Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission

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Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission Yue Long, Chao Xu∗ School of Physics, Liaoning University, Shenyang 110036, People’s Republic of China Received 30 June 2016; received in revised form 14 March 2017; accepted 28 May 2017 Available online xxx

Abstract In this paper, the fault detection problem is studied in finite frequency domain for constrained networked systems under multi-packet transmission. The considered transmission mechanism is that only one packet including parts of the measured information can be transmitted through the communication channel and their accessing is scheduled by a designed stochastic protocol. Then by virtues of the introduced performance indices in finite frequency domain, a novel effective fault detection scheme is presented, in which the fault detection filters completing the task with partially available measurements are designed to make sure that the residual is sensitive to the reference input and the fault in faulty cases and robust to the reference input in fault-free case. Further, convex conditions in terms of time-domain inequalities are developed to handle the proposed fault detection scheme. The theoretical results are validated by the simulation to detect the sensor fault on a lateral-directional aerodynamic model. © 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction The rapid advances in computer and communication technologies revolutionized the scale of the control systems. A class of control systems termed as networked control systems (NCSs), in which large spatially distributed components such as plants, controllers, sensors and actuators can be connected together [1–4], has gained a great deal of research attention because ∗

Corresponding author. E-mail address: [email protected] (C. Xu).

http://dx.doi.org/10.1016/j.jfranklin.2017.05.043 0016-0032/© 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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of their growing applications in practical areas such as aircrafts, ventilation, automobiles etc. Although the NCSs introduced many advantages such as reduction of system wiring, simple system diagnosis and low cost they also bring some new phenomena including networkedinduced delay, packet losses, quantization and communication constraint, one can see [5,6] for more details. Fault detection (FD) problem has been another challenging and active field of research due to the increasing demands on safety, higher performance and reliability in industrial process. Consequently, a great deal of efforts have been paid to this issue, and the results are mainly divided into two categories. One is the model-based methods, see e.g. [7,8] and the other is data-based methods, see e.g. [9,10]. For more details, one can see [11] for a survey. However, among these FD methods [12], the celebrated frequency domain approach is accepted as a significant one because the signals of the practical systems are often in certain frequency domain, see e.g., the finite frequency FD method have been developed for linear uncertain discrete-time systems in [13] and for linear parameter-varying systems in [14], respectively, with the aid of the generalized Kalman–Yakubovich–Popov (GKYP) lemma. Further, a FD scheme based on the steady-state method has been proposed in finite frequency domain in [15] for linear output feedback control systems. Though the mentioned methods were proven to be effective to detect certain types of faults, they were proposed for linear systems. In recent years, some finite frequency FD method have been carried out for nonlinear systems, i.e. [12] for T-S fuzzy systems and [16] for multi-delay uncertain systems, etc.. However, the FD method for stochastic systems has not been mature. Due to the wide utilization of NCSs, it is naturally to study the FD methods for this class of systems. And thus a lot of researches have been carried out, i.e., FD method which is robust to the network-induced time delay has been developed in [18–20] and to packet dropouts in [21–23]. However, it should be noticed in above literatures and most other existing results concerned with the FD problem for NCSs, that the system data are assumed to be delivered in one packet. Nevertheless, the outputs of the system are often measured by multiple and distributed sensors, and thus the sampled data cannot be transmitted by one packet. In such cases, the mentioned FD methods will be invalid. Furthermore, as mentioned in [24,25], one of the communication constraints called medium access constraint, which is coming after the insert of the networks, also degrades the performance of the FD system. Under such constraint, not all the transmission nodes are allowed to transmitted through the communication channel simultaneously. Consequently, it is nature and important to study the FD approach for the NCSs under multi-packet transmission and such constraint. Long and Yang [26] have proposed a FD scheme for NCSs subject to time-varying transmission intervals and delays, packet dropouts, and communication constraints in the framework of switched stochastic parameter systems, but the frequency of the signals has not been considered yet. Motivated by the above statements, the FD problem in finite frequency domain for NCSs with communication constraints under multi-packet transmission is addressed in this paper against to sensor failures. Specifically, the main contributions of this manuscript are lied in the following aspects: (i) the communication constraint, that is only one network node can access the communication channel, and a novel sensor fault model are considered. The whole FD system is modeled as multiple Markovian jump systems, and sufficient conditions guaranteeing the whole system with desired performances in finite frequency domain are derived. (ii) A new FD scheme, which is effective to detect the sensor faults with either large or small magnitudes, is proposed by making the generated residuals sensitive to both reference inputs and fault signals in faulty cases while robust to the reference inputs in fault-free case. Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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Fig. 1. Structure of the considered networked systems.

This paper is organized as follows. In Section 2, the considered NCS is modeled and the FD filter design objectives are presented. The conditions to parameterize the FD filters are derived in Section 3. In Section 4, the residual evaluator and the threshold are given. Finally, an example are provided to illustrate the results of this paper in Section 5 and some conclusions end this manuscript in Section 6. Notations: The following notations are used throughout this paper. L2 denotes the Hilbert space of square integrable functions; E(·) denotes the expectation operator; I (Iv ) represents the identity matrix (row vector) with compatible dimensions. ∗ is utilized to represent a term that is induced by symmetry; Denoting He(A ) := A + AT be the sum of a square matrix A and its transposition AT . 2. Problem formulation 2.1. System model The considered networked system in this paper is depicted in Fig. 1, in which the plant is a continuous-time one described by  x˙(t ) = Ac x(t ) + Bc d (t ) + Ec s(t ) (1) y(t ) = Cc x(t ) + Dc s(t ) where x(t) ∈ Rn is the system state vector, y(t) ∈ Rm is the measured output, d(t) is the unknown exogenous disturbance signal that belongs to L2 [0, ∞}, s(t) represents the reference input in certain frequency domain. The system matrices are known and with appropriate dimensions. It is assumed that the sampling period is h, and then the discrete-time form of system (1) can be derived as follows  x(k + 1) = Ad x(k) + Bd d (k) + Ed s(k) (2) y(k) = Cd x(k) + Dd s(k) Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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Table 1 Illustration of the faulty model. Fault (l = 1, 2, . . . , Q)

