Int. J. Appl. Math. Comput. Sci., 2008, Vol. 18, No. 4, 525–537DOI: 10.2478/v10006-008-0046-3

FAULT DIAGNOSIS OF NETWORKED CONTROL SYSTEMS

CHRISTOPHE AUBRUN, DOMINIQUE SAUTER, JOSEPH YAMÉ

Centre de Recherche en Automatique de NancyCRAN-UMR 7039, Nancy-Université, CNRS, F-54506 Vandoeuvre-lès-Nancy Cedex, France

e-mail: christophe.aubrun@cran.uhp-nancy.fr

Networked Control Systems (NCSs) deal with feedback control systems with loops closed via data communication net-works. Control over a network has many advantages compared with traditionally controlled systems, such as a lower im-plementation cost, reduced wiring, simpler installation and maintenance and higher reliability. Nevertheless, the network-induced delay, packet dropout, asynchronous behavior and other specificities of networks will degrade the performance ofclosed-loop systems. In this context, it is necessary to develop a new theory for systems that operate in a distributed andasynchronous environment. Research on Fault Detection and Isolation (FDI) for NCSs has received increasing attention inrecent years. This paper reviews the state of the art in this topic.

Keywords: networked control systems, fault diagnosis, fault tolerant control, network-induced time delays, packet losses,limited communication.

1. Introduction

In recent years, an increasing amount of research ad-dresses the problem of NCSs. For this class of systems, acommunication network is used as a feedback medium asshown in Fig. 1. The role of the communication networkis to ensure data transmission and coordinating manipula-tion among spatially distributed components. Comparedwith conventional point-to-point control systems, the ad-vantages of NCSs are less wiring, a lower installation cost,as well as greater flexibility in diagnosis and maintenance.Thanks to these distinctive benefits, typical applications ofthese systems range over various fields, such as automo-tive, mobile robotics, advanced aircraft, and so on. How-ever, the introduction of communication networks in thecontrol loops makes the analysis and synthesis of NCSscomplex. There are several network-induced effects thatarise when dealing with an NCS, such as time delays,packet losses and limited bandwidth. Because of the in-herent complexity of such systems, an increasing amountof research addresses the problem of distributed controlof NCSs by taking into account network-induced effects.For instance, the stability and stabilization problems ofNCSs were investigated in (Halevi and Ray, 1988; Nilssonet al., 1998; Branicky et al., 2000; Zhang et al., 2001; Li etal., 2005) for network-induced delays, in (Ling and Lem-mon, 2002; Seiler and Sengupta, 2005) for packet losses,

in (Hu and Zhu, 2003; Yue et al., 2005; Li et al., 2006b)for network-induced delays and packet losses, in (Nairand Evans, 1997; Hristu, 1999; Ishii and Francis, 2002)for limited communication. The decision, coordinationand task schedulings were addressed in (Tipsuwan andChow, 2003; Hokayem and Abdallah, 2004; Yang, 2006).

Fig. 1. NCS architecture.

Due to an increasing complexity of dynamic systems,as well as the need for reliability, safety and efficient oper-ation, model-based fault diagnosis has became an impor-tant subject in modern control theory and practice, see,e.g., (Willsky, 1976; Frank, 1990; Gertler, 1998; Chenand Patton, 1999; Mangoubi and Edelmayer, 2000; Zhangand Jiang, 2003) and the references therein. Owing tothe network-induced effects, the theories for traditional

526 C. Aubrun et al.

point-to-point systems should be revisited when deal-ing with NCSs. However, only a few studies of theimpact of the communication network on the diagnosisperformances have been recently published (Ding andZhang, 2006; Llanos et al., 2006). The main idea ofthese approaches is to minimize the false alarms causedby transmission delays. In this case, a network-induceddelay is considered when designing the FDI filter.

The general configuration of NCSs considered in ourworks is shown in Fig. 2, wherein an NCS consists of aplant and a spatially distributed sensor, a controller andan actuator. When sampling and control data are trans-mitted over the network, many network-induced effectssuch as time delays and packet losses will naturally arise.Our work addresses the issues of modeling, analysis andsynthesis of the NCS and takes into account the network-induced effects from the viewpoint of fault diagnosis andfault-tolerant control. In the sequel, the main ideas andresults on these topics will be summarized.

The paper is organized as follows: Section 2 stud-ies the fault diagnosis problems of NCSs with network-induced effects focusing on time delays, packet losses andlimited communication. Fault-tolerant control of NCSs isaddressed in Section 3. Section 4 gives the conclusionsand indicates some future work.

Fig. 2. General configuration of an NCS.

2. Fault diagnosis of NCSs withnetwork-induced effects

2.1. Fault diagnosis of NCSs with network-inducedtime delays. Time delays in an NCS consist of: (a)a communication delay between sensors and controllersτsc, (b) a communication delay between controllers andactuators τ ca, (c) computational time in controllers τ c.Generally speaking, the computational time of controllerscan be included in the communication delay between con-trollers and actuators. Different industrial networks havedifferent communication delay features and real-time per-formances, e.g., the delay feature of the Ethernet is an un-certain stochastic delay; the delay feature of a token-type

field bus is a deterministic bounded delay. These delayswith different features can degrade the performance ofcontrol systems and even destabilize the systems. Thus,fault diagnosis for NCSs, taking into account network-induced time delays have gained attention from many re-searchers.

