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Fault detection in finite frequency domain for Takagi-Sugeno fuzzy systems with sensor faults Xiao-Jian Li and Guang-Hong Yang, Senior Member, IEEE
Abstract—This paper is concerned with the fault detection (FD) problem in finite frequency domain for continuous-time Takagi-Sugeno fuzzy systems with sensor faults. Some finitefrequency performance indices are initially introduced to measure the fault/reference input sensitivity and disturbance robustness. Based on these performance indices, an effective FD scheme is then presented such that the generated residual is designed to be sensitive to both fault and reference input for faulty cases, while robust against the reference input for fault-free case. As the additional reference input sensitivity for faulty cases is considered, it is shown that the proposed method improves the existing FD techniques and achieves a better FD performance. The theory is supported by simulation results related to the detection of sensor faults in a tunnel-diode circuit. Index Terms—Fault detection, finite frequency domain, sensor faults, T-S fuzzy systems.
I. Introduction
T
HE fault detection (FD) problem is a very important part of automatic control because of increasing requirements on safety, reliability and low maintenance costs [1]. Among the existing fault detection schemes, the celebrated frequency domain approach is accepted as a useful one, and many related results have been reported in the past two decades, e.g., [2]–[6]. Although these methods with FD performance indices defined in full frequency domain have been proven to be capable of successfully detecting and diagnosing certain
Manuscript received May 9, 2012; revised August 16, 2013; accepted October 11, 2013. This work was supported in part by the National 973 Program of China under Grant 2009CB320604, in part by the Funds of National Science of China under Grant 60974043 and Grant 61273148, in part by the Funds of Doctoral Program of Ministry of Education, China (No. 20100042110027), in part by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 201157), in part by the China Post-Doctoral Science Foundation (No. 2013M541241), in part by the Fundamental Research Funds for the Central Universities (No. N110804001 and No. N110305006), in part by the Nature Science of Foundation of Liaoning Province (No. 201202063), and in part by Fundamental Research Funds of State Key Laboratory of Synthetical Automation for Process Industries (No. 2013ZCX01). This paper was recommended by Associate Editor E. Santos. (Corresponding author G.-H. Yang). Xiao-Jian Li is with the the College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110004, China. (e-mail: [email protected]). Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China and also with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, 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/TCYB.2013.2286209
types of system faults, they are not completely compatible with practical requirements. Indeed, FD performance indices are often given not for the entire frequency range but rather for a certain frequency range of relevance. For instance, a FD system design typically requires high fault sensitivity in a low frequency range. Based on this observation, the finite frequency FD problems were investigated in [7]–[9] by introducing weighting functions. However, the design iterations to search for good weighting functions can be time consuming, and the FD systems complexity (order) tends to increase with the complexity of the weighting functions. To solve this problem, generalized Kalman-Yakubovic-Popov (GKYP) lemma [10] based finite frequency FD methods were given in [11]–[13]. Note that, all references previously cited deal with linear systems, whereas in many practical situations, nonlinear properties of the monitored system cannot be neglected for the purpose of FD. For this reason, nonlinear system FD has also become an active research topic. Typical examples of such researches are nonlinear geometric and observer-based approaches [14]–[17], and algebraic approaches [18], etc. However, it is hard to use these existing techniques to address the finite frequency FD problem for nonlinear systems, which motivates this paper. Recently, it has been shown that the Takagi-Sugeno (T-S) fuzzy models described by a set of IF-THEN rules could approximate any smooth nonlinear function to any specified accuracy within any compact set. Thus it is expected that the T-S fuzzy systems can be used to represent a large class of nonlinear systems, and many important results on the T-S fuzzy systems have been reported in the literature, see [19]–[24]. Furthermore, a FD filter design approach was developed in [25] for uncertain T-S fuzzy systems; in [26], the robust FD filter design problem was formulated as an H∞ filtering problem for T-S fuzzy Itˆo stochastic systems; under the assumption that the communication links between the plant and the FD filter are imperfect, the problem of FD was investigated in [27] for T-S fuzzy systems with intermittent measurements; in [28], a switching-type FD filter with varying gains was designed for T-S fuzzy systems with unknown membership functions. Especially, based on the work of [29], [30] has investigated the finite frequency FD approach for discretetime T-S fuzzy systems. However, the results in [30] still leave room for further improvement and completeness. First, the definitions of finite frequency FD performance indices and the corresponding FD systems design conditions cannot be applied
c 2013 IEEE 2168-2267
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for continuous-time T-S fuzzy systems. Second, as only the fault sensitivity is considered in [30], it is required that the magnitudes of faults should be large enough such that they can be detected. Third, the proposed FD method in [30] cannot be used to address the T-S fuzzy systems with multiple faulty models. This paper is concerned with the FD problem in finite frequency domain for continuous-time T-S fuzzy systems subject to sensor stuck faults. Some FD performance indices are initially defined in finite frequency domain, which extend the known ones defined in full frequency range. Then, considered all the possible sensor stuck faults, the T-S fuzzy systems are modeled via multimodels, i.e., fault-free model and faulty models. A new fault detection scheme is then developed such that the generated residual is designed to be sensitive to both finite frequency fault signals and reference inputs for the faulty cases, while robust against the reference inputs for the fault-free case. Since the additional reference input sensitivity for the faulty cases is considered, it is shown that both sensor faults with large magnitudes and those with small magnitudes can be detected within the proposed FD framework. In addition, the FD filter design conditions given in this paper improve the existing results in [25], where some FD filter gains have to be fixed beforehand. The following notations are used throughout this paper. A block diagonal matrix with matrices X1 , X2 , · · · Xn on its main diagonal is denoted as diag(X1 , X2 , · · · Xn ). For a symmetric matrix, A > 0 and A < 0 denote positive definiteness and negative definiteness. The symmetric terms in a symmetric matrix are denoted by ∗. For a matrix A, its transpose and complex conjugate transpose are denoted by AT and A∗ , ∗ He(A) =: A + AT , He(A) =: A+A . L2 represents the Hilbert 2 space of square integrable functions with the following norm ∞ 1 : v(t) 2 = { 0 vT (t)v(t)dt} 2 .
II. Preliminaries and Problem Statement A. System description The T-S fuzzy model is described by fuzzy IF-THEN rules, whose collection represent the approximation of the nonlinear system. The ith rule of the T-S fuzzy model is of the following form. Plant Rule i: IF z1 (t) is μi1 and, · · · , and zs (t) is μis , THEN x˙ (t) = Ai x(t) + Bi d(t) + Bri yr (t) y(t) = Ci x(t) + Di d(t)
(1) (2)
where x(t) ∈ Rn is the state space vector, y(t) ∈ Rny is the system output vector, d(t) ∈ Rnd and yr (t) ∈ Rnr denote the disturbance and finite frequency reference input, which belong to L2 . Ai , Bi , Bri , Ci and Di are known matrices with appropriate dimensions. z1 (t), z2 (t), · · · zs (t) are premise variables and μij (i = 1, 2, · · · q, j = 1, 2, · · · s) are fuzzy sets, q is the number of IF-THEN rules, and s is the number of premise variables. The overall fuzzy model achieved by
fuzzy-blending of each individual plant rule (local model) is given by x˙ (t) =
q
hi (z(t))[Ai x(t) + Bi d(t) + Bri yr (t)]
(3)
hi (z(t))[Ci x(t) + Di d(t)]
(4)
i=1 q
y(t) =
i=1
where σi (z(t)) μij (zj (t)) , σi (z(t)) = q j=1 σi (z(t)) s
hi (z(t)) =
(5)
i=1
and μij (zj (t)) is the grade of the membership of zj (t) in μij . We assume σi (z(t)) ≥ 0, i = 1, 2, · · · q,
q
σi (z(t)) > 0
(6)
i=1
for any z(t). Therefore, for all t we have hi (z(t)) ≥ 0, i = 1, 2, · · · q,
q
hi (z(t)) = 1.
(7)
i=1
For simplicity, we introduce the following notations: hi = hi (z(t)), A(h) =
q
hi Ai , B(h) =
i=1
Br (h) =
q i=1
hi Bri , C(h) =
q
h i Bi
i=1 q
hi Ci , D(h) =
i=1
q
hi Di .
i=1
Then, the T-S fuzzy models (3) and (4) can be rewritten as x˙ (t) = A(h)x(t) + B(h)d(t) + Br (h)yr (t) y(t) = C(h)x(t) + D(h)d(t).