ρ lj

ρ lj

σ jl

Normal drift Outage Stuck Loss of effectiveness

1 1 0 0 >0

1 1 0 0 <1

0 1 0 1 0

h h where Ad = eAc h , Bd = 0 eAc t Bc dt, Ed = 0 eAc t Ec dt, Cd = Cc and Dd = Dc . It is also assumed that the measured outputs sampled by the sensors are packaged into m¯ (m¯ ≤ m) ones and then spread to the FD center through a constrained network wherein only one packet node is admitted to access the shared channel at each sampling period. When the sensors of the system experience faults, the jth ( j ∈ {1, 2, . . . , m}) sampled output ysl j (k) at time kth can be characterized by the following model: ysl j (k) = ρ lj (k)y j (k) + σ jl f jl (k),

j = 1, . . . , m, l = 0, 1, 2, . . . , Q.

ρ lj (k)

(3)

σ jl

where is the known efficiency factor of the jth sensor, is an unknown constant taking l value of 0 or 1, and f j (k) denotes the unknown bounded drift or stuck value of the jth sensor. Q is the quantity of the faulty modes while l = 0 represents the fault-free situation. From the practical systems, one can derive 0 ≤ ρ lj (k) ≤ 1. From above statement, we have  l l  l T (4) ysl (k) = ys1 , ys2 , . . . , ysm = ρ l (k)y(k) + σ l f l (k), l = 0, 1, 2, . . . , Q. where ρ l = diag{ρ1l , ρ2l , . . . , ρml } and σ l = diag{σ1l , σ2l , . . . , σml }. Remark 1. Compared with the faulty models considered in relevant researches, the model (4) is more general, since which can describe the fault of drift, loss of effectiveness, stuck together with outage case besides normal situation. That is illustrated by Table 1. Further, as mentioned above, the constraint of the transmission network under consideration is that only one packet can access to the communication channel at one time slot. Accordingly, it is supposed that the accession to the communication channel of each packet is governed by a specific designed protocol, as that in [27], which is determined by a Markov process taking values in a finite state space S = {0, 1, 2, . . . , m¯ } with the transition probability generator ⎡ ⎤ λ00 λ01 λ02 . . . λ0m¯ ⎢ λ10 λ11 λ12 . . . λ1m¯ ⎥ ⎢ ⎥   ⎢ ⎥  = (πij )i,j∈S = P Nk+1 = j| Nk = i i,j∈S = ⎢ λ20 λ21 λ22 . . . λ2m¯ ⎥ ⎢ .. .. .. . . .. .. ⎥ ⎣ . ⎦ . . λm¯ 0

λm¯ 1

λm¯ 2

...

λm¯ m¯

wherein Nk = i implies the ith packet transmitted through the network at the kth sampling period while Nk = 0 indicates the situation of packet loss, that is all the packets cannot be transmitted. Remark 2. Recent years, many research attentions, such as [27,28], have been paid to the problem of the network protocol due to the wide application of the NCSs. The results contained the accessing priority to the channel of each transmission node and the choice of their Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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accessing probability. A class of widely utilized stochastic protocol, as that in [28,29], determined by a Markov chain was adopted. This is to say, the transition probability λij is prior designed and known to us. Besides, denoting the signals received by the FD center by yr (k), it can be easily derived that yr (k) = ysl (k), and for i ∈ 1, 2, . . . , m, we can obtain  ysil (k), ∀i ∈ Nk yri (k) = (5) κyri (k − 1), ∀i ∈ / Nk where 0 ≤ κ ≤ 1 is a weighting factor. Further, for i ∈ S, defining i = diag{Nk (1), Nk (2), . . . , Nk (i), . . . Nk (m)} K = diag{κ, . . . , κ}m×m where Nk (i) = 1 if the ith measured output is transmitted in the packet Nk = i, otherwise, ¯ Nk (i) = 0. One can seen that m  = I and 0 = 0. i i=0 Remark 3. To compensate the signals which are not updated in certain time slot, two mechanisms, named as the zero input mechanism and the hold input mechanism, were generally utilized [25]. In those two mechanisms, the outputs not updated are assumed to be zero or hold at the latest updated values, respectively. As stated in [25], the compensation scheme (5) can be equal to the zero input and the hold input ones by setting κ = 0 and κ = 1, respectively. Therefore the compensation mechanism in this paper is more general than those two widely-applied ones. Moreover, such a mechanism can be achieved by adding a zero-order holder designed beforehand. Consequently, by denoting ζ (k) = [x T (k)yrT (k − 1)]T , the dynamics of the system (1) together with the network-induced phenomenon at the (k + 1)th sampling period can be characterized by  ζ (k + 1) = Ali ζ (k) + Bil d (k) + Eil s(k) + Fil f (k) (6) yr (k) = Cil ζ (k) + Dil s(k) + Gli f (k) where  l Ai Eil Cil Dil

Fil Gli

Bil



⎡

Ad = ⎣i ρ l Cd i ρ l Cd

0 K(I − i ) K(I − i )

Ed i ρ l Dd i ρ l Dd

0 i σ l i σ l

⎤ Bd 0⎦

After the above treatments, the network-induced phenomenon is converted into the jumping behavior of the Markov jump system (6) with the range set S and the transition probability generator . 2.2. Preliminaries and problem formulation To detect the possible sensor faults, the following fault detection filters (FDFs) are constructed as residual generators: x˜(k + 1) = A f i x˜(k) + B f i yr (k) + Bs f i s(k) r(k) = C f i x˜(k) + D f i yr (k)

(7)

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where x˜ ∈ Rn is the state vector of the FDFs, A f i , B f i , Bs f i , C f i and D f i are filter matrices to be designed. Further, the dynamics of the whole FD system can be presented by ε(k + 1) = Ali ε(k) + Bil d (k) + Eil s(k) + Fil f (k)

