A Fault Detection Scheme for Time-delay Systems using Minimum-order

Functional Observers

H. M. Tran and H. Trinh

School of Engineering, Faculty of Science, Engineering and Built Environment, Deakin University,

Geelong VIC 3217, Australia

Keywords:

Time-delay Systems, Fault Detection, Reduced-order Functional Observers, Residual Generators.

Abstract:

This paper presents a method for designing residual generators using minimum-order functional observers to

detect actuator and component faults in time-delay systems. Existence conditions of the residual generators

and functional observers are ﬁrst derived, and then based on a parametric approach to the solution of a gener-

alized Sylvester matrix equation, we develop systematic procedures for designing minimum-order functional

observers to detect faults in the system. The advantages of having minimum-order observers are obvious from

the economical and practical points of view as cost saving and simplicity can be achieved, particularly when

dealing with high-order complex systems. Extensive numerical examples are given to illustrate the proposed

fault detection scheme. In all the numerical examples, we design minimum-order residual generators and

functional observers to detect faults in the system.

1 INTRODUCTION

Time-delay systems are commonly encountered in

various engineering complex systems. As cited in

(Duan and Patton, 2002; Fu et al., 2004; Wu and

Duan, 2007) time delays appear in practical processes

such as aviation industries, chemical processes, long

transmission lines and rolling mill systems. In fact,

when time delays appear in high complex systems,

they can cause the systems to be more vulnerable

to unexpected faults. Faults can enter the systems

via input or state delay. Normally, faults can cause

malfunctions of the system operations such as partly

break down or even whole system shutdown (Teh

and Trinh, 2013; Chen and Patton, 2012). Hence,

the problems of fault detection have been extensively

considered over the last several decades to improve

reliability and safety of system performance.

Accounting from the last several decades, there is

a wide range of approaches which are based on the

foundation of mathematics to build models in order to

detect system faults. One of the theories is applying

Kalman ﬁlter method to generate a residual based on

the difference between the ideal output and real out-

put of the system, this method can be seen in (Wang

et al., 2006; Zhong et al., 2003). Another trend in this

ﬁeld is using geometric approach which can be seen

in (Meskin and Khorasani, 2009). Meanwhile, a very

common approach, which is used for fault detection,

is observer-based strategy. The basic idea behind the

observer-based approach is to estimate the state and

the output of the system from the measurements by

using some types of state observers, and then con-

struct a residual by a properly weighted output esti-

mation error. The residual is then examined for the

likelihood of faults (Duan and Patton, 1998; Duan and

Patton, 2001; Huong et al., 2014).

In this paper, the work on reduced-order func-

tional observers for dynamical systems (Darouach

et al., 1999; Darouach, 2001; Trinh and Ha, 2007;

Fernando et al., 2010; Trinh et al., 2004; Trinh and

Fernando, 2012; Fernando and Trinh, 2013) and (Fer-

nando and Trinh, 2014) are used to design a simple

and effective scheme to detect faults for time-delay

systems. We construct residual generators based on

the system outputs and minimum-order functional ob-

servers to trigger faults occurring in the systems. The

salient feature of our work is that the order of our de-

signed residual generators and observers is very low

and hence our fault detection scheme is very practical

and easy to implement. In the next section, we present

system description and problem statement. This is

then followed by preliminaries results and our pro-

posed fault detection scheme for time-delay systems.

Finally, three examples in Numerical Examples sec-

tion to illustrate the proposed theory can be seen.

Tran H. and Trinh H..

A Fault Detection Scheme for Time-delay Systems using Minimum-order Functional Observers.

