network-based fault detection for discrete-time state-delay systems: a new measurement model

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2008; 22:510–528Published online 3 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.1000

TECHNICAL NOTE

Network-based fault detection for discrete-time state-delaysystems: A new measurement model

Xiao He1, Zidong Wang2, Y. D. Ji3 and D. H. Zhou1,∗,†

1Department of Automation, Tsinghua University, Beijing 100084, People’s Republic of China2Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K.

3Research Institute of Information Technology, Tsinghua University, Beijing 100084, People’s Republic of China

SUMMARY

In this paper, the fault detection problem is studied for a class of discrete-time networked systems withmultiple state delays and unknown input. A new measurement model is proposed to account for both therandom measurement delays and the stochastic data missing (package dropout) phenomenon, which aretypically resulted from the limited capacity of the communication networks. At any time point, one of thefollowing cases (random events) occurs: measurement missing case, no time-delay case, one-step delaycase, two-step delay case, . . . , q-step delay case. The probabilistic switching between different cases isassumed to obey a homogeneous Markovian chain. We aim to design a fault detection filter such that, forall unknown input and incomplete measurements, the error between the residual and weighted faults ismade as small as possible. The addressed fault detection problem is first converted into an auxiliary H∞filtering problem for a certain Markovian jumping system (MJS). Then, with the help of the bounded reallemma of MJSs, a sufficient condition for the existence of the desired fault detection filter is establishedin terms of a set of linear matrix inequalities (LMIs). A simulation example is provided to illustrate theeffectiveness and applicability of the proposed techniques. Copyright q 2007 John Wiley & Sons, Ltd.

Received 1 March 2007; Revised 18 July 2007; Accepted 18 July 2007

KEY WORDS: fault detection; networked systems; random measurement delay; data missing; Markovianjumping system; LMI

∗Correspondence to: D. H. Zhou, Department of Automation, Tsinghua University, Beijing 100084, People’s Republicof China.

†E-mail: [email protected], [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 60574084Contract/grant sponsor: National 863 Project of China; contract/grant number: 2006AA04Z428Contract/grant sponsor: National 973 Program of China; contract/grant number: 2002CB312200

Copyright q 2007 John Wiley & Sons, Ltd.

FAULT DETECTION FOR DISCRETE-TIME STATE-DELAY SYSTEMS 511

1. INTRODUCTION

Fault detection and isolation (FDI) and fault-tolerant control (FTC) have been active fields ofresearch over the past decades due to an increasing demand for higher performance, as well as dueto higher safety and reliability standards [1–7]. The main purpose of FDI is to construct a residualsignal and compare it with a predefined threshold. When the residual exceeds the threshold, analarm is generated. Recently, the model-based approaches to FDI problems for dynamic systemshave received more and more attention, and the basic idea is to introduce a performance index inorder to recast the FDI problem into an associated optimization problem. For example, in [8, 9],the H∞ norm of transfer function matrix from unknown input to residual has been designed to besmall, whereas the H∞ norm (or the smallest nonzero singular value) of transfer function matrixfrom fault to residual has been guaranteed to be large. In [10], the error between the residual andweighted faults has been made as small as possible, and then the FDI problem can be solved byusing the H∞ filtering approach.

The recent advances in network technologies have led to more and more feedback control systemswith control loops closed via digital communication channels. Compared with the traditional point-to-point wiring, the use of the communication channels can reduce the costs of cables and power,simplify the installation and maintenance of the whole system and increase the reliability. Network-based analysis and designs have many industrial applications in automobiles, manufacturing plants,aircrafts, signal system of high-speed railway and HVAC systems. However, the insertion of thecommunication channels creates discrepancies between the data records to be transmitted and itsassociated remotely transmitted images and hence raises new interesting and challenging problemssuch as network-induced delays or packets dropout, see [11–16] for some representative works.

Since network-induced delays are inherently random and time varying [17–19], they have beenmodeled in various probabilistic ways [13, 20, 21]. One of the attractive approaches is to use binaryswitching sequence viewed as a Bernoulli distributed white sequence taking on values of 0 and 1,since such a representation is very effective to describe network-induced delays [22–24]. Anotherproblem in networked systems is the data missing (dropout) phenomenon [25], which can also bedescribed using a binary switching sequence [26]. Very recently, in [27], the network-induced delayand data dropout problems have been investigated in an integrated way within a unified framework,and the robust H∞ filtering problem with polytopic uncertainties has been thoroughly studied.In all the aforementioned results, it has been assumed that the delay or missing characteristicsare statistically mutually independent from transfer to transfer. However, such an independenceassumption may have limitations since, in network-based signal transmissions, time delays anddata dropouts typically occur in a batch mode, and the transition from one mode to another mayobey certain probability distribution. Therefore, a seemingly more realistic way is to assume thatthe switching between difference modes abides by a Markovian chain. On the other hand, the faultdetection problem in a networked environment has recently gained some initial research attention[7, 28–32]. So far, to the best of the authors’ knowledge, the fault detection problems for NCSswith simultaneous measurement delays and data missing described by a Markovian chain have notbeen fully investigated, which motivates our present study.