(8)

r(k) = Cil ε(k) + Dil s(k) + Gil f (k) where ε(k) = [ζ T (k)  l Ai Cil

Eil Dil

Fil Gil

x˜T (k)]T and ⎡ l  Ai 0 l Bi = ⎣ B f iCil A f i D f iCil C f i

Eil l B f i Di + Bs f i D f i Dil

Eil B f i Gli D f i Gli

⎤ Bil 0⎦

In order to propose the FD scheme clearly, we introduce the following notation of finite frequency range

ω := {ς (ω − 1 )(ω − 2 ) ≤ 0|ω ∈ R}

(9)

where ω is the frequency of the relevant signal, ϖ1 and ϖ2 are known scalars reflecting the frequency range, and ς = ±1. Remark 4. According to [17], when ς = +1, ω in Eq. (9) characterizes middle frequency ranges. Further, when ς = +1 and choosing 1 = − l , 2 = l , ω will be equal to lω := {|ω| ≤ l |ω ∈ R}, which defines the low frequency range, and when ς = +1, choosing 1 =

h , 2 = 2π − h , Eq. (9) will be transformed into hω := {|ω| ≥ h |ω ∈ R}, which reflects the high frequency range. Moreover, if ς = −1 and 1 = 2 = 0, the set ω in Eq. (9) will be reduced to full frequency domain. Let ωs and ωf represent the frequency of the signals s(k) and f(k), the finite frequency ranges ωs and ω f with the form of

ωs := {ς (ωs − s1 )(ωs − s2 ) ≤ 0 |ωs ∈ R}

(10)

ω f := {ς (ω f − f1 )(ω f − f2 ) ≤ 0 |ω f ∈ R}

(11)

can be given according to Eq. (9), where s1 , s2 , f1 and f2 are given scalars reflecting the frequency range. And then the definitions of finite frequency stochastic H− index and finite frequency stochastic H∞ index will be given for system (8). Definition 1. The system (8) is said to be with a finite frequency stochastic H− index β s to s(k), if the following inequality r(k) E2 ≥ βs2 s(k) E2

(12)

holds for all solutions under zero initial condition with s(k) ∈ L2 such that e j s

∞ 

(ε(k + 1) − e j s1 ε(k))(ε(k + 1) − e− j s2 ε(k))T ≤ 0

(13)

k=0

where s = ( s1 + s2 )/2 and · E2

=

∞ 

1

{E| ·|2 } 2 .

0

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Definition 2. The system (8) is said to be with a finite frequency stochastic H∞ index γ s to s(k), if the following inequality r(k) E2 ≤ γs2 s(k) E2

(14)

holds for all solutions under zero initial condition with s(k) ∈ L2 such that e

j s

∞    T ε(k + 1) − e j s1 ε(k) ε(k + 1) − e− j s2 ε(k) ≤ 0.

(15)

k=0

Based on the above statements, the FD scheme to be addressed in this manuscript can be formulated as follows: for the constrained networked system together with the sensor failure, i.e. the system (6), design a FD filter in the form of Eq. (7) such that for the augmented filter error system (8), the following requirements are satisfied: (1) System (8) is mean-square stable (MSS). (2) The effect of the reference input s(k) on the residual r(k) is maximized by satisfying r(k) E2 ≥ βs2 s(k) E2

(16)

in faulty cases (l = 1, 2, . . . , Q), while the effect of the reference input s(k) on the residual r(k) is minimized by satisfying r(k) E2 ≤ γs2 s(k) E2

(17)

in fault free case (l = 0) with s(k) ∈ L2 such that e j s

∞    T ε(k + 1) − e j s1 ε(k) ε(k + 1) − e− j s2 ε(k) ≤ 0

(18)

k=0

where β s and γ s are prescribed finite frequency stochastic performance indices. (3) The effect of the fault f(k) on the residual r(k) is maximized by satisfying the following performance r(k) E2 ≥ β 2f f (k) E2

(19)

with f(k) ∈ L2 such that e j f

∞    T ε(k + 1) − e j f1 ε(k) ε(k + 1) − e− j f2 ε(k) ≤ 0

(20)

k=0

when σ jl = 1 in faulty cases (l = 1, 2, . . . , Q), where β f is a prescribed finite frequency stochastic H− index. (4) The effect of the exogenous disturbance d(k) on the residual r(k) is minimized by satisfying r(k) E2 ≤ γd2 d (k) E2

(21)

with d(k) ∈ L2 in fault free and faulty cases, i.e. l = 0, 1, . . . Q, where γ d is a prescribed finite frequency stochastic H∞ index. Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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3. Main results In this section, the required FD filters will be designed. Note that in practice, faults usually occur in low frequency domain, e.g. 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 the low frequency domain [12]. And thus without loss of generality, in this paper, it is assumed that the fault and the reference input belong to the low frequency range ls and lf , which are defined as follows.

ls := {|ωs | ≤ sl |ωs ∈ R}

(22)

lf := {|ω f | ≤ f l |ω f ∈ R}

(23)

Before proceeding further, the following lemma will be given firstly to help us to derive our results. Lemma 1 (Finsler’s lemma). Let x ∈ Rn , F ∈ Rn×n , U ∈ Rn×m . Let U⊥ be any matrix such that U⊥ U = 0. The following statements are equivalent: (i) (ii) (iii) (iv)

x T Fx < 0, ∀UT x = 0, x = 0, T U⊥ FU⊥ < 0, ∃μ ∈ Rn : F − μUUT < 0, ∃Y ∈ Rm×n : F + UY + YT UT < 0.

3.1. Finite frequency stochastic performance analysis The following lemmas provide the analysis conditions for system (8) satisfying the performances stated in the above requirements. Lemma 2. Assume the system (8) is MSS, let β s > 0 be a given scalar, system (8) is said to be with a stochastic H− index β s for s(k) in low frequency domain (22) when d (k) = 0 and f (k) = 0, if there exist mode-dependent matrices Psli = (Psli )T , Qsl i = (Qsl i )T > 0, i ∈ S such that the following inequality holds.  l T  l   l T  l  Eil Ai Eil Ci Dil Ci Dil l Ai + <0 s i s (24) I 0 I 0 0 I 0 I where

 l lsi = −Pl si Qs i and



−I s = 0 l

where Psi =

Qsl i l Pi − 2cos sl Qsl i 0 βs2 I 

 (25)



j∈ S

(26) λij Pslj .