DOI: 10.5220/0005506300640071

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 64-71

ISBN: 978-989-758-122-9

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

2 SYSTEM DESCRIPTION AND

PROBLEM STATEMENT

We consider the following time-delay system











˙x(t) = Ax(t) + A

x(t − τ) + Bu(t)

+E f(t), t ≥ 0,

y(t) = Cx(t),

x(t) = φ(t), t ∈ [−τ, 0),

(1)

where τ is a positive real number presenting the

known time delay in the state, x(t) ∈ R

, u(t) ∈ R

and y(t) ∈ R

are the state, input and output vector,

respectively. Unpredictable fault signals f(t) ∈ R

enter from the system input. A ∈ R

n×n

, A

∈ R

n×n

B ∈ R

n×m

, C ∈ R

p×n

and E ∈ R

n×l

are known con-

stant matrices. We assume that the pair (A,C) is ob-

servable, rank(C) = p, rank(E) = l. Furthermore, the

faults f(t) are assumed to be linear independent to

avoid vagueness situations which may appear when

some faults occur simultaneously. Hence, the resid-

ual generator may not detect faults because of zero

overall effects of faults.

Let us introduce a functional observer ω(t) of a

general q-order,1 ≤ q ≤ n. Here, ω(t) is an estimation

of a function Lx(t), L ∈ R

q×n

, where











ω(t) = Nω(t) + Gy(t) + G

y(t − τ)

+Hu(t), t ≥ 0,

ω(t) = ϕ(t), t ∈ [−τ, 0),

(2)

ω(t) ∈ R

, N ∈ R

q×q

, G ∈ R

q×p

, G

∈ R

q×p

and

H ∈ R

q×m

are constant observer parameters which

will be determined such that ω(t) is an asymptotic es-

timation of the function Lx(t) when there is no fault

appearing in the system, i.e., f(t) = 0. Matrix L will

be determined for the purpose of fault detection.

Let us consider a residual generator r(t) which is

used to trigger faults in system (1) whenever the faults

appear, i.e., f(t) 6= 0,

r(t) = Tω(t) + Fy(t), (3)

where T ∈ R

1×q

and F ∈ R

1×p

are constant matrices

which will be detemined to satisfy the residual func-

tion such that

lim

t→∞

r(t) =

(

0 if f(t) = 0

c or undeﬁned if f(t) 6= 0,

(4)

where c 6= 0, and f(t) = 0 implies a faultless condi-

tion, f (t) 6= 0 implies a faulty condition (Trinh et al.,

2013).

3 PRELIMINARIES RESULTS

Obviously, the reduced-order functional observers

and the residual generator can detect faults in systems

(1) if all the unknownparameters satisfy the following

conditions in Theorem 1.

Theorem 1. Under faultless conditions, ω(t) is an

asymptotic estimate of Lx(t) and residual genera-

tor r(t) function as (4) for any initial condition

φ(t), x(0), ω(0) and any u(t) if and only if

N is Hurwitz, (5)

NL+ GC− LA = 0, (6)

C− LA

= 0, (7)

H − LB = 0, (8)

TL+ FC = 0. (9)

Under fault conditions, residual r(t) satisﬁes (4)

if and only if all the parameters satisfy the conditions

(5)-(9) and

LE 6= 0. (10)

Proof. Let us deﬁne an error vector e(t) ∈ R

which

is the difference between the estimate ω(t) and the

function Lx(t) as follows

e(t) = ω(t) − Lx(t), t ≥ −τ. (11)

It follows from (1), (2) and (11), we obtain

˙e(t) = Ne(t) + (NL+ GC− LA)x(t)

+ (G

C− LA

)x(t − τ)

+ (H − LB)u(t) − LE f(t), t ≥ −τ.

(12)

Sufﬁciency: In the case of faultless conditions, i.e.,

f(t) = 0, if conditions (6)-(8) are satisﬁed then equa-

tion (12) is reduced to ˙e(t) = Ne(t). As a result, if

condition (5) is satisﬁed then the error e(t) asymptot-

ically tends to zero. Hence, the reduced-order func-

tional observer ω(t) is an estimation of the functional

Lx(t) as expected. Furthermore, by (1), (3) and (11),

residual r(t) can be expressed as follows

r(t) = Te(t) + (TL+ FC)x(t). (13)

It is clear that under no fault condition, the error

e(t) is expected to asymptotically tends to zero, thus

the residual is proposed to be zero and it happens if

condition (9) holds.