In this paper, we aim at solving the fault detection problem for a class of networked systems withmultiple state delays and unknown inputs. The measured data are transmitted over communicationnetwork, and a unified representation is proposed to describe data missing and measurementdelays simultaneously. Different from the conventional assumption on the mutual independence,we assume that the stochastic variable in the measurement description varies in a Markovian

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2008; 22:510–528DOI: 10.1002/acs

512 X. HE ET AL.

jumping fashion. By properly augmenting the states of the original system and the fault detectionfilter, the addressed fault detection filter design problem can be transformed into an auxiliaryH∞ filtering problem [33, 34] for the resulting Markovian jumping system (MJS). Using thebounded real lemma (BRL) for MJSs, a sufficient condition for the existence of the desired faultdetection filter is established in terms of certain linear matrix inequalities (LMIs). When these LMIsare feasible, the explicit expression of the desired fault detection filter can also be determined.A simulation example is provided to demonstrate the usefulness of the present methods.

Notation: The notations used throughout the paper are fairly standard. Rn and Rn ×m denote,respectively, the n dimensional Euclidean space and the set of all n ×m real matrices. P>0 meansthat P is real symmetric and positive definite. The superscript ‘T’ denotes the matrix transpose.Pr{·} represents the occurrence probability of the event ‘·’, and when x and y are both stochasticvariables, E{x} stands for the mathematical expectation of x . l2[0, ∞) is the space of all square-summable vector functions over [0,∞), with ‖x‖ the standard l2 norm of x , i.e. ‖x‖= (xTx)1/2.RH∞ is the set of proper and stable rational functions with real coefficients. In symmetric blockmatrices, we use ‘∗’ to represent a term that is induced by symmetry, and diag{· · ·} stands for ablock-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to becompatible for algebraic operations and, sometimes, the arguments of a function will be omittedin the analysis when no confusion can arise.

2. PROBLEM FORMULATION AND PRELIMINARIES

There are many ways to define quality-of-service (QoS) for networked systems [35]. In this paper,communication network is inserted between the remote system and the fault detection filter, andthree of the most popular QoS measurements determining the performance of fault detection aretaken into account: (1) the point-to-point maximal allowable delay bound (MADB) of the packetthat is used to denote how long a packet is expected to be delivered from the remote system to thefault detection filter; (2) the point-to-point network allowable data dropout rate that is employed tocharacterize the probability of data packet dropout in data transmission; and (3) the point-to-pointnetwork throughput that is used to indicate how fast the signal can be sampled and sent as packetthrough the network.

For the networked system considered in this paper, we assume that the measurement at time kis transmitted in a single packet, and different data packets have the same length L . We introducea certain MADB q , and packets with delays more than q are discarded (considered as dropout).As will be seen later, we will use different measurement delays and data dropout (missing) rates toquantify the random packet delay and losses in the communication channel. On the other hand, fora fixed number of sensors, the sampling period h is decided by network throughput Qik , and weassume that a sampled-data model can be obtained through online measurement such as sendingprobing data packet to measure network characteristics and QoS scheduling.

Consider the following state-delayed networked system after sampling:

xk+1 = A0xk +q∑

i=1Ai xk−i + Bwwk + B f fk

yk = �(�k, 0)C0xk +q∑

i=1�(�k, i)Ci xk−i + Dwk

xk = �k, k =−q,−q + 1, . . . , 0

(1)



where xk ∈ Rn is the state vector; wk ∈ Rp is the unknown input belonging to l2[0, ∞); fk ∈ Rl isthe fault to be detected; 1�i�q (q�1) are integer time delays. yk ∈ Rm is the measured outputvector, which may contain random communication delays and stochastic data missing. �k is agiven real initial sequence on [−q, 0], and �(·, ·) stands for the Kronecker delta, i.e.

