Statistical χχχ2 Testing Based Fault Detection for Linear

Discrete Time-delay Systems

LIU Bo-Ang1 YE Hao2

Abstract This paper is concerned with statistical χ2 testing based fault detection (FD) for a class of linear discrete time-varying(LDTV) stochastic systems with delayed state. Different from the traditional residual based FD, we propose to construct theevaluation function by directly using measurement observations. Then an equivalent solution can be given in terms of Riccatirecursion by utilizing projection and innovation analysis technique. Moreover, the fault free case evaluation function is with centralχ2 distribution and the heavy computational burden is reduced. Furthermore, strategies of χ2 statistic testing on evaluation functionare also discussed. Finally, a numerical example is given to illustrate the proposed method.

Key words Fault detection (FD), linear discrete time-delay system, evaluation function, χ2 distribution, statistical testing

Citation Liu Bo-Ang, Ye Hao. Statistical χ2 testing based fault detection for linear discrete time-delay systems. Acta Automatica

Sinica, 2014, 40(7): 1278−1284

Research on model-based fault detection and isolation(FDI) has received much attention during the past threedecades; see for example [1−10] and references therein.Generally speaking, model-based FD consists of two stages,i.e., the residual generation and the residual evaluation. Assummarized in [1−2], the first and important task is to de-sign a fault detection filter (FDF) by maximizing the sen-sitivity of residual to fault and simultaneously minimizingthe robustness of residual to unknown input. Then a faultindicator can be delivered by comparing a chosen residualevaluation function with a prescribed threshold.

Under the assumption of unknown input being Gaussiannoise, the well known Kalman filter is a suitable choice ofresidual generator. Then a statistical testing of whiteness,mean and covariance of the residual can be used to detectthe occurrence of a fault[3]. In [11], a least square basedresidual generation and matching approach was proposedfor a class of discrete time-varying networked sensing sys-tems with incomplete measurements.

For linear systems subject to l2-norm bounded unknowninput, much attention has been paid to residual genera-tion in the framework of H∞ filtering or H∞ optimization.The former one was to formulate the design of FDF as anstandard H∞ filtering in the sense of minimizing the L2-induced gain from the unknown input to the error betweenthe residual and the fault[12−16] . The another one was todefine the sensitivity and robustness as an H− index or H∞

norm and formulated the underlying FD problem as an H∞

optimization; see e.g. [17−25]. In [18], a unified solutionwas proposed to the H∞/H∞ and/or H−/H∞ optimizationproblem and, through co-inner-outer factorization, a resid-ual generator was given in terms of Riccati equation. In [19]and [24], optimal solutions of H−/H∞ and H∞/H∞ wereobtained for LDTV systems. An Krein space approach wasproposed for robust H∞ FD and a solution of FDF wasgiven in terms of Riccati recursion[25].

On the other hand, time-delays are frequently en-

Manuscript received October 23, 2013; accepted November 23, 2013Supported by National Natural Science Foundation of China

(61290324)Recommended by Associate Editor HU Chang-Hua1. Department of Mathematical Sciences, Tsinghua University, Bei-

jing 100084, China 2. Department of Automation, Tsinghua Uni-versity, Beijing 100084, China

countered in practical control systems and efforts havebeen paid to the investigation of H∞ FD for time-delaysystems[26−31]. In [27] and [30−31], the problem of FD,estimation, and compensation for time-delay systems weredealt with. Reference [32] dealt with the problem of fi-nite horizon H∞ fault estimation for linear discrete time-varying (LDTV) systems with delayed states by using linearprojection in Krein space, while [33] concerned only the de-lay of measurement output. It should be pointed out thatthe existing results of FD for linear time-delay systems con-cern with norm bounded unknown input and most of thefocuses are on residual generation. The problem of FD fornetworked systems subject to time-delays was investigatedin [34−38]. By considering the main purpose of FD, i.e.,delivering a fault indicator after its occurrence, a natureidea is to generate evaluation function directly in the firststage and perform FD as statistical testing in the secondstage, and this motivates the present study.