Necessity: Under no fault conditions, if condition

(5) is unsatisﬁed then even conditions(6)-(8) hold, the

error e(t) 6→ 0 with any initial condition of φ(t) and

x(0). Contrarily, if one of the conditions (6)-(8) is not

satisﬁed then even (5) holds, it is possible to ﬁnd x(0)

to make e(t) 6→ 0.

AFaultDetectionSchemeforTime-delaySystemsusingMinimum-orderFunctionalObservers

Under fault conditions, if conditions (5)-(9) hold,

by (12) and (13), the residual is then governed by the

following equations

(

˙e(t) = Ne(t) − LE f(t),

r(t) = Te(t).

(14)

It follows from (14), r(t) can detect faults f(t) if

condition (10) is satisﬁed. This completes the proof

of Theorem 1.

The design of the functional observers and the

residual generator is now reduced to ﬁnding matrices

L, N, G, G

, H, T and F which satisfy Theorem 1.

4 FAULT DETECTION SCHEMES

This section is to determine the possible order of the

observer and the necessary parameters for designing

reduced-order functional observers and residual gen-

erator to detect faults in system (1). Note that, based

on the conditions in Theorem 1, whenever matrix L

is found, matrix H is determined from (8) and con-

dition (10) is checked. Moreover, other unknown pa-

rameters are solutions of equations (6), (7), (9) and

condition (5) holds. In order to simplify the three ma-

trix equations (6), (7) and (9), we employ the partition

method introduced in (Trinh and Fernando, 2012).

Let P ∈ R

n×n

be deﬁned by

P =



⊥



, (15)

where C

∈ R

n×p

is the Moore-Penrose inverse of

matrix C, CC

= I

, and C

⊥

∈ R

n×n−p

is an orthog-

onal basis for the null space of matrix C, CC

⊥

= 0.

Then P is an invertible matrix (see, (Trinh and Fer-

nando, 2012)).

Now we deﬁne the following partitions

CP =





, (16)

LP =





, (17)

−1

AP =





, (18)

−1

P =



d11

d12

d21

d22



, (19)

where submatrices L

∈ R

q×p

, L

∈ R

q×(n−p)

, A

∈

p×p

, A

∈ R

p×(n−p)

, A

∈ R

(n−p)×p

, A

∈

(n−p)×(n−p)

, A

d11

∈ R

p×p

, A

d12

∈ R

p×(n−p)

, A

d21

∈

(n−p)×p

and A

d22

∈ R

(n−p)×(n−p)

Post-multiply both sides of equations (6), (7) and

(9) by matrix P, we have following equations

NLP+ GCP− LPP

−1

AP = 0, (20)

CP− LPP

−1

P = 0, (21)

TLP+ FCP = 0. (22)

Substitute equations (15)-(19) into (20)-(22), we

obtain

+ L

− NL

= G, (23)

d11

+ L

d21

= G

, (24)

F + TL

= 0, (25)

− L

= 0, (26)

d12

+ L

d22

= 0, (27)

= 0. (28)

It is clear from equations (23), (24) and (25) that

matrices G, G

and F are computed when matrices

N, L

, L

and T are found. Consequently, the de-

sign of the observers and residual generator is now

reduced to determine matrices N, L

, L

and T such

that three conditions (26)-(28) are satisﬁed and con-

dition (5) holds.

4.0.1 Case 1: p >

In this case, we consider the design of only ﬁrst-order

functional observers and residual generators to detect

faults in the systems (1). We show that indeed it

is possible provided that the number of outputs

satisfying p >

and that condition (10) is satisﬁed.

Theorem 2. If p >

, ﬁrst-order observers and

residual generators always exist to detect faults in the

system if the condition (10) is satisﬁed.