2.1.1. Low-pass post-filtering. The plant to be con-trolled through the digital communication network, andwhich may be subject to faults, is described by

·x (t) = Acx (t) + Buu(t) + Bdd(t) + Bff(t),y(t) = Cx(t),

(1)

where x(t), u(t), y(t) and d(t) are respectively the statevector, the control and output signals, and the distur-bances. The vector f(t) represents the faults which mayact on the process. We assume that the signals have ap-propriate dimensions and that the matrices Ac, Bu, Bd,Bf , C have accordingly compatible dimensions but arenot endowed with a particular structure. If we further as-sume that the unknown delay induced by the digital net-work is random and shorter than one sampling period,then the network-based controlled plant with unknown in-puts d and faults f can be modeled as a discrete-time sys-tem (Aström and Wittenmark, 1984):

x(k + 1) = Ax(k) + Γ0u(k) + Γ1u(k − 1)+ Bdd(k) + Bff(k),

y(k) = Cx(k), (2)

where the matrices of the discrete-time model are easilyobtained from those of the continuous time model. Thediscrete-time model can be further written as (Li et al.,2006a; Wang et al., 2006a; Ye et al., 2006)

x(k + 1)= Ax(k) + Bu(k) + g(k) + Bdd(k) + Bff(k), (3)

where

g(k) = −Γ1Δuk, Δuk = u(k) − u(k − 1). (4)

There exists a time-varying term g(k) in the state evolu-tion equation of the system (3) and (4). When the total de-lay τk combining τsc and τ ca is random, the variable g(k)can be regarded as a random disturbance in (3). There-fore, it is natural to adopt a low-pass filter to reduce theimpact of g(k) on the residual signal. However, the tech-nique cannot be applied by simply designing a traditionaloptimal residual generator and adding a low-pass filter toits output. The optimality of the global fault detectionfilter, which consists of a residual generator and a postfilter, is not ensured anymore when the system comes tobe networked. So it is necessary to consider the residual

Fault diagnosis of networked control systems 527

generator and the low-pass filter when designing the faultdetection system (Ye and Ding, 2004).

As an extension of the results in (Ye et al., 2004), afault detection approach based on the parity space and theStationary Wavelet Transform (SWT) for an NCS with arandom network-induced delay was introduced in (Ye andDing, 2004), which is briefly presented as follows:

Let

�s,k =[

�T (k − s) �T (k − s + 1) · · · �T (k)]T

,

(5)where ‘�’ may represent u, y, d or f .

Set

Hu,s =

⎡⎢⎢⎢⎢⎢⎣

0 0 · · · 0

CB 0. . .

......

. . .. . . 0

CAs−1B · · · CB 0

⎤⎥⎥⎥⎥⎥⎦

, (6)

and define Hd,s, Hf,s, Hg,s as the matrices obtained byreplacing B in (6) with Bd, Bf and the identity matrix I ,respectively. Let

Ho,s =[

CT AT CT · · · (As)T CT]T

.

Then a parity space and an SWT based residual generatoris defined as the SWT of the output of a traditional parityspace based residual generator, i.e.,

rs,k = vs(ys,k − Hu,sus,k), (7)

rWTs,k = WT a

rs(jm, k), (8)

whose dynamics are governed by

rs,k = vs(Hd,sds,k + Hf,sfs,k + Hg,sgs,k), (9)

rWTs,k = WT a

rs(jm, k), (10)

where vs is the parity vector to be designed, which shouldbe selected from the parity space Ps defined by Ps ={vs|vsHo,s = 0}, and WT a

rs(jm, k) denotes the approx-

imation coefficients of the SWT of rs,k, under scale jm,which can be regarded as a kind of low-pass filtering ofrs,k. It can be proved that the dynamics (9) and (10) can bewritten in the following explicit form (Ye and Ding, 2004):

rWTs,k =vs(Hd,sN

dl,jm

ds+iset,k + Hf,sNfl,jm

fs+iset,k

+ Hg,sNgl,jm

gs+iset,k),

where Ndl,jm

, Nfl,jm

, Ngl,jm

are known and constant ma-trices determined by the SWT filter, whose definitions canbe found in (Ye and Ding, 2004).

Similarly to traditional parity space-based methods,the following optimization problem taking into account

the influence of the delay can be defined and solved todetermine the optimal parity vector vs:

minvs∈Ps

JWTs = min

vs∈Ps

[vsHd,sN

dl,jm

(Ndl,jm

)T HTd,sv

Ts

vsHf,sNdl,jm

(Nfl,jm

)T HTf,sv

Ts

+vsHg,sN

gl,jm

(Ngl,jm

)T HTg,sv

Ts

vsHf,sNdl,jm

(Nfl,jm

)T HTf,sv

Ts

]. (11)

Finally, the residual signal can be calculated according to(7) and (8). The approach is robust to network-induceddelays due to the utilisation of the SWT-based low-passfilter. Moreover, it has optimal robustness to d and sensi-tivity to f in the sense of (11).