�( j, l) ={0 if j �= l

1 if j = l

Furthermore, �k is a stochastic variable whose role is to determine, at time k, the size of theoccurred delay as well as the possibility of data missing. In this paper, {�k} is assumed to be adiscrete-time homogeneous Markov chain taking values in the finite state space

� := {−1, 0, . . . , q} (2)

and stationary transition probability matrix �= [�i j ], where�i j :=Pr{�k+1 = j |�k = i} (3)

Finally, all system matrices in (1) have appropriate dimensions and are assumed to be known realconstant matrices.

Remark 1The state in the state space of the Markov chain corresponding to the measurement missingsituation is set to −1. This will not lose any generality but will bring us convenience in denotingthe transition probability matrix. It should be noted that, when �k = −1, the measurement yk willconsist of pure noise only.

Remark 2The system measurement described in (1) can be regarded as a substantial extension of the delayedsensor model in [22, 24] and the missing measurement model in [23, 26, 36]. The main advantageof such a representation is that it can account for both the random communication delays andthe stochastic data missing in a unified way. Our results cover the measurement mode in [22] byimposing Pr{�k = 0} + Pr{�k = 1} = 1, and Pr{�k = i}= 0, ∀2�i�q .

Case study: To further explain the real applications of the measurement model, we give thiscase study. For a certain time instant k, the measurement is ideally (without any delays or datamissing) transmitted over the network with a probability 0.5, one-step delay occurs with prob-ability 0.25, two-step delay happens with probability 0.15, and the measurements are missingwith the probability 0.1. The measurement model in (1) with Pr{�k =−1}= 0.1, Pr{�k = 0}= 0.5,Pr{�k = 1}= 0.25 and Pr{�k = 2} = 0.15 can properly describe this situation.

Remark 3Different from [22, 24, 26], we assume that {�k} is a discrete-time homogeneous Markov chainwith finite state space and stationary transition probability matrix. Compared with the mutualindependence assumption of �k from transfer to transfer, the current assumption is more reasonablewhen describing the influence introduced by the limited-bandwidth communication network. Sincethe status of the network varies slower than the sampling period and the characteristics of thenetwork at a certain time are usually dependent on the superior time instant, our ‘Markov jumping’assumption of {�k} is able to describe this phenomenon properly. Generally speaking, transition


514 X. HE ET AL.

probability matrix depends on the network status. By a number of offline experiments, the transitionprobability matrix under a certain network status can be determined priorly. Furthermore, theMarkov state at a certain time can also be determined by introducing timestamps along with themeasurements that are transmitted over the network.

Consider a full-order fault detection filter of the following form:

xk+1 = G(�k)xk + K (�k)yk

rk = L(�k)xk(4)

where xk ∈ Rn is the filter state vector, and rk ∈ Rl is the so-called residual that is compatible withthe fault vector. For �k = i ∈ �, we denote matrices G(�k), K (�k) and L(�k) as Gi =G(�k = i),Ki = K (�k = i) and Li = L(�k = i). Our main aim is to make the error between the residual andfault signals as small as possible.

For the purpose of fault detection, it is not necessary to estimate the fault fk . Sometimes oneis more interested in the fault signal of a certain frequency interval, which can be formulated asthe weighted fault:

f (z) = T f (z) f (z) (5)

where T f (z) ∈RH∞ is a prescribed weighting matrix.

Remark 4Similar to [10], the introduction of a suitable weighting matrix T f (z) can limit the frequencyinterval of interest, and the system performance can then be improved. In fact, the use of weightedfault f (z) is more general than using the original fault f (z), because if we impose T f (z) = I , wecan obtain f (z) = f (z).

Suppose a minimal realization of f (z) = T f (z) f (z) is

xk+1 = At xk + Bt fk

fk =Ct xk + Dt fk(6)

where xk ∈ Rn is the weighted fault state, fk ∈ Rl is the original fault and fk ∈ Rl is the weightedfault. At , Bt , Ct and Dt are assumed to be known real constant matrices with appropriate dimen-sions.