(8) (9)
Remark 1: Note that the reference input yr (t) is directly applied to the open-loop system (8). In fact, the system given in (8) and (9) represents a general tracking system, which has been studied in [31] and [32], and can be derived as follows. the system (8) T with Br (h) = 0, define a new variable xˆ (t) = For t e(τ)dτ x(t) , where e(t) = yr (t) − y(t) is the tracking 0 error. Let 0 −C(h) ˆ 0 ˆ A(h) = B(h) = 0 A(h) B(h)
I ˆ ˆ r (h) = B C(h) = 0 C(h) 0 then we have the following augmented system ˆ x(t) + B(h)d(t) ˆ ˆ r (h)yr (t) xˆ˙ (t) = A(h)ˆ +B ˆ y(t) = C(h)ˆx(t) + D(h)d(t) which is in the same form of (8) and (9). Definition 1: (Sensor stuck fault) when sensor stuck faults occur, the sensor output is given by
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LI AND YANG et al.: FAULT DETECTION IN FINITE FREQUENCY DOMAIN FOR TAKAGI-SUGENO FUZZY SYSTEMS WITH SENSOR FAULTS
yF (t) = Fm y(t) + (I − Fm )fm , m = 0, 1, · · · l
¯ m (h)η(t) + B ¯ dm (h)d(t) + B ¯ Kr (h)yr (t) η˙ (t) =A
(10)
¯ fm (h)f (t) +B ¯ m (h)η(t) + D ¯ dm (h)d(t) + D ¯ Kr (h)yr (t) r(t) =C
where l denotes the total number of fault models, and
T fm = fm1 · · · fmk · · · fmny
¯ fm (h)f (t) +D
with f mk ≤ fmk ≤ f mk (k = 1, 2, · · · ny ) denoting the stuck value of the kth sensor, f mk and f mk being known scalars. Fm are diagonal matrices in the form of Fm = diag Fm1 , · · · Fmk , · · · Fmny Fmk = 0 or 1, k = 1, 2, · · · ny .
(11)
Remark 2: If Fmk = 0 for 1 ≤ k ≤ ny , the fault model (10) corresponds to the case that the kth sensor gets faulty. If Fmk = 1, the kth sensor is fault-free. Without loss of generality, we assume that F0 = I, namely, m = 0 corresponds to the fault-free case. From (8)–(10), we have x˙ (t) = A(h)x(t) + B(h)d(t) + Br (h)yr (t)
(12)
yF (t) = Fm C(h)x(t) + Fm D(h)d(t) + (I − Fm )fm
(13)
m = 0, 1, · · · l. B. Fault detection filter In order to detect the sensor faults, a fuzzy FD filter is designed as follows. Plant Rule i: IF z1 (t) is μi1 and z2 (t) is μi2 and,· · · , and zs (t) is μis , then x˙ f (t) = Afi xf (t) + Bfi yF (t) + Bfri yr (t) F
r(t) = Cfi xf (t) + Dfi y (t) + Dfri yr (t)
F
r(t) = CK (h)xf (t) + DK (h)y (t) + DKr (h)yr (t)
AK (h)n×n =
hi Afi BK (h)n×ny =
q
i=1 q
CK (h)1×n =
(16) (17)
hi Cfi , DK (h)1×ny =
i=1 q
BKr (h)n×nr =
hi Bfi
i=1 q
i=1
A(h) 0 BK (h)Fm C(h) AK (h) B(h) Br (h) ¯ ¯ Bdm (h) = BKr (h) = BKr (h) BK (h)Fm D(h) 0 ¯ fm (h) = ¯ Kr (h) = DKr (h) B D BK (h)(I − Fm )fm ¯ fm = DK (h)(I − Fm )fm , D ¯ dm (h) = DK (h)Fm D(h) D
¯ m (h) = DK (h)Fm C(h) CK (h) , f (t) = 1. C (20)
¯ m (h) = A
C. Finite frequency performance indices In this section, some performance indices are defined in finite frequency domain for the purpose of FD. In practice, the effects of faults and reference inputs may occupy in different frequency ranges. Therefore, we consider the following two finite frequency interval for frequency ω both in fault f (t) and reference input yr (t): f : = {ω ∈ R | τ(ω − f1 )(ω − f2 ) ≤ 0} yr : = {ω ∈ R | τ(ω − r1 )(ω − r2 ) ≤ 0}
hi Dfi
(21) (22)
where f1 , f2 , r1 , r2 are given real scalars. Note that, when τ = +1, f1 = −f2 = f and r1 = −r2 = r , f and yr denote the low frequency range lf : = {ω ∈ R || ω |≤ f }
(23)
: = {ω ∈ R || ω |≤ r }
(24)
lyr
where f and r are assumed to be known positive real scalars. If τ = −1 and f1 = −f2 (r1 = −r2 ), or τ = +1 and f1 ≤ f2 (r1 ≤ r2 ), f (yr ) denotes the high frequency range or middle frequency range. In the sequel, we give the following two definitions on finite frequency H∞ and H− performance indices for the fuzzy system (18), (19). Definition 2: The fuzzy system (18), (19) has a finite frequency H∞ index bound γ, if under zero initial condition, the following inequality
+∞
r (t)r(t)dt ≤ γ T
i=1 q
hi Bfri , DKr (h)1×nr =
(19)
where
(15)
where q
(18)
(14)
where xf (t) ∈ Rn is the filter state, r(t) ∈ R denotes the residual signal which carries information on the time of the occurrence of a fault, Afi , Bfi , Bfri , Cfi , Dfi , Dfri are filter gains to be determined. Thus, the FD filter can be represented by the following form: x˙ f (t) = AK (h)xf (t) + BK (h)yF (t) + BKr (h)yr (t)
3
hi Dfri .
i=1
T Let η(t) = xT (t) xfT (t) , the augmented dynamics can be described by
0
2 0
+∞
yrT (t)yr (t)dt
(25)
holds for all solutions of (18) with yr (t) ∈ L2 such that
+∞
τ(r1 η + j η˙ )(r2 η − j η˙ )T dt ≤ 0
(26)
0
where τ, r1 and r2 reflecting the frequency range of reference input yr (t) are given in (22), and γ is a given real
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positive scalar which denotes the worst case criterion for the effect of reference input yr (t) on the residual r(t). The smaller the γ is, the more robust the generator becomes. Definition 3: The fuzzy system (18), (19) has a finite frequency H− index bound β, if under zero initial condition, the following inequality:
+∞
rT (t)r(t)dt ≥ β2
0
+∞
f T (t)f (t)dt
(27)
0
holds for all solutions of (18) with f (t) ∈ L2 such that
+∞ τ(f1 η + j η˙ )(f2 η − j η˙ )T dt ≤ 0 (28)
2) For both fault-free and faulty cases (m = 0, 1, · · · l), to minimize the effect of the disturbance d(t) on the residual output r(t), it needs to satisfy
+∞
+∞ r T (t)r(t)dt ≤ γ 2 d T (t)d(t)dt, m = 0, 1, · · · l 0
(31)
with d(t) ∈ L2 . 3) For faulty cases (m = 1, · · · l), to maximize the effects of the reference input yr (t) and fault f (t) on the residual output r(t), it needs to satisfy the following H− performances:
0
where τ, f1 and f2 reflecting the frequency range of fault f (t) are given in (21), β is a given real positive scalar, which is a measurement of the fault sensitivity in the worst case from f (t) to residual r(t). The larger the β is, the more sensitive the generator becomes. Remark 3: Definitions two and three are motivated by [33], which can be regarded as an extension of the FD performance indices defined in full frequency domain [25]. In fact, when τ = −1 and f1 = f2 = r1 = r2 = 0, the constraints (26) and (28) are automatically satisfied and the sets f and yr , in (21) and (22), characterize the full frequency domain. In this case, the Definitions two and three reduce to the ones given in [25]. Remark 4: The physical meaning of Definition two can become clear when special cases are considered. For example, in the low frequency range lyr , we have τ = 1 and r1 = −r2 = r , which implies that (26) is equivalent to +∞ +∞ η˙ η˙ T dt ≤ r2 0 ηηT dt. In this case, by [33], the Defini0 tion two means that the system (18) and (19) possesses property (25) with respect to the reference input yr (t) that does not drive the states too quickly where the bound on the quickness +∞ +∞ is given by r in the sense of 0 η˙ η˙ T dt ≤ r2 0 ηηT dt. Similar interpretations can be drawn for the Definition three.