Proof. From the notion of the finite frequency range, Remark 4 and Eq. (10), we choose ς = +1, s2 = − s1 = sl . And then, pre- and post-multiplying Eq. (24) by [ε T (k)sT (k)] and its transpose, it follows that Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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l

ε T (k)Psli ε(k) − ε T (k + 1)Psi ε(k + 1) − r T (k )r(k ) + βs2 sT (k )s(k )   + tr{Qsl i ε (k)ε T (k + 1) − ε(k + 1)ε T (k) − 2cos sl ε (k)ε T (k) } ≤ 0

(27)

Summing it from 0 to ∞ and utilizing the stability property, one can obtain ∞ 

(E(−r T (k)r(k) + βs2 sT (k)s(k))) + tr{Qsl i S} ≤ 0

(28)

k=0

where S :=

∞ 

(ε(k)ε T (k + 1) + ε(k + 1)ε T (k) − 2cos sl ε(k)ε T (k))

(29)

k=0

It is easily seen that −S is equal to the left-hand side of Eq. (18), thus one can obtain l l S ≥ 0. On  the other Thand, since Q2siT > 0, the term tr{Qsi S} ≥ 0 while Eq. (18) holds. Thus we have ∞ (E (−r (k) r(k) + β s (k) s(k)) ≤ 0, which is equivalent to Eq. (16). The proof s k=0 is completed.  Lemma 3. Assume the system (8) is MSS, let γ s > 0 be a given scalar, system (8) is said to be with a stochastic H∞ index γ s for s(k) in low frequency domain (22) when d (k) = 0 and f (k) = 0, if there exist mode-dependent matrices Psi = PsTi , Qsi = QsTi > 0, i ∈ S such that the following inequality holds.  l T  l   l T  l  Eil Dil Dil Ai Eil A C C + i <0 lsai i sa i (30) I 0 I 0 0 I 0 I where

 l lsai = −Pl sai Qsai and



I sa = 0 l

with Psai =

l Qsa i l l Psai − 2cos sl Qsa i

0 −γs2 I 

 (31)



j∈S

(32) l λij Psa j.

Proof. The main idea to prove Lemma 3 is similar as that of Lemma 2, so it is omitted here.  3.2. Fault detection filter design In this subsection, the design conditions of finite frequency fault detection filter will be derived based on the Lemmas 2 and 3. Theorem 1. Let β s be a given scalar, for all Nk = i ∈ S, consider the system (8) in faulty cases (l = 1, 2, . . . , Q), it is with a stochastic H− index β s in low frequency domain (22) if there exist matrices Xi , Yi , Ni , Mi , A f i , B f i , Bs f i , C f i , D f i and  l   l  ∗ ∗ Pssi1 Qssi1 lT l lT l , Qssi = Qssi = >0 Pssi = Pssi = (33) Pssl i2 Pssl i3 Qlssi2 Qlssi3 Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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such that the following LMIs hold, ⎡ l −P ssi1 ∗ ∗ ⎢ ⎢ l ⎢ −P lssi2 −P ssi3 ∗ ⎢ ⎢ l lT ⎢Q 33 ⎢ ssi1 − Xi Qssi2 − Ni ⎢ l l ⎢Q 43 ⎢ ssi2 − Yi Qssi3 − Ni ⎢ ⎢ 0 0 −MTi Iv ⎣ −α1 Iv Xi

−α1 Iv Ni

l

where P ssi =

 j∈ S

63

∗

∗

∗

∗

∗

∗

44

∗

0

−I

64

−α2 Mi

∗

⎤

⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥<0 ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎦ 66

(34)

λij Pssl j and

33 = Pssl i1 − 2cos sl Qlssi1 + He(Xi Ali + B f iCil + IvT D f iCil ), 43 = Pssl i2 − 2cos sl Qlssi2 + Yi Ali + B f iCil + ATf i + C Tf i Iv , 44 = Pssl i3 − 2cos sl Qlssi3 + He(A f i ), 63 = EilT XiT + DilT B Tf i + BsTf i + α1 Iv Xi Ali + α1 Iv B f iCil + DilT D Tf i Iv + α2 D f iCil , 64 = EilT YiT + DilT B Tf i + BsTf i + α1 Iv A f i + α2 C f i , 66 = βs2 I + He(α1 Iv Xi Eil + α1 Iv B f i Dil + α1 Iv Bs f i + α2 D f i Dil ). Proof. It is easily derived from Lemma 2 that the finite frequency stochastic H− index β s of system (8) can be captured by satisfying Eq. (24). And then, we rewrite Eq. (24) to the form of ⎡ l ⎡ l ⎤T ⎤ Ai Eil  Ai Eil  ⎣I (35) 0 ⎦ J lsi J T + Hil s HilT ⎣ I 0⎦ < 0 0 I 0 I 

T  T I 0 0 0 Cil Dil l where J = and Hi = . 0 I 0 0 0 I On the other hand, the following null space bases calculations can be easily obtained ⎡ ⎤⊥ ⎡ ⎤T −I Ali Eil ⎢AlT ⎥ (36) 0⎦ ⎣ i ⎦ =⎣ I lT 0 I E i

Then from Lemma 1, one can see that the inequality (35) is equivalent to J lsi J T + Hil s HilT + He(Lli Wi ) < 0

(37)

 T where Wi is an auxiliary variable introduced by Lemma 1 and Lli = −I Ali Eil . We decompose the matrix Wi into the form of Wi = 0 WiT α1 WiT Iv T , where Wi is a variable, α 1 is a weighting scalar, Iv = Iv 0 and Iv ∈ R2n , Iv ∈ Rn . Note that this partition of Wi introduced some conservatism because of its special structure, however, it will lead to an easily solvable condition. Then, the following inequality will provide a sufficient condition Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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for Eq. (37). ⎡ l −Pssi ⎣e− j sc Ql − Wi ssi −α1 Iv Wi

∗ os22 EilT WiT + α1 Iv Wi Ali − DilT Cil

11

⎤ ∗ ⎦<0 ∗ 2 l lT l βs I + He(α1 Iv Wi Ei ) − Di Di (38)

l l lT l where os22 = Pssl i − 2cos sl Qss i + He (Wi Ai ) − Ci Ci .