Proof. To design a ﬁrst-order functional observer,

i.e., q = 1, N ∈ R

1×1

can be chosen to be any neg-

ative real number, i.e.,

N = s, s < 0. (29)

Note that s < 0 ensures the satisfaction of the con-

dition (5), i.e., N is Hurwitz. By letting L

= 0, equa-

tion (26) and (27) are reduced to

= 0, (30)

d12

= 0. (31)

We can express (30) and (31) as



d12



= 0. (32)

Since p >



d12



∈ R

p×2(n−p)

is a col-

umn matrix, i.e., p > 2(n− p), thus a solution to (32)

where L

6= 0 always exists. Let N (X) be a matrix

of row basis vectors for the row-nullspace of X, i.e.,

N (X)X = 0. The solution to L

6= 0 according to (32)

exists when N (X) 6= 0. Therefore, solutions for L

can be computed by ﬁrst ﬁnding

according to

= N



d12



, (33)

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

and then choosing any row of

as L

In (28) and since L

= 0, T 6= 0 can be arbitrarily

chosen to be any scalar, say, α where α 6= 0. Finally,

if the condition (10), LE 6= 0, is satisﬁed, where L =





−1

is obtained according to (17), matrices

H, G, G

and F can then be obtained from (8), (23),

(24) and (25), respectively. This completes the proof

of Theorem 2.

The effectiveness and the simplicity of this

scheme can be seen in Example 1 of the Numeri-

cal Examples section, in which a system with n = 4,

p = 3, m = 1 and l = 1 is taken under considera-

tion. For this system, according to Theorem 2, we

only need to design ﬁrst-order observer and residual

generator to detect faults in the system.

4.0.2 Case 2: 1 ≤ p ≤

In this case, ﬁrst-order functional observers do not ex-

ist and we have to consider observers of higher or-

der. We present a solution to the three matrix equa-

tions (26)-(28) with the requirement that N has pre-

scribed distinct eigenvalues. For completeness, let us

ﬁrst present a parametric solution (Duan, 1993) to the

generalized Sylvester matrix equation (26).

Let N ∈ R

q×q

with q distinct eigenvalues be de-

ﬁned as follows

N = Q

−1

ΛQ, (34)

where Q ∈ R

q×q

is any freely chosen invertible ma-

trix, Λ = diag(s

, s

, ..., s

), s

6= s

for i 6= j and

Re(s

) < 0 for all i = 1, 2, ..., q.

With N as deﬁned in (34), L

and L

satisfy-

ing (26) are given in the following parametric forms

(Duan, 1993)

= Q



U(s

... U(s



⊤

,(35)

= Q



Z(s

... Z(s



⊤

,(36)

where b

∈ C

(i= 1, 2, ..., q) are free parameters satis-

fying b

if s

= ¯s

, ¯s

denotes the complex conju-

gate of s

. Matrices U(s) ∈ R

p×p

and Z(s) ∈ R

(n−p)×p

are coprime polynomial matrices satisfying the fol-

lowing coprime factorization

(sI

n−p

− A

⊤

)

−1

⊤

= Z(s)U

−1

(s). (37)

The reader can refer to (Duan, 1993) for a numer-

ically reliable algorithm to compute Z(s) and U(s).

Also, as suggested in (Trinh and Fernando, 2012),

U(s) and Z(s) can be conveniently computed accord-

ing to the following equations

U(s) = det(sI

n−p

− A

, (38)

Z(s) = adj(sI

n−p

− A

⊤

, (39)

where det(.) and adj(.) denote the determinant and

the adjugate matrix of matrix (.), respectively. For

any given A

, the characteristic polynomial can be

obtained as follows

a(s) , det(sI

n−p

− A

)

= s

n−p

+ a

n−p−1

+ a

n−p−2

+ ... + a

n−p

, (40)

where the coefﬁcients a

, i = 1, 2, ..., (n − p), are real

constants.

The adjugate matrix adj(.) is then obtained as fol-

lows

adj(sI

n−p

− A

⊤

) = ϒ

n−p−1

+ ϒ

n−p−2

+ ϒ

n−p−3

+ ... + ϒ

n−p

, (41)

where ϒ

, i = 1, 2, ..., n− p, are computed by using the

coefﬁcients of a(s) and matrix A

as given below











= I

n−p

= ϒ

⊤

+ a

n−p

= ϒ

⊤

+ a

n−p

= ϒ

n−p−1

⊤

+ a

n−p−1

n−p

(42)

Theorem 3. A functional observer always exists with

q-order where q is the lowest order that matrix M (M

is deﬁned in equation (45)) has row basis vectors for

the row-nullspace of M, N (M) 6= 0. Furthermore, the

proposed residual generator (3) can detect the faults

in systems if condition (10) holds.