0
+∞
0
r T (t)r(t)dt ≥ βr2
+∞
0
yrT (t)yr (t)dt, m = 1, · · · l (32)
with yr (t) ∈ L2 such that
+∞ τ(r1 η + j η˙ )(r2 η − j η˙ )T dt ≤ 0
(33)
0
and
+∞
0
r T (t)r(t)dt ≥ βf2
+∞
f T (t)f (t)dt, m = 1, · · · l
0
(34)
with f (t) ∈ L2 such that
+∞
τ(f1 η + j η˙ )(f2 η − j η˙ )T dt ≤ 0
(35)
0
where τ, f1 and f2 are given in (21), βr and βf are prescribed H− indices. E. Comparison With the Existing FD Techniques In the existing FD techniques, the sensor faults are always represented by the discrepancy of sensor outputs and inputs
D. Fault detection scheme Based on the Definitions two and three, the FD problem to be addressed in this paper is formulated as follows: given the fuzzy system (8) and (9) with sensor faults described in (10), determine a FD filter in the form of (16) and (17) such that the following requirements are satisfied. 1) For fault-free case (m = 0), minimize the effect of the reference input yr (t) on the residual output r(t). Then, given a prescribed suitable H∞ index γ, the following H∞ performance should be satisfied for the fuzzy system (18) and (19):
+∞
+∞ 2 T r (t)r(t)dt ≤ γ yrT (t)yr (t)dt, m = 0 (29) 0
0
with yr (t) ∈ L2 such that
+∞ τ(r1 η + j η˙ )(r2 η − j η˙ )T dt ≤ 0 0
where τ, r1 and r2 are given in (22).
(30)
f1 (t) =: yF (t) − y(t) = (Fm − I)(y(t) − fm ).
(36)
Then the fuzzy system (12), (13) with sensor faults can be rewritten in the following form which is widely studied in the literature: ˜ ˜ d(t) x˙ (t) = A(h)x(t) + B(h) ˜ + f1 (t) ˜ d(t) yF (t) = C(h)x(t) + D(h)
(37) (38)
where
˜ ˜ = D(h) 0 B(h) = B(h) Br (h) , D(h)
˜ = d(t) yr (t) T . d(t) To detect the fault f1 (t), an FD filter was introduced in [25] to generate a residual r(t) such that the following requirements are satisfied:
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+∞
0 +∞
r T (t)r(t)dt ≤ γ 2 r (t)r(t)dt ≥ β T
0
+∞
˜ d˜ T (t)d(t)dt
(39)
f1T (t)f1 (t)dt
(40)
0 +∞
2 0
where γ and β are the H∞ and H− indices, which should be minimized and maximized, respectively. Although the above FD techniques considered in full frequency domain can be used to detect sensor faults, the FD result in [25] will be conservative in some degree if the faults lie in finite frequency ranges. In addition, as only the fault sensitivity performance (40) is considered in [25], the fault f1 (t) with small magnitude cannot be detected, and this conclusion has been demonstrated by [34]. Recently, the following discrete-time T-S fuzzy system with actuator and sensor faults was investigated in [30] x(k + 1) = A(h)x(k) + Bd (h)d(k) + Bf (h)f (k)
(41)
y(k) = C(h)x(k) + Dd (h)d(k) + Df (h)f (k)
(42)
where x(k) is the state variable, y(k) is the measurement output, d(k) represents the disturbance input, and f (k) denotes the fault signal. A(h), Bd (h), Bf (h), C(h), Dd (h), Df (h) are known constant real matrices with appropriate dimensions. A FD observer in the form of (43)–(45) xˆ (k + 1) = A(h)ˆx(k) + L(h)(y(k) − yˆ (k)) yˆ (k) = C(h)ˆx(k) r(k) = y(k) − yˆ (k)
(43) (44) (45)
was designed in [30] to satisfy the following finite frequency performance indices: r(k) 2 ≤ γ d(k) 2 , ωd1 ≥ ω ≥ ωd2 r(k) 2 ≥ β f (k) 2 , ωf1 ≥ ω ≥ ωf2
(46) (47)
where L(h) in (43) is an observer gain matrix to be determined, r(k) in (45) is the generated residual, ωd1 , ωd2 , ωf1 and ωf2 in (46) and (47) are given real positive scalars that reflect the individual frequency ranges of disturbance d(k) and fault f (k), γ and β are the prescribed H∞ and H− indices. The simulation results have illustrated the advantages of the finite frequency FD approach given in [30]. However, similar to [25], the method of [30] applied to an ideal case where the magnitude of actuator or sensor fault should be large enough such that it can be detected. Furthermore, the FD performance indices and the corresponding FD observer design conditions in [30] cannot be applied to the continuous-time T-S fuzzy systems. In this paper, we initially define some finite frequency FD performance indices for continuous-time T-S fuzzy systems. Based on these performance indices, a novel FD scheme is then developed such that the generated residual is designed to be sensitive to both fault signal and reference input for the faulty cases, while robust against reference input for the fault free case. As the additional reference input sensitivity
5
is considered, the proposed method improves the existing FD techniques and guarantees that either fault with large magnitude or the one with small magnitude can be detected. The theory will be supported by simulation results related to the detection of sensor faults in a tunnel-diode circuit in Section V.
III. Main Results A. Finite Frequency H∞ and H− Performances Analysis In practice, faults usually emerge in low frequency domain, e.g., for an incipient signal, the fault information is always contained within a low frequency band as the fault development is slow, and the constant stuck fault just belongs to low frequency domain [36]. Therefore, the fault in this paper is assumed to belong to low frequency range lf in (23), in addition, the reference input is also assumed to lie in low frequency range lyr in (24). Based on the definitions of lf and lyr , the following lemmas will be given to provide the analysis conditions for the fuzzy system (18), (19) satisfying the finite frequency H∞ performance (29), (30) and H− performance (32), (33). Lemma 1: Assume that the system (18) is asymptotically stable. Then the system (18), (19) satisfies the H∞ performance (29), (30) in low frequency domain lyr , if there exist symmetric matrices P = P T , Q = QT such that τQ ≥ 0 and
¯ 0 (h) B ¯ 0 (h) B ¯ Kr (h) T ¯ Kr (h) A A
+ I 0 I 0 ¯ 0 (h) D ¯ 0 (h) D ¯ Kr (h) T ¯ Kr (h) C C 0, P m = Pb 11 Pb 12 Qm = m a b Qm Pbm 12 Pbm 22 b 12 Qb 22 ˆ fi , B ˆ fi , Cfi , Dfi , Y, M, N, A such that, for fmk ∈ {f mk , f mk }(k = 1, 2, · · · ny ), the following inequality holds: ⎡ ⎢ ⎢ ⎢ b ij = ⎢ ⎢ ⎢ ⎣
such that the following inequality holds: ⎡ ⎢ ⎢ ⎢ ij = ⎢ ⎢ ⎢ ⎣
m −Qm a 11 −Qa 12 ∗ −Qm a 22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 T CiT FmT Dfj − l11 2T Cfj − l12 2
−2I ∗
Pam 11 − Y T Pam 12 + M T Pam T12 − N T Pam 22 − N T a33 a34 ∗ a44 ∗ ∗ ∗ ∗ ⎤ −Y T v −N T v ⎥ ⎥ ⎥ a36 ⎥ ⎥ 0 are weighting factors which should be given beforehand. Then, matrices Cfi , Dfi , Dfri are obtained, and other filter parameters can be determined according to the following equalities: ˆ fi , Bfi = N −1 B ˆ fi , Bfri = N −1 B ˆ fri Afi = N −1 A i = 1, 2, · · · q.