 Wi11 Wi12 It is supposed that, without loss of generality, Wi has the partition of Wi = Wi21 Wi22 with Wi11 and Wi22 being nonsingular.  I 0 Defining Ti = and performing the congruence transformation 0 Wi−T WiT12 22 diag(Ti Ti I ) on Eq. (38), one can derive ⎡

l

−P ssi

⎢ ⎢e− j sc Ql − U i ⎣ ssi −α1 Iv Ui



∗

∗

l − 2cos Ql + He (U A˜ l ) − C˜lT C˜l Pss sl ssi i i i i i

∗

E˜ilT UiT + α1 Iv Ui A˜ li − DilT C˜il

where Pssl i = TiT Pssl i Ti ,

l Qlssi = TiT Qss i Ti ,

βs2 I

⎤

+ He(α1 Iv Ui E˜il )

 C˜il = Cil Ti = D˘ f i Cil

⎥ ⎥<0 ⎦

(39)

− DilT Dil

C˘f i

 ,

    Wi11 Wi12 Wi−T WiT12 Xi Ni 22 = , Ui = = Yi Ni Wi12 Wi−1 W Wi12 Wi−T WiT12 22 i21 22  l    Ai Eil 0 −1 l −1 l l l ˜ ˜ , Ei = Ti Ei = ˘ , Ai = Ti Ai Ti = ˘ B f iCil A˘f i B f i Dil + B˘ s f i TiT Wi Ti

and  A˘f i C˘f i

  B˘ s f i Wi−T WiT22 12 = 0 0

B˘ f i D˘ f i

On the other hand, ⎤T ⎡ I 0 0 ⎢0 I ⎢ 0⎥ ⎢ ⎥ l⎢ ⎣0 C˜l D l ⎦ i ⎣ i i 0 0 I ⎡

where

⎡

⎢ ⎢ li = ⎢ ⎣

l

−P ssi Qli − Ui 0 −α1 Iv Ui

 0 Afi I Cf i

Bfi Dfi

⎡  W −T W T Bs f i ⎣ i22 i12 0 0 0

0 I 0

⎤ 0 0 ⎦. I

Eq. (39) can be rewritten to ⎤ I 0 0 0 I 0 ⎥ ⎥<0 l ˜ 0 Ci Dil ⎦ 0 0 I

∗ Pssl i − 2cos sl Qlssi + He(Ui A˜ li ) 0 lT T ˜ Ei Ui + α1 Iv Ui A˜ li

(40)

(41)

∗ ∗ −I 0

∗ ∗ ∗ 2 βs I + He(α1 Iv Ui E˜il )

⎤ ⎥ ⎥ ⎥ ⎦

It can be easily seen that the following inequality with some appropriate Xi provides a necessary and sufficient condition to Eq. (41) by applying Lemma 1 on it  T  li + He 0 C˜il −I Dil MiT < 0 (42) Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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where Mi is introduced by Lemma 1. In order to make the problem tractable, the matrix Mi is restricted to the form of  MiT = 0

MTi Iv

0

 α2 MTi .

where α 2 is a given constant. Therefore, Eq. (42) can be expressed as ⎡

l

−P ssi

⎢ ⎢ l ⎢Q i − U i ⎢ ⎢ ⎢ 0 ⎣ −α1 Iv Ui

∗

∗

Pssl i − 2cos sl Qlssi + He(Ui A˜ li + Iv T Mi C˜il )

∗

−MTi Iv

−I

E˜ilT UiT + α1 Iv Ui A˜ li + DilT MTi Iv + α2 Mi C˜il

−α2 Mi

∗

⎤

⎥ ⎥ ∗⎥ ⎥<0 ⎥ ∗⎥ ⎦ o44

(43)

where o44 = βs2 I + He(α1 Iv Ui E˜il + α2 Mi Dil ). Partitioning the matrices Pssl i and Qlssi into the form of Eq. (33) and defining the new variables as follows, A f i = Ni A˘f i ,

B f i = Ni B˘ f i ,

Bs f i = Ni B˘ s f i ,

C f i = Mi C˘f i ,

D f i = Mi D f i . (44)

Eq. (34) can be reached, which completes the proof.  Remark 5. If Qli , α 1 in Theorem 1 are chosen Qli = 0, α1 = 0 , respectively, also the following equation holds, Theorem 1 can be equivalent to the Theorem 4 in [30] by applying Lemma 1. 

  T  T T  lT l Iv T Xi D˜ sil + α2 C˜ilT XiT −Iv Xi −Iv Xi −C˜i C˜i + = lT T −α2 Xi −α2 Xi He(α2 D˜ si Xi ) ∗

He(C˜ilT XiT Iv ) ∗

−C˜ilT D˜ sil −D˜ lT D˜ l si



si

On the other hand, in order to linearize the matrix inequality condition (39), some given matrices are necessary in Theorem 1 of [12] to cope with the terms −C˜ilT C˜il , −C˜ilT D˜ il , −D˜ ilT C˜il and −D˜ ilT D˜ il . Nevertheless, for the systems with high-dimensions or the systems with many subsystems, it is so hard to choose these matrices. Instead of these given l· · , the matrix variables Xi are introduced to deal with those nonlinear terms. From the above statement, one can see the method in this manuscript is with less conservatism than those in [12,30]. Theorem 2. Let β f be a given scalar, for all Nk = i ∈ S, consider the system (8) when the fault signature σ jl = 1 in faulty cases (l = 1, 2, . . . , Q), it is with a stochastic H− index β f in low frequency domain (23) if there exist matrices Xi , Yi , Ni , Mi , A f i , B f i , Bs f i , C f i , D f i and  l P lT fi = Pfi =