This paper is concerned with FD for a class of LDTVstochastic systems with delayed state. Different from tra-ditional residual based FD, we will first consider to directlygenerate an evaluation function by using a quadratic formof measurement observations. Under the assumption of un-known input being jointly normal distributed, the evalua-tion function will be proved to be central χ2 distributionand, by using projection and innovation analysis, the evalu-ation function for LDTV systems with delayed state will begiven in terms of Riccati recursion. Similar to the statisticaltesting of residual, the χ2 statistical testing of evaluationfunction will also be discussed for detecting the occurrenceof a fault. A numerical example will be given to show theeffectiveness of the proposed method.

Notation. Through out the paper, E [·] denotes expec-tation of [·]. Superscripts “−1” and “T ” stand for theinverse and transpose of a matrix, respectively. R

n×m isthe set of all n × m real matrices. I is the identity ma-trix with appropriate dimensions. For a real symmetricmatrix P , P > 0 (respectively, P < 0) means that P isa real positive definite (respectively, negative definite) ma-trix. diag{·} denotes a block-diagonal matrix. L{·} denotesthe linear space spanned by sequence {·}. For zero meanstochastic vectors α and β, 〈α, β〉 stands for the covariance

No. 7 LIU Bo-Ang et al.: Statistical χ2 Testing Based Fault Detection for Linear Discrete Time-delay Systems 1279

matrix and Rα = 〈α, α〉.

1 Problem formulation

Consider the following LDTV system with delayed state

x(k + 1) = A(k)x(k) + Aτ (k)x(k − τ )+

Bd(k)d(k) + Bf (k)f(k)

y(k) = C(k)x(k) + v(k) + Df (k)f(k)

x(0) = x0, x(k) = 0, for k < 0

(1)

where x(k) ∈ Rn, y(k) ∈ R

m, d(k) ∈ Rq , v(k) ∈ R

m

and f(k) ∈ Rp are the state, measurement output, pro-

cess noise, measurement noise and fault to be detected,respectively; τ > 0 is the state delay which is an inte-ger; f(k) ∈ l2 [0, N ]; A(k), Aτ (k), Bd(k), Bf (k), C(k) andDf (k) are known matrices with appropriate dimensions.The initial state x0 and d(k), v(k) are assumed to be zeromean and normally distributed vectors with

⟨

x0

d(k)

v(k)

,

x0

d(j)

v(j)

⟩

= diag{P0, R(k)δkj , Q(k)δkj}

(2)

where δkj = I for k = j, δkj = 0 for k 6= j.Define

fN =[

fT(0) fT(1) · · · fT(N)]T

dN =[

dT(0) dT(1) · · · dT(N)]T

vN =[

vT(0) vT(1) · · · vT(N)]T

yN =[

yT(0) yT(1) · · · yT(N)]T

The state transition matrix of system (1) can be given by

Φ(i + 1, j) = A(i)Φ(i, j) + Aτ (i)Φ(i − τ, j),

i = 0, 1, · · · , N − 1, j ≥ 0

Φ(m, n) = 0, m < n

Φ(m, n) = I, m = n

(3)

Let

H0 =

C(0)

C(1)Φ(1, 0)

C(2)Φ(2, 0)...

C(N)Φ(N, 0)

HdN = [hdN (i, j)](N+1)×(N+1)

HfN = [hfN (i, j)](N+1)×(N+1)

with

hdN (i, i) = 0, hfN (i, i) = Df (i − 1)

for i = 1, 2, · · · , N + 1, and

hθN (i, j) = C(i − 1)Φ(i − 1, j)Bθ(j − 1)

for i > j = 2, 3, · · · , N + 1; θ = d, f in sequence. We canrewrite (1) in the following form

yN = H0Nx0 + HdNdN + HfNfN + vN (4)

Under the assumption of x0, d(k), v(k) being jointly nor-mal distribution with (2) and f(k) = 0, the yN is withnormal distribution and has the following properties:

E [yN ] = E [H0Nx0 + HdNdN + vN ] = 0 (5)

RyN=

[

H0N HdN I]

×

P0 0 0

0 RN 0

0 0 QN

HT0

HTdN

I

(6)

where

RN = diag{R(0), R(1), · · · , R(N)}

QN = diag{Q(0), Q(1), · · · , Q(N)}

Without loss of generality, it is assumed that RyNis

invertible. Introduce

JN = yTNR−1

yNyN (7)

Recall that yN is normally distributed and satisfies (5) and(6) if the fault is not taken into account. So, in the case ofno fault, the JN follows central χ2 distribution with free-dom degrees of (N + 1)m. In the faulty case, however, themean of yN may not be zero and the distribution of yN

may be changed also. Thus the faulty case JN does nothave the central χ2 distribution. Therefore, we can use JN

as an evaluation function and develop FD strategies basedon statistical χ2 testing of JN .