By defining

vk =[wTk f Tk ]T, rk = rk − fk, xk =[xTk−1 · · · xTk−q ]T, �k =[xTk xTk xTk xTk ]T (7)

and, again, denoting matrices A(�k), B(�k), C(�k) and D(�k) as Ai = A(�k = i), Bi = B(�k = i),Ci = C(�k = i) and Di = D(�k = i), we have the overall fault detection dynamics governed by thefollowing system:

�k+1 = Ai�k + Bivk

rk = Ci�k + Divk

(8)



where

Ai :=[Ai 0

0 At

], Bi :=

[Bi

Bt

], Ci := [Ci − Ct ], Di := [0 − Dt ]

Ai :=⎡⎢⎣

A0 Ad 0

A21 A22 0

�(i, 0)K0C0 KiCiei Gi

⎤⎥⎦ , Bi :=

⎡⎢⎣

Bw B f

0qn×p 0qn×l

Ki D 0

⎤⎥⎦ , Bt := [0 Bt ]

A21 :=[

In

0(q−1)n×n

], A22 :=

[0 0

I(q−1)n×(q−1)n 0

], Ad := [A1 · · · Aq ]

ei := [�(i, 1)In×n · · · �(i, q)In×n], Ci := [0l×n 0l×qn Li ], Di := Di

After the above manipulations, the admissible sensor delays and data missing can be reformulatedas the jumping parameters of an MJS (8). Recall the following definition of mean square stabilityfor MJSs.

Definition 1 (Costa and Fragoso [37])System (8) with vk = 0 is said to be mean square stable if

E{‖�k‖2} → 0 as k → ∞for any initial condition �0 and initial distribution �0 ∈ �.

Assumption 1System (1) with wk = 0 and fk = 0 is assumed to be mean square stable.

In this paper, we do not consider any control issues for networked systems but focus on thefault detection filter design problem. Hence, Assumption 1 is a prerequisite for the fault detectiondynamics (8) to be mean square stable, since xk is one part of �k and the fault detection filter (4)cannot affect the state of the original system (1).

With Definition 1, we can further transform the fault detection filter design problem of system(1) into an H∞ filtering problem for an MJS. What we need to do here is to find the filter parametersGi , Ki and Li (i ∈ �) such that the fault detection dynamics in (8) is mean square stable and theinfimum of � is made small in the feasibility of

supvk �=0

E{‖rk‖2}‖vk‖2 <�2, �>0 (9)

We adopt a residual evaluation stage including an evaluation function J (k) and a threshold Jthof the following form as in [10]:

J (k) ={

k∑s=0

rTs rs

}1/2

(10)

Jth = supw∈l2, f = 0

E{J (L)} (11)

where L denotes the maximum time step of the evaluation function.


516 X. HE ET AL.

On the basis of (10), the occurrence of faults can be detected by comparing J (k) with Jthaccording to the following rule:

J (k)>Jth �⇒with faults�⇒ alarm (12)

J (k)�Jth �⇒ no faults (13)

3. FAULT DETECTION FILTER DESIGN

In this section, we shall discuss the fault detection filter design problem using an LMI approach.The following BRL is useful in deriving our main results in the sequel.

Lemma 1 (Costa and Marques [38] (Discrete BRL for MJS))Consider the following MJS:

xk+1 = A(�k)xk + B(�k)wk

yk =C(�k)xk + D(�k)wk

x0 = x0, �0 = j0 ∈ {1, . . . , N }(14)

where xk is the state vector; wk is the unknown input belonging to l2[0,∞); and yk is the outputvector. {�k} is a discrete-time homogeneous Markov chain with finite state space {1, . . . , N } andstationary transition probability matrix �= [�i j ], where �i j := Pr{�k+1 = j |�k = i}. A(�k), B(�k),C(�k) and D(�k) are known real matrices of appropriate dimensions for all �k ∈ {1, . . . , N }; the ma-trices associated with �k = i are denoted by A(�k) = Ai , B(�k) = Bi , C(�k) =Ci and D(�k) = Di .Let �>0 be a given scalar, then the system is mean square stable with wk = 0 and, under zeroinitial conditions, satisfies:

supwk �=0

E{‖yk‖2}‖wk‖2 <�2

if there exist matrices Pi>0 (i = 1, . . . , N ) such that the following LMIs⎡⎢⎢⎢⎢⎣

−Pi ATi Pwi 0 CT

i

∗ −Pwi Pwi Bi 0

∗ ∗ −�2 I DTi

∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎦<0 (15)

hold for i = 1, . . . , N , where

Pwi =N∑j=1

�i j Pj (16)

In Section 2, we have converted the fault detection filter design problem for a class of state-delaynetworked systems with missing and delayed measurements into the H∞ filtering problem of anMJS. With the help of Lemma 1, we are now in the position to present our main result in thispaper.