2.1.2. Structure matrix of a network-induced time de-lay. With respect to (3) and (4), (Wang et al., 2006a; Yeet al., 2006; Ye and Ding, 2004; Liu et al., 2005) proposedthe so-called structure matrix of τk to address the fault di-agnosis for NCSs. The procedure is decomposed into twosteps:

(a) decompose g(k) into two parts: (knownpart)×(unknown part), where the “known part”,expressed as the known information (such as Ac,Bu, Δuk), is extracted from g(k) and the “unknownpart” includes the unknown information related toτk;

(b) use traditional robust fault detection methods toachieve robustness to τk.

These results are further summarized as the Taylorapproximation (Ye and Ding, 2004), eigendecompositionand the Padé approximation (Wang et al., 2006a), theaccurate structure matrix of τk and PCA (Ye et al., 2006).

A. Taylor approximation. Consider a simpler NCS modeldefined as follows:

x(k + 1) = Ax(k) + Bu(k) + g(k) + f(k),y(k) = Cx(k).

(12)

When the sampling period h is sufficiently small com-pared with the system’s time constants, by using the Tay-lor approximation of eAch, g(k) will be approximated by

g(k) ≈ Eτ,kτk, Eτ,k = −BuΔuk. (13)

So g(k) has been transformed into an approximate form inwhich the first part is a known structure vector Eτ,k andthe second part is unknown τk (Ye and Ding, 2004). Atime-varying parity space-based residual generator is de-fined as

rs,k = vs,k(ys,k − Hu,sus,k), (14)

528 C. Aubrun et al.

whose dynamics are governed by

rs,k = vs,k(Hτ,s,kτs,k + Hf,sfs,k) (15)

when vs,k ∈ Ps, whereHτ,s,k =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 · · · 0 0

CEτ,k−s 0 0 0 0

CAEτ,k−s CEτ,k−s+1

. . . 0 0...

.... . .

......

CAs−1Eτ,k−s CAs−2Eτ,k−s+1 · · · CEτ,k−1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)To satisfy vs,k ∈ Ps and to decouple the residual

signal from the vector τs,k consisting of network-induceddelays, the parity vector is determined in each samplingperiod by solving

vs,kHo,s = 0, vs,kHτ,s,k = 0. (17)

It is shown that the approach has good robustness tounknown network-induced delays only if both h andτk are small enough. In addition, since τk in (13) is ascalar signal, the existence condition of vs,k in (17) is notdifficult to be satisfied in most cases.

B. Eigendecomposition and the Padé approximation. TheNCS model considered in (Wang et al., 2006a) is assumedto be similar to (12) and the matrix Ac in the continuous-time plant model is assumed to be diagonalizable. Basedon eigendecomposition and the first-order Padé approxi-mation of eλit, where λ1, . . . , λn are the eigenvalues ofAc, g(k) will be approximated by

g(k) ≈ Eτ,kτk, (18)

where the structure vector Eτ,k is defined as

Eτ,k

= −Pdiag(P−1BuΔuk)[2 + λ1h

2 − λ1h. . .

2 + λnh

2 − λnh

]τ

,

(19)

P being obtained through the eigendecomposition of Ac,i.e., Ac = PΛP−1.

The matrix diag(P−1BuΔuk) denotes the diagonalmatrix which is composed of the elements of the vectorP−1BuΔuk.

Comparing (18) with (13), it is seen that the twoterms have the same form. Thus, the residual generationand its design are quite similar to the approach based onthe Taylor approximation. Moreover, the full decouplingproblem (17) does not need a strong condition in mostcases since τk in (18) is still a scalar signal.

As demonstrated in (Wang et al., 2006a), sincethe structure matrix of the network-induced delay (i.e.,

Eτ,k) in (Wang et al., 2006a) has a much better accuracythan that in (Ye and Ding, 2004), the method in (Wanget al., 2006a) is much more robust to the unknownnetwork-induced delay than that in (Ye and Ding, 2004).

C. Accurate structure matrix of τk and PCA. (Ye etal., 2006) proposed an approach to fault detection forNCSs which includes not only the unknown network-induced delay but also the ordinary unknown disturbanceinput d.

By the Cayley-Hamilton theorem,

eAct = I + Act + · · · + 1n!

Anc tn + · · ·

=n−1∑i=0

αi(t)Aic,

(20)

where Ac is the matrix in the continuous NCS modeland n is the dimension of the state x. Then g(k) canbe transformed into the following form accurately (Ye etal., 2006):

g(k) = Eτ,kβτ,k, (21)

where

Eτ,k =[

Bu AcBu · · · An−1c Bu

]

·

⎡⎢⎢⎣

Δuk

. . .

Δuk

⎤⎥⎥⎦ ∈ R

n×n,

βτ,k =[

ητk0 ητk

1 · · · ητkn−1

]T

∈ Rn×1,

ητki = −

∫ h

h−τk

αi(t) dt ∈ R, i = 0, . . . , n − 1.

Thus, in (21), g(k) is separated into a known struc-ture matrix Eτ,k and an unknown vector βτ,k determinedby the network-induced delay τk. The structure matrixEτ,k in (21), different from the form expressed in (13) or(18), is accurate. A time-varying parity space based resid-ual generator is defined as

rs,k = vs,k(ys,k − Hu,sus,k).

