 and 4. Then, the residual outputs are shown in Fig. 5 with the initial condition r(0) = 0, from which we can see that the faults are well discriminated from disturbances.

To detect the fault, we choose the residual evaluation function as stated in (31), and the residual evaluation output is shown in Fig. 6, where Jr (n) and Jth are denoted by Jrn and Jth, respectively.

YANG et al.: FAULT DETECTION FOR T–S FUZZY DISCRETE SYSTEMS

TABLE I C OMPARISONS OF THE O BTAINED R ESULTS W ITH OTHERS

In this figure, the dashed line denotes the threshold which is 0.6. We can conclude that, using the logic rule (33), the fault can be detected as k = 108. Remark 4: In this paper, we make a bridge between fault detection and fuzzy modeling in finite-frequency domain. The effects of modeling uncertainties and faults may occupy in different frequency ranges. Both weighting functions and frequency gridding have a hard time avoiding computational burden and guaranteeing gain property performances simultaneously. By using the GKYP Lemma, the fault detection problem can be dealt with in finite-frequency domain for linear invariant model but not any nonlinear cases. Some comparison results are given in Table I. Therefore, there exist some interesting contents in this paper.

V. C ONCLUSION This paper has presented a new approach to study the problem of fault detection for T–S fuzzy systems in finite-frequency ranges. We constructed a fuzzy fault detection filter system and dynamics of filtering error generator by means of the T–S fuzzy model. A fault detection observer has been designed by employing the two performance indices (H∞ and H− ) which are used to increase the fault sensitivity and attenuate the effects of disturbances in finite-frequency ranges. With the aid of the GKYP Lemma in local linear system model, the design method has been presented in terms of solutions to a set of linear matrix inequalities, which were readily solved via standard numerical software. A numerical example was given to illustrate the effectiveness and potential of the developed techniques. R EFERENCES [1] Y. Wang, W. Wang, and D. Wang, “LMI approach to design fault detection filter for discrete-time switched systems with state delays,” Int. J. Innov. Comput., Inf. Control, vol. 6, no. 1, pp. 387–398, Jan. 2010. [2] Q. Ding and M. Zhong, “On designing H∞ fault detection filter for Markovian jump linear systems with polytopic uncertainties,” Int. J. Innov. Comput., Inf. Control, vol. 6, no. 3(A), pp. 995–1004, Mar. 2010. [3] Z. Gu, D. Wang, and D. Yue, “Fault detection for continuous-time networked control systems with non-ideal QoS,” Int. J. Innov. Comput., Inf. Control, vol. 6, no. 8, pp. 3631–3640, Aug. 2010. [4] P. M. Frank and X. Ding, “Frequency domain approach to optimally robust residual generation and evaluation for model based fault diagnosis,” Automatica, vol. 30, no. 5, pp. 789–804, May 1994. [5] M. Kinnaert and Y. Peng, “Residual generator for sensor and actuator fault detection and isolation: A frequency domain approach,” Int. J. Control, vol. 61, no. 6, pp. 1423–1435, Jun. 1995. [6] D. Sauter and F. Hamelin, “Frequency domain optimization for robust fault detection and isolation in dynamic systems,” IEEE Trans. Autom. Control, vol. 44, no. 4, pp. 878–882, Apr. 1999. [7] P. Zhang, S. X. Ding, G. Z. Wang, and D. H. Zhou, “A frequency domain approach to fault detection in sampled-data systems,” Automatica, vol. 39, no. 7, pp. 1303–1307, Jul. 2003.

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Hongjiu Yang was born in Hebei Province, China, in 1981. He received the B.S. degree in mathematics and applied mathematics and the M.S. degree in applied mathematics from Hebei University of Science and Technology, Shijiazhuang, China, in 2005 and 2008, respectively. He is currently working toward the Ph.D. degree in control science and engineering at Beijing Institute of Technology, Beijing, China. His research interests include robust control/filter theory, delta operator systems, networked control systems, and stochastic systems.

Yuanqing Xia was born in Anhui Province, China, in 1971. He graduated from the Department of Mathematics, Chuzhou University, Chuzhou, China, in 1991, the M.S. degree in fundamental mathematics from Anhui University, Hefei, China, in 1998, and the Ph.D. degree in control theory and control engineering from Beijing University of Aeronautics and Astronautics, Beijing, China, in 2001. From 1991 to 1995, he was with Tongcheng Middle School, Tongcheng, China, where he was a Teacher. During January 2002–November 2003, he was a Postdoctoral Research Associate at the Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, where he worked on navigation, guidance, and control. From November 2003 to February 2004, he was with National University of Singapore, Singapore, as a Research Fellow, where he worked on variable structure control. From February 2004 to February 2006, he was with the University of Glamorgan, Pontypridd, U.K., as a Research Fellow, where he worked on networked control systems. From February 2007 to June 2008, he was a Guest Professor at Innsbruck Medical University, Innsbruck, Austria, where he worked on biomedical signal processing. Since July 2004, he has been with the Department of Automatic Control, Beijing Institute of Technology, Beijing, first as an Associate Professor and then as a Professor since 2008. His current research interests are in the fields of networked control systems, robust control, active disturbance rejection control and navigation, and guidance and control.

Bo Liu was born in the Ningxia Hui Autonomous Region, China, in 1982. He received the B.S. degree in automation from North China University of Technology, Beijing, China, in 2005, and the M.S. degree in automatic control from Beijing Institute of Technology, Beijing, in 2008, where he is currently working toward the Ph.D. degree in control science and engineering. His research interests include networked control systems, active disturbance rejection control, process control, and fault diagnosis.