Moreover, it should be pointed out that the online com-putation burden of such an evaluation function is heavy.Especially, it is not an easy task to implement online FDfor LDTV systems with delayed state with the increasingof N . To overcome this problem, techniques of orthogonalprojection and innovation analysis in [39] will be applied.

Given measurement observations {y(k)}Nk=0, we now for-

mulate the problem of FD as: 1) to find a projection ofx(k) on space spanned by {y(j)}k

j=0 and calculate JN on-line in terms of Riccati recursion; 2) to perform FD byusing statistical χ2 testing of JN .

Remark 1. Different from the traditional FD for LDTVsystems with time delay, it is novel to generate the evalu-ation function by directly using quadratic form (7) insteadof residual generation as usual. There is no doubt that theonline calculation of JN is a heavy burden for time-delaysystems with the increasing of N . So, it will be of signif-icance to derive the recursive computation of JN by usingprojection and innovation analysis.

2 Main results

To overcome the heavy computation problem of JN

in (7), orthogonal projection on linear space spanned by{y(i)}k

i=0 will be first considered. Denote by x(j|k) theprojection of x(j) (j = k − τ, k − τ + 1, · · · , k, k + 1) ontothe linear space spanned by L{y(i)}k

i=0. Let

x(k + 1) = x(k + 1|k)

Define innovations as

e(j) = x(j) − x(j|k) (8)

P (k) = 〈e(k), e(k)〉 (9)

P (j, i) = 〈e(j), e(i)〉, for i 6= j (10)

y(k) = y(k) − C(k)x(k) (11)

1280 ACTA AUTOMATICA SINICA Vol. 40

Ry(k) = 〈y(k), y(k)〉 (12)

It is known from [39] that the innovation sequence {y(i)}ki=0

forms an orthogonal basis of linear space L{y(i)}ki=0 and

L{y(i)}ki=0 = L{y(i)}k

i=0

Hence, the projection x(k + 1) can be calculated by

x(k + 1) =k

∑

i=0

〈x(k + 1), y(i)〉R−1y (i)y(i) =

k−1∑

i=0

〈x(k + 1), y(i)〉R−1y (i)y(i) +

〈x(k + 1), y(k)〉R−1y (k)y(k) =

A(k)x(k) + Aτ (k)x(k − τ ) + Kp(k)y(k) +

Aτ (k)

k−1∑

i=k−τ

Kτ (i)y(i) (13)

x(i) = 0, i ≤ 0 (14)

where

Kp(k) = 〈x(k + 1), y(k)〉R−1y (k) =

(A(k)Θ(k) + Aτ (k)Θτ (k))R−1y (k)

Kτ (i) = 〈x(k − τ ), y(i)〉R−1y (i) =

Θτ (k, i)R−1y (i)

Ry(i) = C(i)P (i)CT(i) + Q(i)

Θ(k) = 〈x(k), y(k)〉 = P (k)CT(k)

Θτ (k) = 〈x(k − τ ), y(k)〉 = P (k − τ, k)CT(k)

Θτ (k, i) = 〈x(k − τ ), y(i)〉 = P (k − τ, i)CT(i),

i = k − τ, k − τ + 1, · · · , k

It follows from (1) and (8)∼ (14) that

e(k + 1) = A(k)e(k) + Aτ (k)e(k − τ ) + Bd(k)d(k)−

Kp(k)y(k) − Aτ (k)k−1∑

i=k−τ

Kτ (i)y(i)

y(i) = C(i)e(i) + v(i), i = k − τ, k − τ + 1, · · · , k

(15)