It can be proved that when vs,k ∈ Ps, the dynamics of theresidual generator are governed by

rs,k = vs,k(Hd,sds,k + Hf,sfs,k + Hτ,s,kΨτ,s,k),

where Ψτ,s,k =[

βTτ,k−s βT

τ,k−s+1 . . . βTτ,k

]T

and

Hτ,s,k takes the same form as (16).In order to achieve the robustness of rs,k with respect

to the network-induced delay vector Ψτ,s,k and to ensure

Fault diagnosis of networked control systems 529

that vs,k belongs to the parity space Ps, it is expected thatvs,k should satisfy

vs,k ∈ Ps, vs,kHτ,s,k = 0. (22)

But since βτ,k in (21) is an n-dimensional vector, the so-lution of (22) may not exist in any case. Thus, (Ye etal., 2006) developed the following objective to determinethe parity vector (22) by Principal Component Analysis(PCA):

vs,k ∈ Ps, vs,kΛmk

τ,s,k = 0, (23)

where Λmk

τ,s,k is defined as the matrix which is composedof the first mk main Principal Component (PC) vectors ofthe matrix Hτ,s,k.

In (Ye et al., 2006) it is argued that due to the goodcharacteristics of PCA, usually suitable mk which is muchsmaller than the column number of Hτ,s,k, can be foundto produce the solution to (23). Moreover, it satisfies (22)with a good accuracy. After solving (23), we may furthertake advantage of the remaining degree of freedom of vs,k

to achieve optimal robustness to d and optimal sensitivityto f in the following sense:

minvs,k∈Ps,vs,kΛ

mkτ,s,k=0

Js,k, (24)

where

Js,k =vsHd,sH

Td,sv

Ts

vsHf,sHTf,sv

Ts

.

The advantages of (Ye et al., 2006) lie in (a) the adoptionof an accurate structure matrix of the network-induceddelay and its inclusion of an ordinary unknown inputd, (b) the known information on the network-induceddelay (i.e., its structure matrix), which makes it differ-ent from the prior work in (Ye and Ding, 2004; Ye andWang, 2006; Wang et al., 2006b).

2.1.3. Robust deadbeat fault filter. In (Li et al.,2006a), the authors assume that the statistical behavior ofthe network-induced delay τk is random and governed bythe Markov chain

θk ∈ S = {1, 2, . . . , s}, ∀k ∈ Z+, (25)

with the transition probabilities λij = Pr[θk+1 = j|θk =i], λij ≥ 0 and

∑sj=1 λij = 1 for any i ∈ S. For no-

tational simplicity, B1,τkis denoted by B1,θk

and Δuk

w (k). Then, the discrete-time model (3) of the network-based controlled plant is replaced by the state space sys-tem with the following particular Markov jump linear sys-tem:

x (k + 1) = Ax (k) + Bu (k) + Ff (k)+ B1,θk

w (k) ,

y (k) = Cx (k) .

(26)

The following filter is presented as the residual gen-erator of the NCS (26):

x (k + 1) = Ax (k) + Bu (k) + K(y (k)− Cx (k)),

αk = L(y (k) − Cx (k)),(27)

where x (k) is the state of the filter, αk the residual gen-erator or the fault indicator. The filter gain K ∈ R

n×m

and the projector L ∈ Rq×m are unknown matrices to be

found for the solution of the fault detection and isolationproblem.

From (26) and (27), the state estimation error e (k) =x (k) − x (k) and the output of the filter αk propagate as

e (k + 1) = (A − KC)e (k)+ Ff (k) + B1,θk

w (k) ,

αk = LCe (k) .

(28)

Let Gfα(z) be the transfer function from f (k) to theoutput residual αk. Then the following theorem is pre-sented to design K and L such that

Gfα(z) = LC(zI − (A − KC))−1F

= diag{z−ρ1 , . . . , z−ρq}, (29)

which ensures the isolation of multiple faults (Li et al.,2006a).

Under the condition rank(Ψ) = q, the solutions of(29) can be parameterized as K = ωΠ + Kθk

Σ, L = Π,with Σ = β(I − ΨΠ),Π = Ψ+, ω = AD and Ψ = CD,where Kθk

∈ Rn×m−q is the vector of free parameters to

be designed, Ψ+ is the pseudo-inverse of Ψ, and β is anarbitrary matrix chosen so that rank(Σ) = m − q.

Then, the filter (27) can be written as

x (k + 1) = Ax (k) + Bu (k) + ωαk

+KθkΣ(y (k) − Cx (k)),

αk = Π(y (k) − Cx (k)),(30)

where αk is a deadbeat filter for the fault f (k) given by

αk = αk +[

n1k−ρ1

· · · nik−ρi

· · · nqk−ρq

]T

,

(31)where αk is the fault indicator signal without faults.It propagates from the fault-free state estimation errore (k) = x (k) − x (k) as

e (k + 1) = (A − KθkC)e (k) + B1,θk

w (k) ,

αk = ΠCe (k) ,(32)

where A = A − ωΠC, C = ΣC and x(k) is the fault-free state. The transfer function from w(k) to αk, whenfreezing θk, is then given by

Gwα(z) = ΠC(zI − (A − KθkC))−1B1,θk

. (33)

530 C. Aubrun et al.

Let αk be the fault indicator signal without distur-bances. From Eqn. (29), the transfer function Gfα(z)from the fault f to the fault indicator αk is a pure delayand

‖Gfα(z)‖∞ := supθ0∈S

sup0 �=f∈�2

‖α‖2

‖f‖2

= 1, (34)

where ‖s‖�2= (

∑∞k=0 ‖s (k)‖)1/2 is the 2 norm of the

signal s (k).Then, the free parameters Kθk

are designed to satisfythe following two constraints:

C1. The H∞-norm of Gwα(z) is less than a prescribedscalar γ > 0.