(60)
Remark 9: Based on the assumption that the measurable output matrices are common for each local model, two effective full frequency fault estimation approaches have been
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developed in [40] and [41] for discrete-time T-S fuzzy systems. Unlike [40] and [41], this paper is concerned with the finite frequency fault detection problem for continuous-time T-S fuzzy systems, and the above mentioned assumption are not required. Especially, the proposed fault sensitivity and reference input sensitivity based FD scheme is valid for both sensor faults with small magnitudes and those with large magnitudes. IV. Detection Threshold Design In this section, the threshold for detecting faults is designed and the detection logic unit is based on the results proposed by [38]. Let the root mean square value which means the average energy of residual signal over a time interval (0, t) 1 t T Jr (t) = r rms := r (τ)r(τ)dτ t 0
(61)
be a residual evaluation function. The threshold Jth is determined by Jth =
sup
Jr (t)
(62)
f (t)=0,d(t)∈L2 ,
and faults can be detected using the following logical relationship : Jr > Jth ⇒ alarm, Jr ≤ Jth ⇒ no faults.
(63)
Fig. 1. yF (t) − y(t) with the stuck value being 0.1.
Sensor 2 stuck fault is considered, therefore, we have 10 0 F1 = in (10), where the stuck value is and f1 = 00 f12 assumed to satisfy −1 ≤ f12 ≤ 1. −0.1 b Given f = 0.01, r = 1, γ = 1, l11 = l11 = , l12 = −0.1 −0.1 0.1 b l12 = and l2 = −l2b = 0.3, v = −vb = 0.3 −0.01 solving the optimization problem (59) with λr = λf = 1, we have
V. Examples In this section, the following example is provided to illustrate the effectiveness of the theoretical results developed. Consider a tunnel diode circuit, whose fuzzy modeling was done in [42] x˙ (t) = y(t) =
2 i=1 2
hi (x1 (t))[Ai x(t) + Bi d(t) + Bri yr (t)] hi (x1 (t))[Ci x(t) + Di d(t)]
(64)
i=1
where x1 (t) = vC (t), x2 (t) = iL (t), and vC (t), iL (t) are the capacitor voltage and inductance current, respectively. The parameter matrices are given by
−0.1 50 −4.6 50 0 A1 = , A2 = , B1 = −1 −10 −1 −10 1 0 10 10 B2 = , C1 = , C2 = 1 01 01 1 1 0 0 Br1 = , Br2 = , D1 = , D2 = . 0 0 0 0 √ √ Assume that x1 (t) ∈ [− 3, 3], and the membership functions for rules one and two are 1 h1 (x1 (t)) = 1 − x12 (t), h2 (x1 (t)) = 1 − h1 (x1 (t)). 3
−43.9654 32.0266 −5.3313 Af 1 = , Bfr1 = −9.0744 −9.9612 −0.9938 −49.3718 35.3985 −5.9094 Af 2 = , Bfr2 = −8.8258 −10.1135 −1.0428
−46.6862 −8.0058 Bf 1 = , Dfr1 = −0.9440 −9.0707 −1.3203
−52.6680 −8.0446 Bf 2 = , Dfr2 = −0.9392 −8.9898 −1.3076
Cf 1 = 0.0766 −1.2544 , Cf 2 = 0.1510 −1.1343
Df 1 = 0.0770 −1.7938 , Df 2 = 0.2120 −1.7198 and the optimal values on the sensitivity performance indexes are βr = 0.5263 and βf = 0.6160. In simulation, the disturbance is assumed to be d(t) = 0.1sin(0.5t) and reference input yr (t) = 1. Two failure cases are considered respectively. Case 1: Sensor 2 is stuck at 0.1 from t = 100s. To show the magnitude of the sensor fault, f1 (t) defined in (36) is plotted in Fig. 1. In addition, the absolutes of the generated residuals using the proposed FD approach (red) and the existing methods (blue) given in Section II are plotted in Fig. 2. It is easy to see that the generated residual in this paper is more sensitive to fault, and the results in Fig. 3 indicate that the fault is detected almost immediately at approximately t = 100s. On the other hand, since the magnitude of f1 (t) is lager than that of disturbance d(t), the generated residual (shown in Fig. 2) using the existing methods without con-
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Fig. 2.
Generated residual with the stuck value being 0.1.
Fig. 3.
Residual evaluation and threshold with the stuck value being 0.1.
sidering reference input sensitivity also has large discrepancy before and after the occurrence of fault f1 (t), which implies that the existing methods are also valid in Case 1. Case 2: Sensor 2 is stuck at −0.01 from t = 100s. The fault f1 (t) defined in (36) is plotted in Fig. 4, from which it can be seen that, the magnitude of f1 (t) in Case 2 is much smaller than that of disturbance d(t). The residual response and residual evaluation response obtained in this paper are displayed in Figs. 5 and 6, respectively. As shown in these two figures, the sensor fault with small magnitude can be detected within the proposed FD framework. Moreover, the comparison results between Figs. 3 and 6 illustrate that the detection time obviously increases as the magnitude of fault decreases. In Case 2, the residual response derived by using the existing FD techniques given in Section II is plotted in Fig. 5 (blue). Combining Figs. 2 and 5, it can be concluded that the existing FD techniques are applicable for the case that the magnitude of fault is large enough, on the contrary, when it is
9
Fig. 4. yF (t) − y(t) with the stuck value being -0.01.
Fig. 5.
Generated residual with the stuck value being -0.01.
very small, the existing FD approaches cannot work. This is because that the strong detectability condition which requests that the magnitude of fault should be large enough is not satisfied in Case 2. In this paper, since the additional reference input sensitive performance is considered for the faulty cases, the fault with relative small magnitude can still be detected even if such strong detectability condition is not satisfied, which further exhibits the significance of our proposed FD approach. VI. Conclusion In this paper, an effective finite frequency FD approach for continuous-time T-S fuzzy systems subject to sensor stuck faults has been investigated, where the generated residual is designed to be sensitive to both fault signals and reference inputs for the faulty cases. The proposed FD scheme has improved the existing FD techniques, more specifically, both sensor faults with small magnitudes and those with large magnitudes can be detected within the proposed FD framework.
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T d T r r − (η Pη) + yr yr dt tr[He((r1 η + j η˙ )(r2 η − j η˙ )T )Q] ≤ 0. Integrating from t = 0 to ∞ and using the stability property, we have
∞
0
T
r yr
r dt ≤ tr[He(S)Q] yr
(67)
(r1 η + j η˙ )(r2 η − j η˙ )T )dt.
(68)
where
S=
∞
0
Fig. 6.
Residual evaluation and threshold with the stuck value being -0.01.
Moreover, the advantages of the proposed FD approach over the available ones have been illustrated through a numerical example.
By the Parseval’s theorem [35], we have
−∞ 1 (r1 − ω)(r2 − ω)ˆηηˆ T dω S= 2π ∞ where ηˆ is the Fourier transform of η. Thus S is Hermitian and the bound on the righthand side of (67) becomes tr(SQ). Since τQ ≥ 0, the bound tr(SQ) is nonpositive whenever τS ≤ 0 holds, then the proof is completed.
Acknowledgment The authors would like to thank the anonymous reviewers and editors for their useful comments and suggestions.
Appendix A Lemma 3: (Projection Lemma) Let U, V, be given. There exists a matrix F satisfying T
T
U FV + V FU + < 0 if and only if the following two conditions hold NUT NU < 0
(65)
NVT NV < 0
(66)
where NU and NV are arbitrary matrices whose columns form a basis of the nullspaces of U and V , respectively. Lemma 4: (Finsler’s Lemma) Let ξ ∈ Cn , Q ∈ Cn×n and H ∈ Cn×m . Let H⊥ be any matrix such that H⊥ H = 0. The following statement are equivalent: i) ξ ∗ Qξ < 0, ∀H∗ ξ = 0, ξ = 0, ∗ ii) H⊥ QH⊥ < 0, iii) ∃μ ∈ R : Q − μHH∗ < 0, iv) ∃X ∈ Rm×n : Q + HX + X ∗ H∗ < 0.