Moreover, P (k + 1) can be given by

P (k + 1) = 〈e(k + 1), e(k + 1)〉 =

A(k)P (k, k + 1) + Aτ (k)P (k − τ, k + 1) +

Bd(k)R(k)BTd (k)−

Kp(k)C(k)P (k, k + 1)−

Aτ (k)

k−1∑

i=k−τ

Kτ (i)C(i)P (i, k + 1) (16)

where

P (i, k + 1) = 〈e(i), e(k + 1)〉 =

P (i, k)AT(k) + P (i, k − τ )ATτ (k)−

P (i, k)CT(k)KTp (k)−

k−1∑

j=k−τ

P (i, j)CT(i)KTτ (i)AT

τ (k) (17)

P (k, k + 1) = 〈e(k), e(k + 1)〉 =

P (k)AT(k) + PT(k − τ, k)ATτ (k)−

Θ(k)KTp (k) (18)

P (k − τ, k + 1) = 〈e(k − τ ), e(k + 1)〉 =

P (k − τ, k)AT(k) +

P (k − τ )ATτ (k) − Θτ (k)KT

p (k)−

k−1∑

i=k−τ

Θτ (k, i)KTτ (i)AT

τ (k) (19)

Recall that

y(k) = Proj{y(k)|L{y(i)}k−1i=0 } =

Proj{y(k)|L{y(i)}k−1i=0 } =

k−1∑

i=0

〈y(k), y(i)〉R−1y (i)y(i)

y(k) = y(k) + y(k) =[

lk,1 lk,2 · · · lk,k−1 I]

yk

where

lk,i = 〈y(k), y(i)〉R−1y (i), k = 1, 2, · · · , N ; i ≤ k − 1

We then have

yN = LN yN (20)

RyN= 〈yN , yN〉 = LN 〈yN , yN〉LT

N = LNRyNLT

N (21)

where LN is given by

LN =

I 0 0 0

l2,1 I 0 0...

. . .. . .

...

lN+1,1 · · · lN+1,N I

(22)

with

li,j = 〈y(i), y(j)〉R−1y (j), i = 2, 3, · · · , N + 1; j < i

and RyNis block diagonal with

RyN= diag{Ry(0), Ry(1), · · · , Ry(N)} (23)

Substituting (20)∼ (23) to (7) yields

JN = yTNLT

NR−1yN

LN yN = yTNR−1

yNyN =

N∑

k=0

yT(k)R−1y (k)y(k) (24)

When a fault is taken into account, the JN can be calcu-lated by (24) with

x(k + 1) = A(k)x(k) + Aτ (k)x(k − τ )+

Aτ (k)k−1∑

i=k−τ

Kτ (i)y(i) + Kp(k)y(k)

y(i) = y(i) − C(i)x(i), k − τ ≤ i ≤ k

(25)

where Kp(k), Kτ (i), i = k − τ, k − τ + 1, · · · , k − 1 arecalculated by

No. 7 LIU Bo-Ang et al.: Statistical χ2 Testing Based Fault Detection for Linear Discrete Time-delay Systems 1281

Kp(k) = (A(k)P (k)CT(k) +

Aτ (k)P (k − τ, k)CT(k))R−1y (k) (26)

Kτ (i) = P (k − τ, i)CT(i)R−1y (i) (27)

Ry(k) = C(k)P (k)CT(k) + Q(k) (28)

and P (k) is given by (16)∼ (19).Based on the above analysis, the determination of eval-

uation function for FD can be summarized as the followingtheorem.

Theorem 1. Given observations {y(k)}Nk=0, if there ex-

ists P (k) given by (16)∼ (19) such that Ry(k) = C(k) ×P (k)CT(k) + Q(k) > 0, then JN can be calculated by

JN =N

∑

k=0

yT(k)R−1y (k)y(k) (29)

where y(k) is generated by (25) with (16)∼ (19) and (26)∼(28).

So far, we have derived a feasible evaluation function forFD. The remaining task is decision making, which consistsof the choice of threshold and logic unit. Similar to thestatistical testing of residuals in [3], the occurrence of afault can be detected based on χ2 testing.