C2. The eigenvalues of (A − KθkC) are located within

a prescribed region in the complex plane so that theresidual dynamical has the given transient properties.

The following theorem solves these two constraints (Liet al., 2006a): For given discs Di(ξi, δi), if there existmatrices Pi = PT

i > 0, Gi and Yi for prescribed scalarsγ > 0, −1 < −ξi + δi < 1, ∀i = θk ∈ S such that

⎡⎢⎢⎢⎣

−Pi 0 AT GTi − CT Y T

i CT ΠT

0 −γ2I BT1,iG

Ti 0

GiA − YiC GiB1,i Pi − Gi − GTi 0

ΠC 0 0 −I

⎤⎥⎥⎥⎦

< 0,(35)[

−δ2i Pi AT GT

i − CT Y Ti − ξiG

Ti

GiA − YiC − ξiGi Pi − Gi − GTi

]< 0,

(36)where A = A − ωΠC, C = ΣC, then the free parame-ters are designed as Ki = G−1

i Yi and ensure the second-moment stability of the error system (32) and the con-straints C1 and C2.

Given discs Di(ξi, δi), i = θk ∈ S, the search prob-lem of the lowest possible value of γ can be formulated asthe following convex optimization problem:

OP : minPi=P T

i >0,Gi,Yi

γ,

s.t. LMI (35), (36),(37)

which can be effectively solved by the existing MatlabLMI toolbox (Gahinet et al., 1995).

2.1.4. Adaptive residual evaluation strategy. In(Sauter and Boukhobza, 2006), the multiple input plantis considered as

x(t) = Ax(t) +m∑

i=1

Biui(t) + Ef(t),

y(t) = Cx(t).

(38)

Assume that the network-induced time delay is shorterthan one sampling period. Then the discrete-time modelof the plant is given by

x(k + 1) = Φx(k) +m∑

i=1

Γui(k)

−m∑

i=1

Γ1iΔui(k) + Ξf(k),

y(k) = Cx(k), (39)

with Δui(k) = ui(k) − ui(k − 1), and the computationof the matrices are straightforward. A classical observer-based residual generator is given as

x(k + 1) = Φx(k) +m∑

i=1

Γiui(k) + L(y(k) − y(k)),

y(k) = Cx(k).(40)

From (39) and (40), the estimation error e(k) =x(k) − x(k) and the residual vector r(k) propagate as

e(k + 1) = (Φ − LC)e(k) +m∑

i=1

Γ1i(τi)Δui(k)

+Ξf(k),r(k) = TCe(k).

(41)Clearly, it appears that the residual signal is cor-rupted with uncertainties, that is, the unknown term∑m

i=1 Γ1i(τi)Δui(k), caused by the network-induced de-lays.

In order to be able to distinguish the faults from theseuncertainties induced by the delays, a threshold is de-fined on the basis of an evaluation function taken as thetime-varying functional Ψ(kh) = ‖r(kh)‖. Note thatfrom the dynamics (41), this functional may be viewedas a continuous function of the unknown vector of time-delays τ = [τ∗

1 , . . . , τ∗m]T , i.e., we may write alternatively

Ψ(kh) = ‖r(kh)‖ = Ψ(τ). As the time-delays are as-sumed to be bounded, that is, 0 ≤ τi ≤ τmax

i for somepositive reals τmax

i , i = 1, . . . ,m, the unknown vector τactually belongs to a compact set T of R

m. The time-dependent variable

Th(kh) = maxf=0;τ∈T

Ψ(kh) (42)

is therefore well defined and is considered as the detectionthreshold, i.e.,

{Ψ(kh) ≥ Th(kh) for f �= 0,

Ψ(kh) < Th(kh) for f = 0.(43)

This threshold can be computed, through an opti-mization problem, via the following continuous-time dy-

Fault diagnosis of networked control systems 531

namical system:

e(t) = (A − LC)e(t)

+m∑

i=1

Bi(ui(t, τi) − ui(t, 0)) + Ef(t),

r(t) = TCe(t),

(44)

where

ui(t, τi) = ui (kh − 1) , kh ≤ t < kh + τi,

ui(t, τi) = ui (kh) , kh + τi ≤ t < (k + 1)h.(45)

The rationale behind this continuous time dynamicsis that the discrete-time system (41) can be seen as re-sulting from a zero-order hold discretization of the system(44) and the optimization problem is more easily handledin the continuous-time framework. With respect to (42),the optimization problem is to find the time-delays τ∗

i inthe control (45) running on the time interval [kh, (k+1)h]such that the performance index Ψ(τ) = ‖r(tk+1)‖ ismaximal on that interval, where we have set tk+1 =(k + 1)h. Note that the problem has been reduced to aterminal-cost optimization with respect to τ over an inter-val of one period. For that purpose, the Hamiltonian

H = λT [(A − LC)e(t) +m∑

i=1

Bi(ui(t, τi) − ui(t, 0))]

+ λT Ef(t) (46)

is introduced, where λ is the co-state vector. The so-lution to the optimization problem is given by the m-dimensional vector of time-delays ((Lawden, 2006))

τ∗ = [τ∗1 , . . . , τ∗

m]T

= arg minτi∈[0, τmax

i ]

(λT

m∑i=1

BiΔui(t, τi)),

(47)

with λ satisfying the adjoint equations

λT = −∂H

∂e= −λT (A − LC) (48)

with terminal conditions

λ(tk+1) = TC

⎡⎢⎢⎣

sign(e1(tk+1))...

sign(en(tk+1))

⎤⎥⎥⎦ . (49)

Since the inputs over the time interval considered are step-wise, the optimization procedure can be iterated over sev-eral sampling intervals.