P lf i1 P lf i2

  l Q f i1 ∗ lT l , Q = Q = l f i f i Qlf i2 P f i3

 ∗ >0 Qlf i3

such that the following LMIs hold, Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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⎡

l

−P f i1

⎢ l ⎢ −P f i2 ⎢ l ⎢Q f i 1 − X i ⎢ ⎢ Ql − Yi ⎢ f i2 ⎣ 0 −α1 Iv Xi l

where P f i

∗

∗

l

−P f i3 ∗ QlT − N  33 i f i2 Qlf i3 − Ni 43 0 −MTi Iv −α1 Iv Ni 63  = j∈S λij P lf j and

∗

∗

∗ ∗ 44 0 64

∗ ∗ ∗ −I −α2 Mi

∗

13

⎤

⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥<0 ∗ ⎥ ⎥ ∗ ⎦ 66

(45)

33 = P lf i1 − 2cos f l Qlf i1 + He(Xi Ali + B f iCil + IvT D f iCil ), 43 = P lf i2 − 2cos f l Qlf i2 + Yi Ali + B f iCil + ATf i + C Tf i Iv , 44 = P lf i3 − 2cos f l Qlf i3 + He(A f i ), T l l lT T l 63 = FilT XiT + GlT i B f i + α1 Iv Xi Ai + α1 Iv B f iCi + Gi D f i Iv + α2 D f iCi , T 64 = FilT YiT + GlT i B f i + α1 Iv A f i + α2 C f i ,

66 = β 2f I + He(α1 Iv Xi Fil + α1 Iv B f i Gli + α2 D f i Gli ). Proof. The proof can be proceeded by following the same process of the proof for Theorem 1 and utilizing corresponding system matrices, thus it is omitted here.  Theorem 3. Let γ s be a given scalar, for all Nk = i ∈ S, consider the system (8) in fault-free case (l = 0), it is with a stochastic H∞ index γ s in low frequency domain (22) if there exist matrices Xi , Yi , Ni , Mi , A f i , B f i , Bs f i , C f i , D f i and     ∗ ∗ Psai1 Qsai1 T T , Qsai = Qsai = >0 Psai = Psai = (46) Psai2 Psai3 Qsai2 Qsai3 such that the following LMIs hold, ⎡ −P sai1 ∗ ∗ ⎢ −P sai2 −P ∗ sai3 ⎢ ⎢ Qsai1 − Xi QT − Ni 33 sai2 ⎢ ⎢ Qsai2 − Yi Qsai3 − Ni 43 ⎢ ⎣ 0 D f iCi0 0 0 63 0  where P sai = j∈S λij Psaj and

∗ ∗ ∗ 44 Cfi 64

∗ ∗ ∗ ∗ I − He(Mi ) Di0T D Tf i

∗ ∗ ∗ ∗ ∗ −γs2 I

⎤ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎦

(47)

33 = Psai1 − 2cos sl Qsai1 + He(Xi A0i + B f iCi0 ), 43 = Psai2 − 2cos sl Qsai2 + Yi A0i + B f iCi0 + ATf i , 44 = Psai3 − 2cos sl Qsai3 + He(A f i ), 63 = Ei0T XiT + Di0T B Tf i + BsTf i , 64 = Ei0T YiT + Di0T B Tf i + BsTf i . Proof. It could be easily seen, from the Lemma 2 that the system (8) is with a stochastic H∞ index γ s to the reference input s(k) in low frequency domain (22) in fault-free case (l = 0) if Eq. (30) holds. Then following the line of the proof for Theorem 1, and fixing Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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the introduced auxiliary matrix variable Mi into the form of [0 derive that Eq. (30) holds if and only if C˜i

sai + He([0

D˜ i ]T [0

−I

0

XiT

0

MTi

0], one can

0]) < 0

(48)

where ⎡

−P sai ⎢Qsai − Ui =⎢ ⎣ 0 0

sai

∗ Psai − 2cos sl Qsai + He(Ui A˜ li ) 0 lT T ˜ Bi Ui

⎤ ∗ ∗ ⎥ ⎥ ∗ ⎦ −γs2 I

∗ ∗ I 0

Then, the inequality (49) can be reached by partitioning the matrices Psai and Qsai as the form of Eq. (46) and defining the variables as Eq. (44), which complete the proof.  As mentioned above, it is reasonable to assume that the reference input and the fault are occurred in low frequency domain in this paper. Moreover, the exogenous disturbance may occurred in different frequency range including low, middle and high ones. And because of this, we consider the situation that the disturbance d(k) lies in full frequency domain. The following theorem will provide a condition to capture the stability and the stochastic H∞ performance (21) for system (8). Theorem 4. Let γ d be a given scalar, for all Nk = i ∈ S, consider the system (8) in faultfree and faulty cases (l = 0, 1, . . . , Q), it is MSS and with a stochastic H∞ index γ d to the disturbance d(k) if there exist matrices Xi , Yi , Ni , Mi , A f i , B f i , Bs f i , C f i , D f i and  PilT = Pil =



Pil 1

∗

Pil 2

Pil 3

 l , QlT i = Qi =

Qli1

∗

Qli2

Qli3

 >0

such that the following LMIs hold, ⎡

l

P d i1 − He(Xi )

⎢ ⎢ l ⎢P d i2 − Yi − NiT ⎢ ⎢ lT T ⎢Ai Xi + CilT B Tf i ⎢ ⎢ ⎢ ATf i ⎢ ⎢ ⎢ BilT XiT ⎣ 0 l

where P d i =

 j∈S

∗

∗

∗

∗

P d i3 − He(Ni )

∗

∗

∗

T lT T AlT i Yi + Ci B f i

−Pdl i1

∗

∗

ATf i

−Pdι i2

−Pdι i3

∗

BilT YiT

0

0

−γd2 I

0

D f iCil

Cfi

∗

l

∗

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ < 0 (49) ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎦ I − He(Mi ) ∗

λij Pdl j .