Proof. Now, by substitute L

and L

from (35) and

(36) into the transpose of (27) and (28), we obtain



d12

d22



⊤



U(s

... U(s

Z(s

... Z(s



⊤

= 0, (43)



Z(s

... Z(s



(QT)

⊤

= 0. (44)

Since Q is an invertible matrix, let (QT)

⊤



. . . t



, t

6= 0, i = 1, 2, ..., q, are arbitrarily

chosen real numbers. It follows (43) and (44), we ob-

tain

Mβ = 0, (45)

where M ∈ R

(q+1)(n−p)×pq

, β ∈ R

pq×1

and

M =







0 . . . 0

. . . 0

0 0 . . .

Z(s

. . . Z(s







= A

⊤

d12

U(s

) + A

⊤

d22

Z(s

), i = 1, 2, . . . , q,

β =



⊤

. . . b

⊤



⊤

AFaultDetectionSchemeforTime-delaySystemsusingMinimum-orderFunctionalObservers

Let N (M) be a matrix of row basis vectors for the

row-nullspace of M, i.e., MN (M) = 0. Therefore, the

solutions for β 6= 0 in (45) exists iff N (M) 6= 0, and β

can be selected as any column of

β, where

β = N (M). (46)

This completes the proof of Theorem 3.

Remark 1: For the case

< p ≤

, if M is a row

matrix, N (M) 6= 0 always exists, thus implies

q >

n−p

2p−n

. (47)

This leads to a method for searching for the or-

der q, we only need to search for the lowest q, which

satisﬁes Theorem 3, in the range of

2 ≤ q ≤

n−p

2p−n

+ 1

. (48)

Remark 2: For the case 1 ≤ p ≤

, M in (45) is

always a column matrix, thus its row basis vectors for

the row-nullspace, N (M) 6= 0, exists if M is not a full

rank matrix, that implies

rank(M) < qp. (49)

Based on Remark 2, q-order can be selected as the

smallest order that M satisﬁes condition (49).

It is concluded that since matrices T is arbitrarily

chosen, matrices L

, L

and N are determined through

this section, the parameters H, G, G

and F are calcu-

lated based on equations (8), (23), (24) and (25), re-

spectively. Matrix L is then achieved from the equa-

tion (17), we can check if condition (10) holds, i.e.,

LE 6= 0. Thus, all the conditions in the Theorem 1

are satisﬁed and the design of the reduced-order func-

tional observers and the ﬁrst-order residual generator

to detect the faults in system (1) is completed. Ex-

amples 2 and 3 in the Numerical Examples section

illustrate the theory of this section.

5 NUMERICAL EXAMPLES

Example 1: In this example, we take consideration

of timely fault detection in a time-delay system with

n = 4, p = 3, m = 1, and l = 1. Since we have the case

where p >

, and as discussed in Case 1 of Section 4,

we only need to design a residual generator based on

ﬁrst-order observer to detect the faults in the system.

For this example, C =





, A, A

, B and E are

as given below

A =







−5 0 1 2

1 −1 0 −2

0 0 −3 −1

−2 2 0 −2







, B =







−2

−1













0 −1 0 0

0 0 0 −1

0 1 0 −2

1 0 0 −1







, E =







−1

−3







Now, the design of a ﬁrst-order functional ob-

server and residual generator can be readily carried

out.

Since C is already in the desired form, i.e., C =





, so P is an identity matrix, i.e., P = I

. Ac-

cording to the partitions (18) and (19), submatrices

, A

d11

, A

d12

, A

d21

and A

d22

are ob-

tained, where











−5 0 1

1 −1 0 −2

0 0 −3

−1

−2 2 0 −2









d11

d12

d21

d22









0 −1 0

0 0 0

−1

0 1 0 −2

1 0 0 −1







It is clear that



d12



is a column matrix

and thus its matrix of row basis vectors for the row-

nullspace exists, i.e., N



d12



6= 0. As a result,

a ﬁrst-order functional observer exists.