2.1.5. Other work. It is worth noting that in the ref-erences cited above the total maximum of the network-induced delays is assumed to be less than one samplinginterval. However, in practice, the delay may be more thanone sampling period. In worse cases, this long time delaymay distort the timing order of the message arriving atthe receiver (Hu and Zhu, 2003; Lincoln and Bernhards-son, 2000; Li et al., 2004; Ray and Halevi, 1988).

In this way, the integrity and sequence of the infor-mation transmission are guaranteed. Then the discretestate model of the system with a network-induced delaycan be described as

x(k + 1) = Ax(k) + B0u(k − 1) + B1u(k − l + 1)+ Bdd(k) + Bffa(k),

y(k) = Cx(k) + fs(k),(50)

which is a familiar discrete time system with input timedelays. An observer-based fault detection method waspresented for the system (50) by comparing the output ofthe observer with the actual output of the actual system(Zheng, 2003). The residual function for this approach is

r(z) = QCP−1Bdd(z) + QCP−1Bffa(z)

− QCP−1(zI − A)V (zI − Λr)−1Lfs(z)+ Qfs(z),

(51)

where P = (zI − A)[I + V (zI − Λr)−1LC].The effect of the disturbance is decoupled from the

residual if the following conditions hold:

QCP−1Bd = H(zI − PT )−1Bd = 0.

The simulation results demonstrating the feasibility of thisapproach can be found in (Zheng, 2003).

In (Wang et al., 2006b), a method for fault detec-tion of an NCS with an unknown network-induced de-lay, which may be greater than h, is also proposed. Inthe method, an NCS model for an unknown network-induced delay which may be greater than h (Ray andHalevi, 1988; Hu and Zhu, 2003) was adopted, and theidea for handling multiplicative faults (Gertler, 1998) wasused to deal with the network-induced delay. However,from another point of view, the method in (Wang etal., 2006b) can also be regarded as an extension of theone-dimensional Taylor approximation used in (Ye andDing, 2004) into a multi-dimensional Taylor approxima-tion.

2.2. Fault diagnosis of NCSs with packet losses.Packet losses happen when packets are dropped due toa link failure or when packets are purposefully droppedin order to avoid congestion or to guarantee the most re-cent data to be sent. Although a single packet loss neither

532 C. Aubrun et al.

deteriorates the system performance nor destabilizes thesystem, the consecutive packet losses have an adverse im-pact on the overall performance. Therefore, it is necessaryto discuss how packet losses influence the fault diagnosisof NCSs. Generally speaking, packet losses can be mod-eled in either a deterministic or stochastic sense. In thefollowing, both cases will be discussed.

2.2.1. Deterministic packet losses. The determinis-tic packet losses were also discussed, either in terms ofswitching systems, by (Zhang et al., 2001) or, in terms ofdelayed differential equations, by (Yue et al., 2005; Yu etal., 2005). As to fault diagnosis of NCSs with determin-istic packet losses, to our best knowledge, no work hasbeen done. However, many existing research results onfault diagnosis for switching and time delay systems canbe extended or applied directly to NCSs. Some of theseresults are briefly introduced as follows:

• Unknown input decoupling. Yang and Saif (1998)addressed fault diagnosis for a class of state-delayeddynamic systems, in which the actuator and sen-sor faults, as well as other effects, such as distur-bances and higher-order nonlinearities, were consid-ered as unknown inputs. More recently, Koenig etal. (2005) dealt with the design problem of full-orderobservers for linear continuous delayed state and in-puts systems with unknown input and time-varyingdelays. A method to design an Unknown Input Ob-server (UIO) for such systems was proposed based ondelay-dependent stability conditions of the state esti-mation error system. A fault diagnosis scheme usinga bank of such UIOs was also presented and tested ona fault diagnosis problem related to irrigation canals.

• H∞-norm model matching formulation. Ding etal. (2000b) developed a weighting transfer functionmatrix to describe the desired behavior of residualswith respect to faults. The observer-based fault de-tection filter for a class of linear systems with time-varying delays was designed such that the error be-tween the generated residual and the fault was assmall as possible in the sense of the H∞-norm. Thedesign was then formulated as an H∞-model match-ing problem, which can be solved by an optimizationtool, such as a linear matrix inequality technique.

• Two-objective optimization approaches. Liu andFrank (1999) regarded the fault detection problemfor linear systems with constant time delays as two-objective nonlinear programming, namely, enhanc-ing the sensitivity of residuals to faults and, at thesame time, suppressing the undesirable effects of un-known inputs and modeling errors. More recently,Jiang et al. (2003) extended the results of (Liu andFrank, 1999) to the case of discrete-time systems.