Theorem 1Consider system (1) and let �>0 be a given scalar. There exists an admissible full-order faultdetection filter of the form (4) ensuring that the overall fault detection dynamics (8) is meansquare stable and the constraint (9) is satisfied, if there exist matrices 0<XT

i = Xi ∈ R(q+2)n×(q+2)n ,Si ∈ Rn×n , Zi ∈ Rn×n , Yi ∈ Rn×n , Gi ∈ Rn×n , Ki ∈ Rn×m , Li ∈ Rl×n , Mi ∈ Rqn×qn , i =−1, . . . , q ,and 0<PT

t = Pt ∈ Rn×n , such that the following LMIs

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Xi �12 0 �14 0 0

∗ �22 �23 0 0 0

∗ ∗ −�2 I DTi 0 BT

t Pt

∗ ∗ ∗ −I −Ct 0

∗ ∗ ∗ ∗ −Pt ATt Pt

∗ ∗ ∗ ∗ ∗ −Pt

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

<0 (17)

hold for i =−1, . . . , q , where

�12 :=

⎡⎢⎢⎢⎢⎢⎣

AT0 Z

Ti [In×n 0n×(q−1)n]MT

i AT0Y

Ti + �(i, 0)CT

0 KT0 + GT

i

ATd Z

Ti

[0(q−1)n×n I(q−1)n×(q−1)n

0n×n 0n×(q−1)n

]MT

i ATdY

Ti + eTi C

Ti K

Ti

AT0 Z


i AT0Y

Ti + �(i, 0)CT

0 KT0

⎤⎥⎥⎥⎥⎥⎦

�14 :=

⎡⎢⎢⎣LTi

0

0

⎤⎥⎥⎦ , �22 :=−

⎡⎢⎢⎣Zi + ZT

i 0 Zi + Y Ti + STi

∗ Mi + MTi 0

∗ ∗ Yi + Y Ti

⎤⎥⎥⎦ + Xi

�23 :=

⎡⎢⎢⎣

Zi Bw Zi B f

0 0

Yi Bw + Ki D Yi B f

⎤⎥⎥⎦ , Xi :=

q∑j=−1

�i j X j

and Bi , Di are defined in (8). Moreover, if (17) is feasible, the parameters of the desired faultdetection filter can be given by

Gi = V−1i Gi S

−1i Vi , Ki = V−1

i Ki , Li = Li S−1i Vi (18)

where Vi ∈ Rn × n is any invertible matrix (for example, Vi could be set as I ).

ProofFrom Lemma 1, it can be concluded that, for a given scalar �>0, system (8) is mean squarestable and the constraint (9) is satisfied under zero initial condition, if there exist positive-definite


518 X. HE ET AL.

matrices

Pi =[Pi 0

0 Pt

]>0 (19)

such that LMIs ⎡⎢⎢⎢⎢⎢⎣

−Pi ATi Pwi 0 CT

i

∗ −Pwi Pwi Bi 0

∗ ∗ −�2 I DTi

∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎥⎦<0 (20)

hold for i =−1, . . . , q , where Ai , Bi , Ci , Di are defined in (8) and

Pwi =q∑

j=−1�i j Pj , i =−1, . . . , q (21)

Therefore, what we need to do afterwards is to prove that the LMIs in (20) hold if the LMIs in(17) hold. Note that the LMIs in (20) can be rewritten as⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Pi ATi Pwi 0 CT

i 0 0

∗ −Pwi Pwi Bi 0 0 0

∗ ∗ −�2 I DTi 0 BT

t Pt

∗ ∗ ∗ −I −Ct 0

∗ ∗ ∗ ∗ −Pt ATt Pt

∗ ∗ ∗ ∗ ∗ −Pt

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

<0 (22)

where

Pwi =q∑

j=−1�i j Pj , i =−1, . . . , q (23)

Following the same steps as in the proof of Theorem 1 in [39], it can be shown that LMIs (22)are feasible if and only if there exist matrices Pi>0, Pt>0 and Qi satisfying LMIs⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Pi ATi Q

Ti 0 CT

i 0 0

∗ −Qi − QTi + Pwi Qi Bi 0 0 0

∗ ∗ −�2 I DTi 0 BT

t Pt

∗ ∗ ∗ −I −Ct 0

∗ ∗ ∗ ∗ −Pt ATt Pt

∗ ∗ ∗ ∗ ∗ −Pt

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

<0, i =−1, . . . , q (24)



Letting Ui ∈ Rn × n and Vi ∈ Rn × n are two nonsingular matrices, we set

QTi =

⎡⎢⎢⎣Y Ti 0 QT

31i

0 MTi 0

V Ti 0 QT

33i

⎤⎥⎥⎦ (25)