Appendix B Proof of Lemma 1. From (24), we have τ = +1, r1 = −r2 = r . Then, pre and postmultiplying (48) by [ηT yrT ] and its transpose, it follows that
Appendix C Proof of Theorem 1. Based on Lemma 2, the system dynamics (18) and (19) satisfy the high gain performance (32), (33) in low frequency domain if there exist matrices mT Pam = Pam T , Qm > 0 such that the following inequality a = Qa holds ¯ m (h) B ¯ m (h) B ¯ Kr (h) T ¯ Kr (h) A A
m + I 0 I 0 T ¯ m (h) D ¯ m (h) D ¯ Kr (h) ¯ Kr (h) C C 1
1
Fault detection in finite frequency domain for Takagi-Sugeno fuzzy systems with sensor faults Xiao-Jian Li and Guang-Hong Yang, Senior Member, IEEE
Abstract—This paper is concerned with the fault detection (FD) problem in finite frequency domain for continuous-time Takagi-Sugeno fuzzy systems with sensor faults. Some finitefrequency performance indices are initially introduced to measure the fault/reference input sensitivity and disturbance robustness. Based on these performance indices, an effective FD scheme is then presented such that the generated residual is designed to be sensitive to both fault and reference input for faulty cases, while robust against the reference input for fault-free case. As the additional reference input sensitivity for faulty cases is considered, it is shown that the proposed method improves the existing FD techniques and achieves a better FD performance. The theory is supported by simulation results related to the detection of sensor faults in a tunnel-diode circuit. Index Terms—Fault detection, finite frequency domain, sensor faults, T-S fuzzy systems.
I. Introduction
T
HE fault detection (FD) problem is a very important part of automatic control because of increasing requirements on safety, reliability and low maintenance costs [1]. Among the existing fault detection schemes, the celebrated frequency domain approach is accepted as a useful one, and many related results have been reported in the past two decades, e.g., [2]–[6]. Although these methods with FD performance indices defined in full frequency domain have been proven to be capable of successfully detecting and diagnosing certain
Manuscript received May 9, 2012; revised August 16, 2013; accepted October 11, 2013. This work was supported in part by the National 973 Program of China under Grant 2009CB320604, in part by the Funds of National Science of China under Grant 60974043 and Grant 61273148, in part by the Funds of Doctoral Program of Ministry of Education, China (No. 20100042110027), in part by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 201157), in part by the China Post-Doctoral Science Foundation (No. 2013M541241), in part by the Fundamental Research Funds for the Central Universities (No. N110804001 and No. N110305006), in part by the Nature Science of Foundation of Liaoning Province (No. 201202063), and in part by Fundamental Research Funds of State Key Laboratory of Synthetical Automation for Process Industries (No. 2013ZCX01). This paper was recommended by Associate Editor E. Santos. (Corresponding author G.-H. Yang). Xiao-Jian Li is with the the College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110004, China. (e-mail: [email protected]). Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China and also with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, 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/TCYB.2013.2286209
types of system faults, they are not completely compatible with practical requirements. Indeed, FD performance indices are often given not for the entire frequency range but rather for a certain frequency range of relevance. For instance, a FD system design typically requires high fault sensitivity in a low frequency range. Based on this observation, the finite frequency FD problems were investigated in [7]–[9] by introducing weighting functions. However, the design iterations to search for good weighting functions can be time consuming, and the FD systems complexity (order) tends to increase with the complexity of the weighting functions. To solve this problem, generalized Kalman-Yakubovic-Popov (GKYP) lemma [10] based finite frequency FD methods were given in [11]–[13]. Note that, all references previously cited deal with linear systems, whereas in many practical situations, nonlinear properties of the monitored system cannot be neglected for the purpose of FD. For this reason, nonlinear system FD has also become an active research topic. Typical examples of such researches are nonlinear geometric and observer-based approaches [14]–[17], and algebraic approaches [18], etc. However, it is hard to use these existing techniques to address the finite frequency FD problem for nonlinear systems, which motivates this paper. Recently, it has been shown that the Takagi-Sugeno (T-S) fuzzy models described by a set of IF-THEN rules could approximate any smooth nonlinear function to any specified accuracy within any compact set. Thus it is expected that the T-S fuzzy systems can be used to represent a large class of nonlinear systems, and many important results on the T-S fuzzy systems have been reported in the literature, see [19]–[24]. Furthermore, a FD filter design approach was developed in [25] for uncertain T-S fuzzy systems; in [26], the robust FD filter design problem was formulated as an H∞ filtering problem for T-S fuzzy Itˆo stochastic systems; under the assumption that the communication links between the plant and the FD filter are imperfect, the problem of FD was investigated in [27] for T-S fuzzy systems with intermittent measurements; in [28], a switching-type FD filter with varying gains was designed for T-S fuzzy systems with unknown membership functions. Especially, based on the work of [29], [30] has investigated the finite frequency FD approach for discretetime T-S fuzzy systems. However, the results in [30] still leave room for further improvement and completeness. First, the definitions of finite frequency FD performance indices and the corresponding FD systems design conditions cannot be applied
c 2013 IEEE 2168-2267
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for continuous-time T-S fuzzy systems. Second, as only the fault sensitivity is considered in [30], it is required that the magnitudes of faults should be large enough such that they can be detected. Third, the proposed FD method in [30] cannot be used to address the T-S fuzzy systems with multiple faulty models. This paper is concerned with the FD problem in finite frequency domain for continuous-time T-S fuzzy systems subject to sensor stuck faults. Some FD performance indices are initially defined in finite frequency domain, which extend the known ones defined in full frequency range. Then, considered all the possible sensor stuck faults, the T-S fuzzy systems are modeled via multimodels, i.e., fault-free model and faulty models. A new fault detection scheme is then developed such that the generated residual is designed to be sensitive to both finite frequency fault signals and reference inputs for the faulty cases, while robust against the reference inputs for the fault-free case. Since the additional reference input sensitivity for the faulty cases is considered, it is shown that both sensor faults with large magnitudes and those with small magnitudes can be detected within the proposed FD framework. In addition, the FD filter design conditions given in this paper improve the existing results in [25], where some FD filter gains have to be fixed beforehand. The following notations are used throughout this paper. A block diagonal matrix with matrices X1 , X2 , · · · Xn on its main diagonal is denoted as diag(X1 , X2 , · · · Xn ). For a symmetric matrix, A > 0 and A < 0 denote positive definiteness and negative definiteness. The symmetric terms in a symmetric matrix are denoted by ∗. For a matrix A, its transpose and complex conjugate transpose are denoted by AT and A∗ , ∗ He(A) =: A + AT , He(A) =: A+A . L2 represents the Hilbert 2 space of square integrable functions with the following norm ∞ 1 : v(t) 2 = { 0 vT (t)v(t)dt} 2 .
II. Preliminaries and Problem Statement A. System description The T-S fuzzy model is described by fuzzy IF-THEN rules, whose collection represent the approximation of the nonlinear system. The ith rule of the T-S fuzzy model is of the following form. Plant Rule i: IF z1 (t) is μi1 and, · · · , and zs (t) is μis , THEN x˙ (t) = Ai x(t) + Bi d(t) + Bri yr (t) y(t) = Ci x(t) + Di d(t)
(1) (2)
where x(t) ∈ Rn is the state space vector, y(t) ∈ Rny is the system output vector, d(t) ∈ Rnd and yr (t) ∈ Rnr denote the disturbance and finite frequency reference input, which belong to L2 . Ai , Bi , Bri , Ci and Di are known matrices with appropriate dimensions. z1 (t), z2 (t), · · · zs (t) are premise variables and μij (i = 1, 2, · · · q, j = 1, 2, · · · s) are fuzzy sets, q is the number of IF-THEN rules, and s is the number of premise variables. The overall fuzzy model achieved by
fuzzy-blending of each individual plant rule (local model) is given by x˙ (t) =
q
hi (z(t))[Ai x(t) + Bi d(t) + Bri yr (t)]
(3)
hi (z(t))[Ci x(t) + Di d(t)]
(4)
i=1 q
y(t) =
i=1
where σi (z(t)) μij (zj (t)) , σi (z(t)) = q j=1 σi (z(t)) s
hi (z(t)) =
(5)
i=1
and μij (zj (t)) is the grade of the membership of zj (t) in μij . We assume σi (z(t)) ≥ 0, i = 1, 2, · · · q,
q
σi (z(t)) > 0
(6)
i=1
for any z(t). Therefore, for all t we have hi (z(t)) ≥ 0, i = 1, 2, · · · q,
q
hi (z(t)) = 1.
(7)
i=1
For simplicity, we introduce the following notations: hi = hi (z(t)), A(h) =
q
hi Ai , B(h) =
i=1
Br (h) =
q i=1
hi Bri , C(h) =
q
h i Bi
i=1 q
hi Ci , D(h) =
i=1
q
hi Di .
i=1
Then, the T-S fuzzy models (3) and (4) can be rewritten as x˙ (t) = A(h)x(t) + B(h)d(t) + Br (h)yr (t) y(t) = C(h)x(t) + D(h)d(t).