Proof. This theorem gives a bounded real conditions for stochastic MJSs in full frequency domain, which can be deduced by proceeding the similar steps of Theorem 1 in [31], and thus it is omitted here.  Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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15

Theorem 5. Combining Theorems 1–4, feasible solutions to the FD problem investigated in this paper can be obtained by solving the optimization problem as follows s.t.

max 1 βs + 2 β f Eq. (34), l = 1, 2, . . . , Q Eq. (45), l = 1, 2, . . . , Q Eq. (47), l = 0 Eq. (49), l = 0, 1, 2, . . . , Q

(50)

for acceptable given scalars γ s and γ d , where  1 and  2 are weighting factors that should be determined in advance. Consequently, matrices Ni , Mi , A f i , B f i , Bs f i , C f i and D f i can be obtained. Further, the filter parameters can be determined according to the following algorithm, f i , B s f i , Cf i and D f i through Eq. (44) S-1. Calculate the matrices Af i , B parameters above. S-2. Calculate Wi12 and Wi22 by Ni = Wi12 Wi−T WiT12 . 22 S-3. Calculate the FDF parameters through Eq. (40) by ⎡ −T    −T T   Wi12 WiT22 ˘ ˘ ˘ A f i B f i Bs f i Wi22 Wi12 0 A f i B f i G f i ⎢ ⎢ 0 = ⎣ Cf i D f i 0 0 I C˘f i D˘ f i 0 0

by obtained

0

0

⎤

I

⎥ 0⎥ ⎦

0

I (51)

Remark 6. It is easily seen from Theorem 5 that (3 × Q × m¯ + Q + m¯ ) LMIs should be solved to capture the filter gains. And that account might be tremendous for the systems with high dimensions. However, the calculation is proceeded off-line. And thus the method proposed in this paper is feasible for practical systems. 4. Detection threshold design In this section, the threshold for detecting fault is designed and the detection logic unit is carried out based on the results in [32]. Let the root mean square value, which means the average energy of corresponding signals, over a time interval (k0 , kτ )   kτ 1  Jr (τ ) =  r T (k)r(k) (52) τ k=k 0

be the residual evaluation function, where k0 implies the initial evaluation time instant and kτ denotes the whole evaluation time steps. The threshold J th is decided by J th =

sup

d (k)∈L2 ,l=0

Jr (τ )

(53)

and consequently, the occurrence of faults can be detected by comparing Jr (τ ) and J th according to the following relationship: Jr (τ ) > J th ⇒ alarm, Jr (τ ) ≤ J th ⇒ no fault.

(54)

Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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16

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2

1

0 0

10

20

30

40

50 time steps k

60

70

80

90

100

Fig. 2. The accessed transmission node Nk .

5. Simulation examples In this section, an application and simulations will be given to illustrate the effectiveness of the proposed FD method. According to some previous researches for the aircraft dynamic model, it can be easily seen that the nonlinear aircraft dynamic model is established on the basis of Newton’s Second Law of motion. Further, for the sake of decoupling the nonlinear dynamics, the model is decomposed of two models along with longitudinal- and lateral-directional motions, respectively. In this example, we consider a lateral-directional aerodynamic model of the NASA High Alpha Research Vehicle utilized in [33], which is given by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −0.166 0.629 −0.9971 −0.13 0.08 −0.27 0.5083 ⎦x(t ) + ⎣ 0.06 −0.12⎦d (t ) + ⎣−0.05 ⎦s(t ) x˙(t ) = ⎣−12.97 −1.761 3.191 −0.1417 −0.1529 −0.07 0.01 0.19     1 0 1 −1.6 x(t ) + s(t ) y(t ) = 1 1 0 0.7 where the states and the output represent ⎡ ⎤   sideslip angle(◦ ) roll rate(◦ /s) ◦ ⎣ ⎦ roll rate( /s) , output = . state = yaw rate(◦ /s) yaw rate(◦ /s) It is assumed that the sampling period is h = 1s, and the measured data of the system is transmitted by two transmission packets through a constraint networked media, that is, each packet involves one output. As we all known, the data packet may lost when transmitting in the channel. Without loss of generality, according to the protocol we adopted, it is assumed that the accessing node together with the situation of packet dropout (i.e. Nk = 0) at each time instant is varying in the range set S = {0, 1, 2} with the transition probability matrix ⎡ ⎤ 0.0 0.4 0.6  = ⎣0.1 0.4 0.5⎦ 0.0 0.2 0.8 Fig. 2 gives one of the jumping cases. Further, choosing the weighting factor in Eq. (5) as κ = 0.9, a discrete-time MJS as the form of Eq. (6) can be derived consequently. Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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the disturbance d1(k)

1

0.5

0

−0.5

−1

0

10

20

30

40

50 time steps k

60

70

80

90

100

Fig. 3. The disturbance d1 (k).