Introducing new matrices

QTi =

⎡⎢⎢⎣Z−Ti 0 QT

31i

0 I 0

UTi 0 QT

33i

⎤⎥⎥⎦ (26)

where the entries QT31i , Q

T33i , Q

T31i and QT

33i are uniquely determined from the following relation:[Y Ti QT

31i

V Ti QT

33i

] [Z−Ti QT

31i

UTi QT

33i

]=

[Z−Ti QT

31i

UTi QT

33i

][Y Ti QT

31i

V Ti QT

33i

]= I (27)

we further have the following relation:

QTi Q

Ti = QT

i QTi =

⎡⎢⎢⎣I 0 0

0 MTi 0

0 0 I

⎤⎥⎥⎦ (28)

Defining

Ti =

⎡⎢⎢⎣ZTi 0 Y T

i

0 I 0

0 0 V Ti

⎤⎥⎥⎦ (29)

we obtain

T Ti Qi AT

i QTi Q

Ti Ti =

⎡⎢⎢⎢⎣

AT0 Z


i AT0Y

Ti + �(i, 0)CT

0 KT0 V

T0 + ZiUiG

Ti V

Ti

ATd Z

Ti

[0(q−1)n×n I(q−1)n×(q−1)n

0n×n 0n×(q−1)n

]MT

i ATd Y

Ti + eTi C

Ti K

Ti V

Ti

AT0 Z


i AT0Y

Ti + �(i, 0)CT

0 KT0 V

T0

⎤⎥⎥⎥⎦

T Ti Qi C

Ti =

⎡⎢⎢⎣ZiUi L

Ti

0

0

⎤⎥⎥⎦ , T T

i Qi (Qi + QTi )QT

i Ti =

⎡⎢⎢⎣Zi + ZT

i 0 Zi + Y Ti + STi

∗ Mi + MTi 0

∗ ∗ Yi + Y Ti

⎤⎥⎥⎦

T Ti Qi Qi Bi =

⎡⎢⎣

Zi Bw Zi B f

0 0

Yi Bw + Vi Ki D Yi B f

⎤⎥⎦


520 X. HE ET AL.

Performing congruence transformations to (24) by diag{QTi Ti , Q

Ti Ti , I, I, I, I } and defining

Xi = T Ti Qi Pi Q

Ti Ti , Xi =

q∑j=−1

�i j X j , Gi = ViGiUTi Z

Ti

Ki = Vi Ki , Li = LiUTi Z

Ti , Si = ViU

Ti Z

Ti , (i =−1, . . . , q)

we can easily obtain (17). Hence, if there exist matrices Xi>0, Si , Zi , Yi , Gi , Ki , Li , Mi ,i =−1, . . . , q , and Pt>0 such that LMIs (17) are feasible, the overall fault detection dynamic (8)is mean square stable and the constraint (9) is satisfied.

Furthermore, from LMIs (17), we can obtain that for i = −1, . . . , q ,⎡⎢⎢⎢⎣Zi + ZT

i 0 Zi + Y Ti + STi

∗ Mi + MTi 0

∗ ∗ Yi + Y Ti

⎤⎥⎥⎥⎦>0 (30)

This indicates that Zi and Yi are nonsingular, and

[I 0 − I ]

⎡⎢⎢⎢⎣Zi + ZT

i 0 Zi + Y Ti + STi

∗ Mi + MTi 0

∗ ∗ Yi + Y Ti

⎤⎥⎥⎥⎦

⎡⎢⎢⎣

I

0

−I

⎤⎥⎥⎦= −Si − STi >0 (31)

which implies that Si is nonsingular and also ensures the existence of parameter matrices Gi , Kiand Li in (18). The proof is completed. �

We focus on the fault detection problem of networked system, and Theorem 1 can be potentiallyapplied to practical networked systems mentioned in Part I whose measurement is transmittedthrough network.

Remark 5In Theorem 1, there are no products of unknown matrices Xi , Si , Zi , Yi , Mi , Pt with filter param-eters Gi , Ki and Li ; hence, the full-order fault detection filter can be obtained from the solutionof convex optimization problems in terms of LMIs, which can be solved using efficient interior-point algorithms [40]. The computational complexity depends mainly on the number of decisionvariables; in practice, this LMI optimization is generally solvable when the system dimension isnot too large. Note that the augmentation of the state vector in our manipulation is according tothe MADB q , and the feasibility of (17) mainly depends on the delay and missing characteristicsof the network, i.e. �k . Generally, we declare that the smaller the MADB q , the lesser the datamissing rate and the better the feasibility of LMI (17).