For this purpose, we rewrite (25) in the following form

e(k + 1) = (A(k) − Kp(k)C(k))e(k) + Aτ (k)e(k − τ )−

Aτ (k)k−1∑

i=k−τ

Kτ (i)C(i)e(i) +

Bd(k)d(k) + (Bf (k) − Kp(k)Df (k))f(k)−

Kp(k)v(k) − Aτ (k)k−1∑

i=k−τ

Kτ (i)v(i)

y(k) = C(k)e(k) + v(k)

It is easy to see that under the assumption of d(k) andv(k) being jointly normally distributed and satisfying (2),the elements of fault free case y(k) are jointly normallydistributed with

E [y(k)] = 0

Ry(k) = C(k)P (k)CT(k) + Q(k)

〈y(i), y(j)〉 = 0, i 6= j

Therefore, in the case of no fault, both the statisticsy(k)TR−1

y (k)y(k) and JN follow the central χ2 distributionwith degrees of freedom m and (N + 1)m, respectively.

On the other hand, if the fault free case evaluation func-tion follows the central χ2 distribution with degree of free-dom β, then a corresponding threshold with the false alarmrate ε can be chosen as Jth = χ2

β,ε. Finally, the fault de-tection test can be performed as the following statistic χ2

testing{

if Je < χ2β,ε, then no fault

if Je ≥ χ2β,ε, then fault alarm

(30)

Similar to the case of statistical testing of residual in [3], thefollowing three alternative strategies of FD are consideredin this paper.

1) Single observation. For the given false alarm rate ε> 0, choose the evaluation function Je(k) and thresholdJth(k) as

{

Je(k) = yT(k)R−1y (k)y(k)

Jth = χ2m,ε

(31)

where Ry(k) is given in (25).2) Observation sequence. Choose the evaluation function

JeN and threshold Jth as{

Jek = yTk R−1

ykyk

Jth(k) = χ2(1+k)m,ε

(32)

3) Window average of observations. Choose the evalua-tion function JeN,a and threshold Jth,a as

Jek,a =1

k + 1yT

k R−1yk

yk

Jth,a = χ2m,ε

(33)

Remark 2. When a residual signal r(k) is chosen as

r(k) = R−1/2y (k)y(k), it is easy to have

JN =

N∑

k=0

rT(k)r(k)

Moreover, the fault free case r(k) is jointly normally dis-tributed with

E [r(k)] = 0, 〈r(k), r(k)〉 = I

In this case, one can further use the strategy in [3] to de-tect the occurrence of the fault, i.e. statistical testing ofresidual. Note that the main focus of this paper is theevaluation function JN , but most of the existing results areconcerned with the residual r(k) or y(k). From this view-point, we call the case of this paper the evaluation functionbased FD, while the conventional one as in [3] the residualbased FD.

Remark 3. It is easy to see that (25) with r(k) =

R−1/2y (k)y(k) can be used as an observer-based FDF for

the LDTV time-delay systems (1). Different from themost existing results for time-delay systems, not only thecurrent innovation y(k), but also the delayed innovations{y(i)}k−1

i=k−τ are considered in (25).

3 A numerical example

Consider system (1) with the following parameters

A(k) =

[

1.8e−k 0.2 sin(k)

0 0.6

]

Aτ (k) =

[

0.7 0.3 cos(k)

0 0.9

]

Bf (k) =

[

0.9

1.8

]

Bd(k) =

[

0.3

0.7

]

C(k) =[

1.4 2.8]

Df (k) = 0

Set τ = 2, P (0) = I , and P (0, i) = 0, i = 1, 2, 3. Supposethat the process noise d(k) and the measurement noise v(k)follow normal distribution with zero mean and variance 1.The fault f(k) is simulated as

1282 ACTA AUTOMATICA SINICA Vol. 40

f(k) =

−1, k = 20 ∼ 40

1, k = 60 ∼ 80

0, otherwise

To detect the occurrence of a fault, the first step is to gener-ate JN online by using Theorem 1. The next step is the socalled χ2 statistical testing. An acceptable false alarm rateis supposed to be 0.1. The above discussed three testingstrategies are considered as follows.