Given γd = γs = 1, solving the optimization problem (50) with 1 = 1, 2 = −1, we can deduce the parameters of the FD filter are ⎡ ⎤ 0.1091 0.1677 −0.4010 0.0384 0.0505 0.0777 0.1848 −2.3090 ⎢−0.0779 −0.4765 1.3165 −0.0450 −0.0537 0.0713 0.0762 2.5535 ⎥ ⎢ ⎥ ⎢ 0.2053 0.0610 0.2813 −0.0490 −0.0473 0. 0014 0. 1421 0. 7401 ⎥ ⎢ ⎥; [•]1 = ⎢ −0.9999 0.0000 0.0000 ⎥ ⎢ 0.0000 0.0000 0.0000 0.0000 0.0000 ⎥ ⎣ 0.0000 −0.0000 0.0000 −0.0000 0.0000 −0.0000 −0.9999 −0.0000⎦ 3.9014 0.7729 −0.5237 0.5756 0.2199 −0.4332 −0.2715 ⎡ ⎤ −0.0116 0.0396 0.0756 −0.0218 −0.0952 0.1959 −0.3401 −0.0946 ⎢ 0.2388 0.1152 0.1558 −0.0650 0.1435 −0.7238 1.3561 −1.3142⎥ ⎢ ⎥ ⎢ 0.2122 0.1954 0.2436 −0.1167 −0.0916 −0.3101 −0.1216 −0.0736⎥ ⎢ ⎥; [•]2 = ⎢ −1.0000 0.0002 0.0000 ⎥ ⎢ 0.0000 0.0000 0.0000 −0.0000 0.0001 ⎥ ⎣ 0.0000 0.0000 0.0000 −0.0000 −0.0000 0.0000 −1.0000 0.0000 ⎦ 1.5130 0.1363 1.8383 0.1718 −0.3199 0.4150 −0.6530 ⎡ ⎤ 0.7953 1.0853 −0.5763 0.1038 −0.0036 0.3312 2.2375 −1.7230 ⎢−0.3557 −1.0719 1.6814 −0.2212 0.0812 −0.5281 −2.1005 0.9929 ⎥ ⎢ ⎥ ⎢ 0.1847 −0.0477 0.3310 0.0387 0.0609 0. 0218 −0. 5756 0.6737 ⎥ ⎢ ⎥; [•]1 = ⎢ −0.9999 −0.0000 0.0000 ⎥ ⎢ 0.0000 −0.0000 0.0000 0.0001 0.0000 ⎥ ⎣ 0.0000 −0.0000 0.0000 −0.0000 0.0000 −0.0000 −1.0000 0.0000 ⎦ 1.6594 1.2129 0.0528 0.0749 0.4238 −0.2165 1.1690   A f i B f i Bs f i and the optimal values on finite frequency sensitivity perwhere [•]3 = Cfi D fi formance are βs = 0.4010 and β f = 0.3527, respectively. In order to show the effectiveness of the proposed FD method, in the following, we will do some simulations in the following three faulty cases with comparison to the existing methods where only the sensitivity to the fault was considered. In the simulation, the disturbance is assumed to be d = [d1T d2T ]T and d1 is a bound-limited white noise with power of 1 as shown in Fig. 3, d2 = 0.5sin(0.5t ) when 0 < t < 100, and the reference input s(t ) = sin(0.2t ). Case 1: The sensor 1 is stuck at 0.5 from t = 40 s. Fig. 4 shows the residual response r(k) and the corresponding Jr (τ ) when sensor 1 is stuck at 0.5 from t = 40s. It can be seen, from the figure, that both the two methods can detect the Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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2.5 |r(k)| in this paper |r(k)| by existing approach

J (τ) in this paper r

J (τ) by existing approach r

5

2

the threshold in this paper the threshold by existing approach

4

Jr(τ)

|r(k)|

1.5 3

1 2

0.5

1

0

0

20

40 60 time steps k

80

0

100

0

20

40 60 time steps k

80

100

Fig. 4. Residual|r(k)| and residual evaluation Jr (τ ) for Case 2.

0.8 2

|r(k)| in this paper |r(k)| by existing approach

Jr(τ) in this paper

0.7

Jr(τ) by existing approach

0.6

1.5

the threshold in this paper

Jr(τ)

|r(k)|

0.5 1

0.4 0.3 0.2

0.5

0.1 0

0

20

40 60 time steps k

80

100

0

0

20

40 60 time steps k

80

100

Fig. 5. Residual |r(k)| and residual evaluation Jr (τ ) for Case 2.

fault effectively. But the method proposed in this paper completes the detection task earlier than the existing approach since Jr (τ ) > Jth when k = 40 s after the fault occurs when t = 40 s by our method and Jr (τ ) > Jth when k = 43 s by existing method, that represents the advantage of our FD method. Case 2: 20% of the effectiveness of sensor 2 is lost from t = 40s. Fig. 5 shows the residual response r(k) and the corresponding Jr (τ ) when 20% of the effectiveness of sensor 2 is lost from t = 40 s. It is easily seen that from this figure, that Jr (τ ) > Jth when k = 41 s after the fault occurs by our FD method, that means in such a faulty case, the FD method proposed in this paper works well. However, the existing method no longer has any effect for this type of fault. Case 3: The sensor 1 is outage from t = 40s. Fig. 6 shows the residual signal r(k) and the corresponding Jr (τ ) in faulty case 3. It is easily seen that from this figure, that the existing method lost their effect for this type of Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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19

0.8 |r(k)| in this paper |r(k)| by existing approach

Jr(τ) in this paper

0.7

J (τ) by existing approach r

0.6

1.5

the threshold in this paper

J (τ)

1

r

|r(k)|

0.5 0.4 0.3 0.5

0.2 0.1

0

0

20

40 60 time steps k

80

100

0

0

20

40 60 time steps k

80

100

Fig. 6. Residual |r(k)| and residual evaluation Jr (τ ) for Case 3.

fault, however, the FD method proposed in this paper works well since Jr (τ ) > Jth when k = 42 s after the fault occurs. In conclusion, the fault even whose magnitude is small such as in Fig. 4 or the outage fault can be detected promptly and appropriately by our FD approach. However, it must be mentioned that the detection time is closely related to the accessing node of the system. That is to say if the fault is occurred on sensor 1 when t = 40 s, but the accessing node does not include the message of sensor 1 from then on, this fault cannot be detected timely and it can be detected till the node including the message of sensor 1 is accessed the channel. This detection delay can be understood easily and we do not simulate in this paper.

6. Conclusion In this paper, the FD problem for NCSs under multi-packet transmission has been addressed in finite frequency domain against sensor failures. The whole dynamics of the considered system have been modeled in the framework of MJS, where the jumping parameters implied the accessing transmission node. Then with the aid of the performance indices given in finite frequency domain, the FD scheme has been proposed, where the generated residual was designed to be sensitive to the fault and the reference input in faulty cases. Most importantly, the FD filters complete the FD task with only part of the available outputs’ information. In such a FD mechanism, the senor faults even the magnitude of it is small or outage ones can be detected effectively. Finally, the advantages of the developed FD approach have been illustrated through a simulation example.

Acknowledgments This work is supported in part by National Natural Science Foundation of China (Grant nos. 61403176 and 61673198), Natural Science Foundation of Liaoning Province (Grant no. 2015020088). Please cite this article as: Y. Long, C. Xu, Fault detection in finite frequency domain for constrained networked systems under multi-packet transmission, Journal of the Franklin Institute (2017), http://dx.doi.org/10.1016/j.jfranklin.2017.05.043

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