Remark 6The possible conservatism in Theorem 1 is threefold: firstly, Lemma 1 is generally a sufficientcondition ensuring the MJS (13) to be mean square stable as well as satisfying the H∞ atten-uation constraint [38]; secondly, block-diagonal Lyapunov matrices (19) and matrices (25) withconstraints in structures are used in our derivation, which will bring conservatism in the design



procedure; thirdly, for different Markov mode i , we impose the same Pt , which will inevitablyadd conservatism to our result. How to further reduce the conservatism would be one of our futuretopics.

Remark 7In most cases, we can know the size of the measurement delay or whether the data are missingat a certain time by using the time-stamp at the system node [12], and therefore the jumpingparameters of the transformed MJS are accessible. In this sense, Theorem 1 provides us witha network-status-dependent fault detection filter design method. On the other hand, if the networkstatus is not accessible, i.e. the jumping parameters of the transformed MJS are unavailable,a network-status-independent result can be easily obtained by imposing

Si = S, Gi = G, Ki = K , Li = L (32)

in Theorem 1.

Note that (17) is LMIs over both the matrix variables and the prescribed scalar �2. This impliesthat the scalar �2 can be included as one of the optimization variables for LMIs (17), which makesit possible to obtain the minimum noise attenuation level bound for the overall fault detectiondynamics (8). Then, a sub-optimal fault detection filter can be readily found by solving thefollowing convex optimization problems:

Problem 1The network-status-dependent sub-optimal fault detection filter design problem for networkedsystems with multiple state delays can be brought forward as follows:

minXi>0,Si ,Zi ,Yi ,Mi ,Gi ,Ki ,Li ,Pt>0,−1�i�q

�2 s.t. (17) (33)

Problem 2The network-status-independent sub-optimal fault detection filter design problem for networkedsystems with multiple state delays can be brought forward as follows:

minXi>0,Si ,Zi ,Yi ,Mi ,Gi ,Ki ,Li ,Pt>0,−1�i�q

�2 s.t. (17) and (32) (34)

For the problems mentioned above, the parameters of the sub-optimal fault detection filter canbe determined using (18), and the sub-optimal H∞ attenuation level for fault detection dynam-

ics is given by �∗ =√

�2opt, where �2opt is the sub-optimal solution of the corresponding convexoptimization problems.

4. A NUMERICAL EXAMPLE

To illustrate the proposed method, we give a simulation example in this section. The parametersof the discrete-time networked system (1) with multiple state delays and unknown input are given


522 X. HE ET AL.

as follows:

A0 =[

0 0.5

0.2 0.3

], A1 =

[0.2 0

0.7 0.1

], Bw =

[0.5

0.3

]

B f =[−1

2

], C0 =C1 =

[0.2 0

0 0.5

], D =

[0.2

−0.1

]

The initial state values �k are set to be �−1 = �0 = 0.We first deal with the case where there is only packet loss. The Markov chain has two states: (1)

data ideally transmitted (�k = 0) and (2) data missing (�k = −1); we give the transition probabilitymatrix as

� :=[�−1,−1 �−1,0

�1,−1 �1,0

]=

[0.6 0.4

0.2 0.8

]

Problem 1 can then be solved by using the Matlab LMI toolbox [40]. As a result, the minimumnoise attenuation level bound of the fault detection dynamic is �opt = 1.0035, and the parametersof the sub-optimal fault detection filter are

G−1 =[0.0204 1.5916

0.0934 0.7113

], K−1 =

[−1.3467 −2.6781

−0.5647 −1.1227

], L−1 = [−0.5222 0.0370]

G0 =[

0.0039 0.9730

−0.0246 0.2386

], K0 =

[0.0006 −0.0063

−0.0012 −0.0013

], L0 =[−0.0605 − 0.4859]

The fault detection filter design procedure in the case only with measurement delays is similar tothe case with only packet loss, which is omitted for the limitation of the space.