(8) (9)
Remark 1: Note that the reference input yr (t) is directly applied to the open-loop system (8). In fact, the system given in (8) and (9) represents a general tracking system, which has been studied in [31] and [32], and can be derived as follows. the system (8) T with Br (h) = 0, define a new variable xˆ (t) = For t e(τ)dτ x(t) , where e(t) = yr (t) − y(t) is the tracking 0 error. Let 0 −C(h) ˆ 0 ˆ A(h) = B(h) = 0 A(h) B(h)
I ˆ ˆ r (h) = B C(h) = 0 C(h) 0 then we have the following augmented system ˆ x(t) + B(h)d(t) ˆ ˆ r (h)yr (t) xˆ˙ (t) = A(h)ˆ +B ˆ y(t) = C(h)ˆx(t) + D(h)d(t) which is in the same form of (8) and (9). Definition 1: (Sensor stuck fault) when sensor stuck faults occur, the sensor output is given by
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yF (t) = Fm y(t) + (I − Fm )fm , m = 0, 1, · · · l
¯ m (h)η(t) + B ¯ dm (h)d(t) + B ¯ Kr (h)yr (t) η˙ (t) =A
(10)
¯ fm (h)f (t) +B ¯ m (h)η(t) + D ¯ dm (h)d(t) + D ¯ Kr (h)yr (t) r(t) =C
where l denotes the total number of fault models, and
T fm = fm1 · · · fmk · · · fmny
¯ fm (h)f (t) +D
with f mk ≤ fmk ≤ f mk (k = 1, 2, · · · ny ) denoting the stuck value of the kth sensor, f mk and f mk being known scalars. Fm are diagonal matrices in the form of Fm = diag Fm1 , · · · Fmk , · · · Fmny Fmk = 0 or 1, k = 1, 2, · · · ny .
(11)
Remark 2: If Fmk = 0 for 1 ≤ k ≤ ny , the fault model (10) corresponds to the case that the kth sensor gets faulty. If Fmk = 1, the kth sensor is fault-free. Without loss of generality, we assume that F0 = I, namely, m = 0 corresponds to the fault-free case. From (8)–(10), we have x˙ (t) = A(h)x(t) + B(h)d(t) + Br (h)yr (t)
(12)
yF (t) = Fm C(h)x(t) + Fm D(h)d(t) + (I − Fm )fm
(13)
m = 0, 1, · · · l. B. Fault detection filter In order to detect the sensor faults, a fuzzy FD filter is designed as follows. Plant Rule i: IF z1 (t) is μi1 and z2 (t) is μi2 and,· · · , and zs (t) is μis , then x˙ f (t) = Afi xf (t) + Bfi yF (t) + Bfri yr (t) F
r(t) = Cfi xf (t) + Dfi y (t) + Dfri yr (t)
F
r(t) = CK (h)xf (t) + DK (h)y (t) + DKr (h)yr (t)
AK (h)n×n =
hi Afi BK (h)n×ny =
q
i=1 q
CK (h)1×n =
(16) (17)
hi Cfi , DK (h)1×ny =
i=1 q
BKr (h)n×nr =
hi Bfi
i=1 q
i=1
A(h) 0 BK (h)Fm C(h) AK (h) B(h) Br (h) ¯ ¯ Bdm (h) = BKr (h) = BKr (h) BK (h)Fm D(h) 0 ¯ fm (h) = ¯ Kr (h) = DKr (h) B D BK (h)(I − Fm )fm ¯ fm = DK (h)(I − Fm )fm , D ¯ dm (h) = DK (h)Fm D(h) D
¯ m (h) = DK (h)Fm C(h) CK (h) , f (t) = 1. C (20)
¯ m (h) = A
C. Finite frequency performance indices In this section, some performance indices are defined in finite frequency domain for the purpose of FD. In practice, the effects of faults and reference inputs may occupy in different frequency ranges. Therefore, we consider the following two finite frequency interval for frequency ω both in fault f (t) and reference input yr (t): f : = {ω ∈ R | τ(ω − f1 )(ω − f2 ) ≤ 0} yr : = {ω ∈ R | τ(ω − r1 )(ω − r2 ) ≤ 0}
hi Dfi
(21) (22)
where f1 , f2 , r1 , r2 are given real scalars. Note that, when τ = +1, f1 = −f2 = f and r1 = −r2 = r , f and yr denote the low frequency range lf : = {ω ∈ R || ω |≤ f }
(23)
: = {ω ∈ R || ω |≤ r }
(24)
lyr
where f and r are assumed to be known positive real scalars. If τ = −1 and f1 = −f2 (r1 = −r2 ), or τ = +1 and f1 ≤ f2 (r1 ≤ r2 ), f (yr ) denotes the high frequency range or middle frequency range. In the sequel, we give the following two definitions on finite frequency H∞ and H− performance indices for the fuzzy system (18), (19). Definition 2: The fuzzy system (18), (19) has a finite frequency H∞ index bound γ, if under zero initial condition, the following inequality
+∞
r (t)r(t)dt ≤ γ T
i=1 q
hi Bfri , DKr (h)1×nr =
(19)
where
(15)
where q
(18)
(14)
where xf (t) ∈ Rn is the filter state, r(t) ∈ R denotes the residual signal which carries information on the time of the occurrence of a fault, Afi , Bfi , Bfri , Cfi , Dfi , Dfri are filter gains to be determined. Thus, the FD filter can be represented by the following form: x˙ f (t) = AK (h)xf (t) + BK (h)yF (t) + BKr (h)yr (t)
3
hi Dfri .
i=1
T Let η(t) = xT (t) xfT (t) , the augmented dynamics can be described by
0
2 0
+∞
yrT (t)yr (t)dt
(25)
holds for all solutions of (18) with yr (t) ∈ L2 such that
+∞
τ(r1 η + j η˙ )(r2 η − j η˙ )T dt ≤ 0
(26)
0
where τ, r1 and r2 reflecting the frequency range of reference input yr (t) are given in (22), and γ is a given real
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positive scalar which denotes the worst case criterion for the effect of reference input yr (t) on the residual r(t). The smaller the γ is, the more robust the generator becomes. Definition 3: The fuzzy system (18), (19) has a finite frequency H− index bound β, if under zero initial condition, the following inequality:
+∞
rT (t)r(t)dt ≥ β2
0
+∞
f T (t)f (t)dt
(27)
0
holds for all solutions of (18) with f (t) ∈ L2 such that
+∞ τ(f1 η + j η˙ )(f2 η − j η˙ )T dt ≤ 0 (28)
2) For both fault-free and faulty cases (m = 0, 1, · · · l), to minimize the effect of the disturbance d(t) on the residual output r(t), it needs to satisfy
+∞
+∞ r T (t)r(t)dt ≤ γ 2 d T (t)d(t)dt, m = 0, 1, · · · l 0
(31)
with d(t) ∈ L2 . 3) For faulty cases (m = 1, · · · l), to maximize the effects of the reference input yr (t) and fault f (t) on the residual output r(t), it needs to satisfy the following H− performances:
0
where τ, f1 and f2 reflecting the frequency range of fault f (t) are given in (21), β is a given real positive scalar, which is a measurement of the fault sensitivity in the worst case from f (t) to residual r(t). The larger the β is, the more sensitive the generator becomes. Remark 3: Definitions two and three are motivated by [33], which can be regarded as an extension of the FD performance indices defined in full frequency domain [25]. In fact, when τ = −1 and f1 = f2 = r1 = r2 = 0, the constraints (26) and (28) are automatically satisfied and the sets f and yr , in (21) and (22), characterize the full frequency domain. In this case, the Definitions two and three reduce to the ones given in [25]. Remark 4: The physical meaning of Definition two can become clear when special cases are considered. For example, in the low frequency range lyr , we have τ = 1 and r1 = −r2 = r , which implies that (26) is equivalent to +∞ +∞ η˙ η˙ T dt ≤ r2 0 ηηT dt. In this case, by [33], the Defini0 tion two means that the system (18) and (19) possesses property (25) with respect to the reference input yr (t) that does not drive the states too quickly where the bound on the quickness +∞ +∞ is given by r in the sense of 0 η˙ η˙ T dt ≤ r2 0 ηηT dt. Similar interpretations can be drawn for the Definition three.