Zhong et al. (2006) dealt with the robust fault de-tection filter problem for linear systems with time-varying delays and model uncertainty.

• Unified optimization approach. Zhong et al.(2005) extended the results of (Ding et al., 2000a)to linear systems with L2-norm bounded unknowninput and multiple constant time delays. Then, anobserver-based fault detection filter was developedsuch that a performance index based on the ratio ofrobustness and sensitivity was minimized. By an ap-propriate choice of a filter gain matrix and post-filter,a solution to the fault detection filter was derived interms of a Riccati equation.

• Adaptive observer-based fault detection and iden-tification. With a structure restriction on the faultdistribution, Jiang et al. (2002) developed an adap-tive observer for fault identification of both linearsystems with multiple state time delays and a class ofnonlinear systems. Jiang and Zhou (2005) proposeda new adaptive observer for robust fault detectionand identification of uncertain linear time-invariantsystems with multiple constant time-delays in bothstates and outputs. Chen and Saif (2006) investi-gated an iterative learning observer based on adaptiveunknown input estimation with considering both thedisturbances and possible faults as unknown inputs.

2.2.2. Stochastic packet losses. The simplest stochas-tic model assumes that losses are realizations of aBernoulli process (Seiler, 2001; Sinopoli et al., 2004).Underlying finite-state Markov chains can be used tomodel correlated packet losses (Smith and Seiler, 2003;Nilsson, 1998; Xiao et al., 2000), and Poisson processescan be used to model stochastic losses in continuous time(Xu, 2006).

In (Zhang et al., 2004), the fault detection problem ofsystems with stochastic packet losses is discussed. First,in order to cope with packet losses, the structure of thestandard model based residual generator is modified anddynamic network resource allocation is suggested as

e(k + 1)

=

{(A − LC)e(k)+(Ef − LFf )f(k)+Lθ(k), γ(k) = 0,

(A − LC)e(k)+(Ef − LFf )f(k), γ(k) = 1,

(52)

and

r(k)

=

{WCe(k) + WFff(k) − Wθ(k), γ(k) = 0,

WCe(k) + WFff(k), γ(k) = 1,

(53)

where θ(k) is the the difference between the real value ofthe measurement y(k) and the value ya(k) used, namely,

Fault diagnosis of networked control systems 533

θ(k) := y(k) − ya(k). γ(k) is a stochastic variable rep-resenting the data communication status. γ(k) = 1 meansthat the measurement at time point k arrives correctly,while γ(k) = 0 means that this measurement is lost. Thedynamics of the residual generator are thus characterizedby a discrete-time Markovian jump linear system.

To reduce the false alarm rate caused by a missingmeasurement, a residual evaluation scheme is then devel-oped as

reval > Jth, a fault is detected,

reval ≤ Jth, no fault is detected,

where reval =(∑∞

j=0 rT (j)r(j))1/2

. To compute the

threshold Jth, a convex optimization problem is then de-veloped to find the minimum of E[‖r‖2]/‖θ‖2, which isformulated as a disturbance attenuation problem of theMarkovian jump linear systems (52) and (53). Further,a co-design approach of a time-varying residual generatorand threshold is proposed to improve the dynamics andsensitivity of the fault detection system to the faults.

It should be noted that there are some research workswhich concern the NCS that take into account simulta-neous time-delays and packet losses, see, e.g., (Yue etal., 2005; Zhang et al., 2005; Yu et al., 2005). How-ever, the obtained results may be somewhat conservativeas they are based on worst-case scenarios. To the best ofour knowledge, there is no previous work analyzing esti-mation where observation packets are subject to a simul-taneous random delay and packet losses in a probabilisticframework.

2.3. Fault diagnosis of NCSs with limited commu-nication. The capacity of the communication networkand its ability to carry a reasonable amount of informa-tion per unit of time play an important role in character-izing the NCS stability. When introducing the networkinto the control loop, issues like the channel/network ca-pacity, encoding/decoding schemes and quantization nat-urally arise. Examples of NCSs with limited communi-cation include unmanned air vehicles owing to stealth re-quirements, wireless sensor networks due to long-lastingenergy limitations, and so on.

Inspired by the Shannon information theory, thereis increasing attention to characterize the minimum bitrate which is needed to stabilize NCSs through feedback,see, e.g., (Sahai, 2000; Tatikonda, 2000; Savkin and Pe-tersen, 2003) and the references therein. In order todescribe the quantization effects on the performance ofNCSs, great research efforts have been devoted to developnew a quantization scheme to achieve lower bit-rates, see,e.g., (Brockett and Liberzon, 2000; Delchamps, 1989; Eliaand Mitter, 2000; Ishii and Francis, 2002; Wong andBrockett, 1997) and the references therein. For more

details on this topic, we refer the reader to the survey(Hokayem and Abdallah, 2004).