1) Single observation. The fault free case and faultycase evaluation functions are calculated by (31), i.e., theJ0(k) and Je(k) shown in Fig. 1. Recall that J0(k) followsthe central χ2 distribution with degrees of freedom 1. Thethreshold is chosen as Jth = χ2

1,0.1 = 2.7055. The result inFig. 1 has shown that both false alarms and missed faultalarm exist in this case.

Fig. 1 The evaluation functions J0(k) and Je(k)

2) Observation sequence. The fault free case and faultycase evaluation functions are calculated online by (32), i.e.,the J0k and Jek shown in Fig. 2. Recall that J0k follows thecentral χ2 distribution with degree of freedom k + 1. Thethreshold is chosen as Jth(k) = χ2

k+1,0.1, i.e., the dash dotline in Fig. 2. It is easy to see that, in the faulty case, thefault alarm has 5 steps of delay and the fault alarm is notcleared away even if f(k) is 0 for 41 ≤ k ≤ 59 and 81 ≤ k≤ 100. Meanwhile, only a few false alarms appear in thefault free case.

Fig. 2 The evaluation functions J0k and Jek

3) Window average of observations. The fault free caseand faulty case evaluation function are calculated onlineby (33), i.e., the J0k,a and Jek,a shown in Fig. 3. Recallthat J0k,a follows the central χ2 distribution with degreeof freedom 1. The threshold is chosen as Jth,a = χ2

1,0.1 =2.7055. It is seen from the result in Fig. 3 that in the faultycase, the fault alarm is delivered at k = 29 and is notcleared away even if f(k) being 0 for 41 ≤ k ≤ 59 and 81≤ k ≤ 100. Meanwhile, there is no false alarm in the faultfree case.

Fig. 3 The evaluation functions J0k,a and Jek,a

Remark 4. It should be pointed out that the availableχ2 statistical testing strategies are not unique. In practi-cal applications, we may choose one of them based on thetrade-off between false alarm rate and missed alarm rate.

Remark 5. The residual mentioned in Remark 2 is alsocalculated and shown in Fig. 4.

Fig. 4 The residuals r0(k) and r(k)

Since the fault free case residual follows zero mean nor-mal distribution with variance 1, one can perform FD byusing the statistical testing of residual r(k) as in [3]. Forsimplicity, more details are not discussed here.

Remark 6. Using the single observation strategy, theoccurrence of a fault can be detected in time, but falsealarm is unavoidable. Meanwhile, the observation se-quences and window average of observations strategies testthe existence of fault over the considered time window, but

No. 7 LIU Bo-Ang et al.: Statistical χ2 Testing Based Fault Detection for Linear Discrete Time-delay Systems 1283

the missed alarm is cleared at the start of fault happen-ing. Moreover, the fault alarm can not stop in time eventhe fault disappears. So, for the purpose of practical appli-cation, a trade-off of false alarm and missed alarm rate isnecessary.

4 Conclusion

In this paper, a statistical χ2 testing based approach toFD has been proposed for a class of LDTV systems withdelayed state and normally distributed noises. Under theassumption of process and measurement noises being withjointly normal distribution, a quadratic form of fault freecase measurement output follows a central χ2 distributionand, therefore, can be used as an evaluation function of FD.To reduce the heavy online computational burden, orthog-onal projection and innovation analysis techniques are ap-plied and, based on this, an available solution is obtained byrecursively computing Riccati recursions. To detect the oc-currence of a fault, three strategies of statistical χ2 testingof evaluation function and decision making are discussed.Finally, a numerical example is given to illustrate the de-veloped method.

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LIU Bo-Ang Master student in the De-partment of Mathematical Sciences, Ts-inghua University. Her research interestcovers statistical analysis and its applica-tions. E-mail: demetria@163.com

YE Hao Professor in the Department ofAutomation, Tsinghua University. He re-ceived his bachelor and Ph. D. degrees inautomation from Tsinghua University, in1992 and 1996, respectively. His researchinterest covers fault detection and diagno-sis of dynamic systems. Corresponding au-thor of this paper.E-mail: haoye@tsinghua.edu.cn