0
+∞
0
r T (t)r(t)dt ≥ βr2
+∞
0
yrT (t)yr (t)dt, m = 1, · · · l (32)
with yr (t) ∈ L2 such that
+∞ τ(r1 η + j η˙ )(r2 η − j η˙ )T dt ≤ 0
(33)
0
and
+∞
0
r T (t)r(t)dt ≥ βf2
+∞
f T (t)f (t)dt, m = 1, · · · l
0
(34)
with f (t) ∈ L2 such that
+∞
τ(f1 η + j η˙ )(f2 η − j η˙ )T dt ≤ 0
(35)
0
where τ, f1 and f2 are given in (21), βr and βf are prescribed H− indices. E. Comparison With the Existing FD Techniques In the existing FD techniques, the sensor faults are always represented by the discrepancy of sensor outputs and inputs
D. Fault detection scheme Based on the Definitions two and three, the FD problem to be addressed in this paper is formulated as follows: given the fuzzy system (8) and (9) with sensor faults described in (10), determine a FD filter in the form of (16) and (17) such that the following requirements are satisfied. 1) For fault-free case (m = 0), minimize the effect of the reference input yr (t) on the residual output r(t). Then, given a prescribed suitable H∞ index γ, the following H∞ performance should be satisfied for the fuzzy system (18) and (19):
+∞
+∞ 2 T r (t)r(t)dt ≤ γ yrT (t)yr (t)dt, m = 0 (29) 0
0
with yr (t) ∈ L2 such that
+∞ τ(r1 η + j η˙ )(r2 η − j η˙ )T dt ≤ 0 0
where τ, r1 and r2 are given in (22).
(30)
f1 (t) =: yF (t) − y(t) = (Fm − I)(y(t) − fm ).
(36)
Then the fuzzy system (12), (13) with sensor faults can be rewritten in the following form which is widely studied in the literature: ˜ ˜ d(t) x˙ (t) = A(h)x(t) + B(h) ˜ + f1 (t) ˜ d(t) yF (t) = C(h)x(t) + D(h)
(37) (38)
where
˜ ˜ = D(h) 0 B(h) = B(h) Br (h) , D(h)
˜ = d(t) yr (t) T . d(t) To detect the fault f1 (t), an FD filter was introduced in [25] to generate a residual r(t) such that the following requirements are satisfied:
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+∞
0 +∞
r T (t)r(t)dt ≤ γ 2 r (t)r(t)dt ≥ β T
0
+∞
˜ d˜ T (t)d(t)dt
(39)
f1T (t)f1 (t)dt
(40)
0 +∞
2 0
where γ and β are the H∞ and H− indices, which should be minimized and maximized, respectively. Although the above FD techniques considered in full frequency domain can be used to detect sensor faults, the FD result in [25] will be conservative in some degree if the faults lie in finite frequency ranges. In addition, as only the fault sensitivity performance (40) is considered in [25], the fault f1 (t) with small magnitude cannot be detected, and this conclusion has been demonstrated by [34]. Recently, the following discrete-time T-S fuzzy system with actuator and sensor faults was investigated in [30] x(k + 1) = A(h)x(k) + Bd (h)d(k) + Bf (h)f (k)
(41)
y(k) = C(h)x(k) + Dd (h)d(k) + Df (h)f (k)
(42)
where x(k) is the state variable, y(k) is the measurement output, d(k) represents the disturbance input, and f (k) denotes the fault signal. A(h), Bd (h), Bf (h), C(h), Dd (h), Df (h) are known constant real matrices with appropriate dimensions. A FD observer in the form of (43)–(45) xˆ (k + 1) = A(h)ˆx(k) + L(h)(y(k) − yˆ (k)) yˆ (k) = C(h)ˆx(k) r(k) = y(k) − yˆ (k)
(43) (44) (45)
was designed in [30] to satisfy the following finite frequency performance indices: r(k) 2 ≤ γ d(k) 2 , ωd1 ≥ ω ≥ ωd2 r(k) 2 ≥ β f (k) 2 , ωf1 ≥ ω ≥ ωf2
(46) (47)
where L(h) in (43) is an observer gain matrix to be determined, r(k) in (45) is the generated residual, ωd1 , ωd2 , ωf1 and ωf2 in (46) and (47) are given real positive scalars that reflect the individual frequency ranges of disturbance d(k) and fault f (k), γ and β are the prescribed H∞ and H− indices. The simulation results have illustrated the advantages of the finite frequency FD approach given in [30]. However, similar to [25], the method of [30] applied to an ideal case where the magnitude of actuator or sensor fault should be large enough such that it can be detected. Furthermore, the FD performance indices and the corresponding FD observer design conditions in [30] cannot be applied to the continuous-time T-S fuzzy systems. In this paper, we initially define some finite frequency FD performance indices for continuous-time T-S fuzzy systems. Based on these performance indices, a novel FD scheme is then developed such that the generated residual is designed to be sensitive to both fault signal and reference input for the faulty cases, while robust against reference input for the fault free case. As the additional reference input sensitivity
5
is considered, the proposed method improves the existing FD techniques and guarantees that either fault with large magnitude or the one with small magnitude can be detected. The theory will be supported by simulation results related to the detection of sensor faults in a tunnel-diode circuit in Section V.
III. Main Results A. Finite Frequency H∞ and H− Performances Analysis In practice, faults usually emerge in low frequency domain, e.g., for an incipient signal, the fault information is always contained within a low frequency band as the fault development is slow, and the constant stuck fault just belongs to low frequency domain [36]. Therefore, the fault in this paper is assumed to belong to low frequency range lf in (23), in addition, the reference input is also assumed to lie in low frequency range lyr in (24). Based on the definitions of lf and lyr , the following lemmas will be given to provide the analysis conditions for the fuzzy system (18), (19) satisfying the finite frequency H∞ performance (29), (30) and H− performance (32), (33). Lemma 1: Assume that the system (18) is asymptotically stable. Then the system (18), (19) satisfies the H∞ performance (29), (30) in low frequency domain lyr , if there exist symmetric matrices P = P T , Q = QT such that τQ ≥ 0 and
¯ 0 (h) B ¯ 0 (h) B ¯ Kr (h) T ¯ Kr (h) A A
+ I 0 I 0 ¯ 0 (h) D ¯ 0 (h) D ¯ Kr (h) T ¯ Kr (h) C C 0, P m = Pb 11 Pb 12 Qm = m a b Qm Pbm 12 Pbm 22 b 12 Qb 22 ˆ fi , B ˆ fi , Cfi , Dfi , Y, M, N, A such that, for fmk ∈ {f mk , f mk }(k = 1, 2, · · · ny ), the following inequality holds: ⎡ ⎢ ⎢ ⎢ b ij = ⎢ ⎢ ⎢ ⎣
such that the following inequality holds: ⎡ ⎢ ⎢ ⎢ ij = ⎢ ⎢ ⎢ ⎣
m −Qm a 11 −Qa 12 ∗ −Qm a 22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 T CiT FmT Dfj − l11 2T Cfj − l12 2
−2I ∗
Pam 11 − Y T Pam 12 + M T Pam T12 − N T Pam 22 − N T a33 a34 ∗ a44 ∗ ∗ ∗ ∗ ⎤ −Y T v −N T v ⎥ ⎥ ⎥ a36 ⎥ ⎥ 0 are weighting factors which should be given beforehand. Then, matrices Cfi , Dfi , Dfri are obtained, and other filter parameters can be determined according to the following equalities: ˆ fi , Bfi = N −1 B ˆ fi , Bfri = N −1 B ˆ fri Afi = N −1 A i = 1, 2, · · · q.
(60)
Remark 9: Based on the assumption that the measurable output matrices are common for each local model, two effective full frequency fault estimation approaches have been
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developed in [40] and [41] for discrete-time T-S fuzzy systems. Unlike [40] and [41], this paper is concerned with the finite frequency fault detection problem for continuous-time T-S fuzzy systems, and the above mentioned assumption are not required. Especially, the proposed fault sensitivity and reference input sensitivity based FD scheme is valid for both sensor faults with small magnitudes and those with large magnitudes. IV. Detection Threshold Design In this section, the threshold for detecting faults is designed and the detection logic unit is based on the results proposed by [38]. Let the root mean square value which means the average energy of residual signal over a time interval (0, t) 1 t T Jr (t) = r rms := r (τ)r(τ)dτ t 0
(61)
be a residual evaluation function. The threshold Jth is determined by Jth =
sup
Jr (t)
(62)
f (t)=0,d(t)∈L2 ,
and faults can be detected using the following logical relationship : Jr > Jth ⇒ alarm, Jr ≤ Jth ⇒ no faults.