In (Zhang and Ding, 2006), the fault detection prob-lem of networked control systems with limited data trans-mission rates is considered. In order to reduce the networkload and thus avoid the uncertainty caused by transmissiondelays and packet losses, the so-called periodic communi-cation sequence is introduced as

y(k) = Nkyp(k), (54)

up(k) = Mku(k), (55)

where y ∈ Rωm represents the sensor signals transmit-

ted from the sensors to the central station through the net-work, Nk ∈ R

ωm×m is a θ-periodic matrix formed byselecting ωm rows of the identity matrix. u ∈ R

p repre-sents the signal generated by the controller, Mk ∈ R

p×p isa θ-periodic diagonal matrix with ωp elements equal to 1on the diagonal. The dynamics of the NCS are then char-acterized by

x(k + 1) = Ax(k) + BMku(k) + Edd(k) + Eff(k),y(k) = Nk(Cx(k) + DMku(k) + Fdd(k))

+ NkFff(k). (56)

The input-output relation of the NCS (2.3) over amoving finite horizon [k − s, k], where s is an integerrepresenting the length of the horizon, can be expressedby

Y (k) = Hs,kx(k − s) + Hu,kU(k) + Hd,kD(k)+ Hf,kF (k).

(57)

Matrices Ho,k,Hu,k,Hd,k,Hf,k in the parity relation (57)are θ-periodic with respect to k. The residual generator isthen constructed as

r(k) = vk(Y (k) − Hu,kU(k)), (58)

where vk ∈ R1×(s+1)ωm is the periodic parity vector to

be designed such that vkHo,k = 0 for any k. The residualdynamics are not influenced by the initial state x(k − s)and are governed by

r(k) = vk(Hd,kD(k) + Hf,kF (k)). (59)

There are two cases to be considered:

• If

rank[

Ho,k Hd,k Hf,k

]> rank

[Ho,k Hd,k

]

for any k, then the residual signal can be decoupledfrom the unknown disturbances by designing vk insuch a way that

vk

[Ho,k Hd,k

]= 0, vkHf,k �= 0

holds for any k.

534 C. Aubrun et al.

• If a full decoupling is not achievable, then a suitablecompromise between the robustness to unknown dis-turbances and the sensitivity to faults can be achievedby solving the optimization problem

minvk

Jk = minvk

vkHd,kHTd,kvT

k

vkHf,kHTf,kvT

k

s.t. vkHo,k = 0

to get an optimal periodic parity vector vk.

Then, the influence of the new communication pattern onfault detection, including a full decoupling and an optimalachievable performance, is analyzed. Finally, the optimalselection of the periodic communication sequence is dis-cussed.

3. Fault-tolerant control of NCSs

Based on the fault diagnosis algorithm for NCSs in Sec-tion 2, fault-tolerant control of NCSs can be obtained.The existing methods of fault tolerant control techniquesagainst actuator faults can be categorized into two groups:passive (Seo and Kim, 1996; Cheng and Zhao, 2004)and active approaches (Zhang and Jiang, 2002; Jiang andZhang, 2006). Zheng (2003) proposed a passive controllerfor NCSs considering random time delays. Although thepassive controllers are easy to implement, their perfor-mances are relatively conservative. The reason is that thisclass of controllers based on the presumed set of compo-nent failures and with a fixed structure and parameters isused to deal with all the possible different failure scenar-ios. If a failure occurs out of those considered in the de-sign, the stability and performance of the closed-loop sys-tem is unanticipated. Such potential limitations of passiveapproaches motivate the research on Active FTC (AFTC).

AFTC procedures require an on-line and real-timefault diagnosis process and a controller reconfigurationmechanism. Because AFTC approaches propose a flex-ibility to select different controllers according to differ-ent component failures, better performance of the closed-loop system is expected. However, the above case holdstrue only if the fault diagnosis process does not providean incorrect or delayed decision. Some preliminary re-sults have been obtained on AFTC which tend to make thereconfiguration mechanism immune from imperfect faultdiagnosis decison, see (Mahmoud et al., 2003; Wu, 1997).Maki et al. (2004) further discussed the above issue by us-ing the guaranteed cost control approach and on-line con-troller switching in such a way that the closed-loop systemwas stable at all times. However, Maki et al. (2004) didnot consider the plant controlled over the network.

Li et al. (2007) addressed the stability guaranteedactive fault tolerant control of NCSs. The design pro-cedures are summarized as follows: (i) design a passive

fault-tolerant controller such that the closed-loop systemstability is maintained for all actuator failure modes; and(ii) under the assumption that a particular actuator is freefrom faults, repeatedly redesign the controller using onlythis actuator alone so that the robust performance is fur-ther improved without affecting the stability property ofthe design in (i). All the design theorems are formulatedin terms of convex optimization problems which can beefficiently solved by existing software, e.g., the MatlabLMI toolbox.

4. Conclusion

In this paper we discussed and summarized model basedFDI approaches to NCSs including observer-based andparity space methods. A fault tolerance principle forNCSs is also presented. The induced effect of the com-munication medium on the performance of the FDI algo-rithm, such as time delays and packet losses or limitedcommunication, is taken into account in the filter design.Directional residual generator decoupling from the distur-bances ensures the treatment of multiple faults occurringsimultaneous or sequentially. It was pointed out that thisdomain is still in progress and the co-design method aim-ing at integrating the control and scheduling for NCS is apromising topic of research.

Acknowledgements

The authors would like to thanks the anonymous review-ers for their constructive comments, which have improvedthe clarity of the paper. This work was supported by theEuropean Union project NeCST under the grant no. EU-IST-2004-004303 and the French Agence Nationale de laRecherche project Safe-Necs under the grant no. ANR-ARA n◦ SSIA_NV_15.

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Received: 30 December 2007