From now on, we consider the case with both measurement delay and data missing. Let q = 1,so that the state-space of the Markov chain {�k} is �= {−1, 0, 1}. The transition probability matrixis given by

� :=⎡⎢⎣

�−1,−1 �−1,0 �−1,1

�0,−1 �0,0 �0,1�1,−1 �1,0 �1,1

⎤⎥⎦=

⎡⎢⎣0.5 0.4 0.1

0.2 0.6 0.2

0.2 0.3 0.5

⎤⎥⎦

and the initial mode is set to be �0 = 0. For k = 0, 1, . . . , 300, the unknown input wk is supposedto be a random noise uniformly distributed over [−0.5, 0.5], and the fault signal fk is given as

fk ={1 for k = 100, 101, . . . , 200

0 others

The weighting matrix is supposed to be T f (z) = (0.5z)/(z − 0.5), with state space realization

xk+1 = 0.5xk + 0.25 fk

fk = xk + 0.5 fkx0 = 0

(35)

where fk and fk are shown in Figure 1.



0 50 100 150 200 250 300

0

0.2

0.4

0.6

0.8

1

time step k

f and

f w

ffw

Figure 1. Fault fk and weighting fault fk( fw).

With the prespecified parameters, from Theorem 1, we can obtain the minimum noise attenuationlevel bound of the fault detection dynamics (8) �opt = 1.0036, and the parameters of the sub-optimalfault detection filter in different modes are given by

G−1 =[−0.0178 1.6075

0.0689 0.7456

], K−1 =

[1.3963 2.8091

−3.2281 −6.4486

], L−1 = [−0.5358 0.0140]

G0 =[−0.0019 0.8511

−0.0304 0.2617

], K0 =

[0.0009 −0.0073

−0.0010 −0.0017

], L0 = [−0.0593 − 0.3227]

G1 =[−0.2452 1.1363

−0.1616 0.6143

], K1 =

[−0.0056 −0.0003

−0.0067 −0.0005

], L1 = [−0.4576 0.1971]

Next, we consider the time-domain simulation using the obtained fault detection filter.Figure 2 shows the measurement mode with random delays and stochastic missing phenomenon.�k =−1, 0, 1 means that the measurement is missing, transmitted over the network ideally andwith one-step delay, respectively.

Figure 3 shows the generated residual signal rk , and the evolution of J (k) = {∑ks=0 r

T(s)r(s)}1/2is presented in Figure 4. We select a threshold as Jth = sup f =0 E {∑300

s=0 rT(s)r(s)}1/2 and, after

400 times simulations, we obtain an average value Jth = 1.252× 10−4. From Figure 4, it can beshown that 0.898× 10−4 = J (105)<Jth<J (106)= 1.305× 10−4; thus, the fault can be detectedin six time steps after its occurrence.


524 X. HE ET AL.

0 50 100 150 200 250 300

0

0.5

1

1.5

time step k

τ k

Figure 2. Measurement mode over network.

0 50 100 150 200 250 300

0

0.005

−0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

time step k

r k

Figure 3. Residual signal rk .



0 50 100 150 200 250 3000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

time step k

J k

Figure 4. Evolution of J (k).

0 50 100 150 200 250 300

0

−1

1

2

3

4

5x 10−3

time step k

r

Figure 5. Residual signal rk .


526 X. HE ET AL.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2x 10−3

time step k

J

Figure 6. Evolution of J (k).

Now, we consider the network-status-independent fault detection filter by solving Problem 2.The minimum noise attenuation level bound of the fault detection dynamics is �opt = 1.0042, andthe parameters of the sub-optimal fault detection filter in different modes are given by

G =[0.5243 1.2985

0.1276 0.2055

], K = 10−3 ×

[0.1849 −0.0900

−0.1829 −0.0037

], L =[−1.0608 − 0.9943]

The generated residual signal rk and the evolution of J (k) are presented in Figures 5 and 6,respectively. A threshold is set to be Jth = 4.7494× 10−6. From Figure 6, we can observe that3.649× 10−6 = J (113)<Jth<J (114)= 4.885× 10−6; thus, the fault can be detected in 14 timesteps after its occurrence.

5. CONCLUSIONS

In this paper, the fault detection filter design problem for a class of discrete-time networked systemswith multiple state delays and unknown input has been studied. The random delay and stochasticmissing phenomenon in the measurements have been simultaneously considered. After properlyaugmenting the states of the original system and the fault detection filter, we have formulated thefault detection filter design problem as an H∞ filtering problem for the resulting MJS. Sufficientcondition for the existence of the desired fault detection filter has been established in terms ofcertain LMIs. A numerical example has been given to illustrate the effectiveness of the proposedmethodologies.



ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China under Grant 60574084,the National 863 Project of China under Grant 2006AA04Z428 and the National 973 Program of Chinaunder Grant 2002CB312200.

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network-based fault detection for discrete-time state-delay systems: a new measurement model

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