(63)
Fig. 1. yF (t) − y(t) with the stuck value being 0.1.
Sensor 2 stuck fault is considered, therefore, we have 10 0 F1 = in (10), where the stuck value is and f1 = 00 f12 assumed to satisfy −1 ≤ f12 ≤ 1. −0.1 b Given f = 0.01, r = 1, γ = 1, l11 = l11 = , l12 = −0.1 −0.1 0.1 b l12 = and l2 = −l2b = 0.3, v = −vb = 0.3 −0.01 solving the optimization problem (59) with λr = λf = 1, we have
V. Examples In this section, the following example is provided to illustrate the effectiveness of the theoretical results developed. Consider a tunnel diode circuit, whose fuzzy modeling was done in [42] x˙ (t) = y(t) =
2 i=1 2
hi (x1 (t))[Ai x(t) + Bi d(t) + Bri yr (t)] hi (x1 (t))[Ci x(t) + Di d(t)]
(64)
i=1
where x1 (t) = vC (t), x2 (t) = iL (t), and vC (t), iL (t) are the capacitor voltage and inductance current, respectively. The parameter matrices are given by
−0.1 50 −4.6 50 0 A1 = , A2 = , B1 = −1 −10 −1 −10 1 0 10 10 B2 = , C1 = , C2 = 1 01 01 1 1 0 0 Br1 = , Br2 = , D1 = , D2 = . 0 0 0 0 √ √ Assume that x1 (t) ∈ [− 3, 3], and the membership functions for rules one and two are 1 h1 (x1 (t)) = 1 − x12 (t), h2 (x1 (t)) = 1 − h1 (x1 (t)). 3
−43.9654 32.0266 −5.3313 Af 1 = , Bfr1 = −9.0744 −9.9612 −0.9938 −49.3718 35.3985 −5.9094 Af 2 = , Bfr2 = −8.8258 −10.1135 −1.0428
−46.6862 −8.0058 Bf 1 = , Dfr1 = −0.9440 −9.0707 −1.3203
−52.6680 −8.0446 Bf 2 = , Dfr2 = −0.9392 −8.9898 −1.3076
Cf 1 = 0.0766 −1.2544 , Cf 2 = 0.1510 −1.1343
Df 1 = 0.0770 −1.7938 , Df 2 = 0.2120 −1.7198 and the optimal values on the sensitivity performance indexes are βr = 0.5263 and βf = 0.6160. In simulation, the disturbance is assumed to be d(t) = 0.1sin(0.5t) and reference input yr (t) = 1. Two failure cases are considered respectively. Case 1: Sensor 2 is stuck at 0.1 from t = 100s. To show the magnitude of the sensor fault, f1 (t) defined in (36) is plotted in Fig. 1. In addition, the absolutes of the generated residuals using the proposed FD approach (red) and the existing methods (blue) given in Section II are plotted in Fig. 2. It is easy to see that the generated residual in this paper is more sensitive to fault, and the results in Fig. 3 indicate that the fault is detected almost immediately at approximately t = 100s. On the other hand, since the magnitude of f1 (t) is lager than that of disturbance d(t), the generated residual (shown in Fig. 2) using the existing methods without con-
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LI AND YANG et al.: FAULT DETECTION IN FINITE FREQUENCY DOMAIN FOR TAKAGI-SUGENO FUZZY SYSTEMS WITH SENSOR FAULTS
Fig. 2.
Generated residual with the stuck value being 0.1.
Fig. 3.
Residual evaluation and threshold with the stuck value being 0.1.
sidering reference input sensitivity also has large discrepancy before and after the occurrence of fault f1 (t), which implies that the existing methods are also valid in Case 1. Case 2: Sensor 2 is stuck at −0.01 from t = 100s. The fault f1 (t) defined in (36) is plotted in Fig. 4, from which it can be seen that, the magnitude of f1 (t) in Case 2 is much smaller than that of disturbance d(t). The residual response and residual evaluation response obtained in this paper are displayed in Figs. 5 and 6, respectively. As shown in these two figures, the sensor fault with small magnitude can be detected within the proposed FD framework. Moreover, the comparison results between Figs. 3 and 6 illustrate that the detection time obviously increases as the magnitude of fault decreases. In Case 2, the residual response derived by using the existing FD techniques given in Section II is plotted in Fig. 5 (blue). Combining Figs. 2 and 5, it can be concluded that the existing FD techniques are applicable for the case that the magnitude of fault is large enough, on the contrary, when it is
9
Fig. 4. yF (t) − y(t) with the stuck value being -0.01.
Fig. 5.
Generated residual with the stuck value being -0.01.
very small, the existing FD approaches cannot work. This is because that the strong detectability condition which requests that the magnitude of fault should be large enough is not satisfied in Case 2. In this paper, since the additional reference input sensitive performance is considered for the faulty cases, the fault with relative small magnitude can still be detected even if such strong detectability condition is not satisfied, which further exhibits the significance of our proposed FD approach. VI. Conclusion In this paper, an effective finite frequency FD approach for continuous-time T-S fuzzy systems subject to sensor stuck faults has been investigated, where the generated residual is designed to be sensitive to both fault signals and reference inputs for the faulty cases. The proposed FD scheme has improved the existing FD techniques, more specifically, both sensor faults with small magnitudes and those with large magnitudes can be detected within the proposed FD framework.
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T d T r r − (η Pη) + yr yr dt tr[He((r1 η + j η˙ )(r2 η − j η˙ )T )Q] ≤ 0. Integrating from t = 0 to ∞ and using the stability property, we have
∞
0
T
r yr
r dt ≤ tr[He(S)Q] yr
(67)
(r1 η + j η˙ )(r2 η − j η˙ )T )dt.
(68)
where
S=
∞
0
Fig. 6.
Residual evaluation and threshold with the stuck value being -0.01.
Moreover, the advantages of the proposed FD approach over the available ones have been illustrated through a numerical example.
By the Parseval’s theorem [35], we have
−∞ 1 (r1 − ω)(r2 − ω)ˆηηˆ T dω S= 2π ∞ where ηˆ is the Fourier transform of η. Thus S is Hermitian and the bound on the righthand side of (67) becomes tr(SQ). Since τQ ≥ 0, the bound tr(SQ) is nonpositive whenever τS ≤ 0 holds, then the proof is completed.
Acknowledgment The authors would like to thank the anonymous reviewers and editors for their useful comments and suggestions.
Appendix A Lemma 3: (Projection Lemma) Let U, V, be given. There exists a matrix F satisfying T
T
U FV + V FU + < 0 if and only if the following two conditions hold NUT NU < 0
(65)
NVT NV < 0
(66)
where NU and NV are arbitrary matrices whose columns form a basis of the nullspaces of U and V , respectively. Lemma 4: (Finsler’s Lemma) Let ξ ∈ Cn , Q ∈ Cn×n and H ∈ Cn×m . Let H⊥ be any matrix such that H⊥ H = 0. The following statement are equivalent: i) ξ ∗ Qξ < 0, ∀H∗ ξ = 0, ξ = 0, ∗ ii) H⊥ QH⊥ < 0, iii) ∃μ ∈ R : Q − μHH∗ < 0, iv) ∃X ∈ Rm×n : Q + HX + X ∗ H∗ < 0.
Appendix B Proof of Lemma 1. From (24), we have τ = +1, r1 = −r2 = r . Then, pre and postmultiplying (48) by [ηT yrT ] and its transpose, it follows that
Appendix C Proof of Theorem 1. Based on Lemma 2, the system dynamics (18) and (19) satisfy the high gain performance (32), (33) in low frequency domain if there exist matrices mT Pam = Pam T , Qm > 0 such that the following inequality a = Qa holds ¯ m (h) B ¯ m (h) B ¯ Kr (h) T ¯ Kr (h) A A
m + I 0 I 0 T ¯ m (h) D ¯ m (h) D ¯ Kr (h) ¯ Kr (h) C C 1