For the design of ﬁrst-order observer and ﬁrst-

order residual generator, let us assign N = −3 and L

is computed according to (33), we obtain



−0.5571 −0.7428 0.3714



Since L

= 0 and according to (17), matrix L is

obtained as

L =



−0.5571 −0.7428 0.3714 0



With L as obtained above, condition (10) is found

to be satisﬁed since

LE = 0.9285 6= 0.

Hence, a ﬁrst-order residual generator exists and

can be constructed to detect faults in the system. By

choosing T = −5, matrices H, G, G

and F are ob-

tained according to equations (8), (23), (24) and (25),

respectively, where

H = 2.4140,

G =



0.3714 −1.4856 −0.5571





0 0.9285 0



F =



−2.7854 −3.7139 1.8570



ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

Figure 1 shows that the residual generator can ef-

fectively detect the faults f(t) in the system. Fault

f(t) appears at time t = 20s and clears from t =

30s. During the time the fault happens, the resid-

ual generator triggers the fault, when there are no

faults, the residual generator converges to zero as ex-

pected. Note also that the residual is insensitive to

the input u(t) as expected. It is clear in this exam-

ple that, a signiﬁcantly lower order (only ﬁrst-order)

residual generator is designed using a ﬁrst-order func-

tional observer. In contrast, existing fault detection

schemes using full-order or reduced-order state ob-

servers would give higher order schemes.

Example 2: This example is given to demonstrate

Case 2 (Section 4), where

< p ≤

. Let us con-

sider a system which hasC =





and matrices A,

, B and E given as

A =







−1 0 0 1 −2

0 −5 3 4 0

1 1 −8 3 0

−4 0 2 −6 0

0 0 0 1 −1







, B =







1 0

0 1

1 −1

0 0

0 1













0 0 2 −1 0

0 −1 0 1 0

0 0 −1 3 0

−1 0 1 −2 0

0 1 0 0 −3







, E =







1 3

−1 1

2 −2

−3 2

2 −3







Thus, for this example, we have n = 5, p = 3, m =

2 and l = 2. Since

< p ≤

, this falls into the Case

2 (Section 4) and thereforewe can carry out the design

of a reduced-order observer and a residual generator

to detect faults in the system.

Since C =





, thus P = I

and according

to (18) and (19), sub-matrices A

, A

d11

, A

d12

, A

d21

and A

d22

are obtained, where











−1 0 0

1 −2

0 −5 3 4 0

1 1 −8

3 0

−4 0 2 −6 0

0 0 0 1 −1









d11

d12

d21

d22









0 0 2

−1 0

0 −1 0 1 0

0 0 −1

3 0

−1 0 1 −2 0

0 1 0

0 −3







According to (40)-(42), we obtain the characteris-

tic polynomial and adjugate matrix as

a(s) = s

+ 7s+ 6,

adj(sI

n−p

− A

⊤

) = ϒ

s+ ϒ

where

= I

, ϒ



1 1

0 6



Time(sec)

0 5 10 15 20 25 30 35 40 45 50

-15

-10

-5

Fault f(t)

Residual r(t)

Input u(t)

Figure 1: Residual generator using ﬁrst-order observer ef-

fectively triggers fault in the system.

The pair of coprime polynomial matricesU(s) and

Z(s) are then calculated based on (38) and (39)

U(s) = (s

+ 7s+ 6)I

Z(s) = (ϒ

s+ ϒ

⊤

As in Theorem 3 and Remark 1, now we search

for the lowest possible order q of the observers ω(t).

It follows equation (48) we have

2 ≤ q ≤ 3.

For the case that q = 2, let us assign the poles of N

to be at s

= −3, s

= −5 and choose Q to be Q = I

and TQ =



1 1



From (45), matrix M and the matrix of row basis

vectors for the row-nullspace of M are obtained

M =







14 10 −6 0 0 0

18 0 0 0 0 0

0 0 0 16 28 12

0 0 0 6 0 0

−4 −8 −6 −6 −16 −12

−6 0 0 −2 0 0







N (M) =



0 0.2339 0.3898 0 0.3509 −0.8187



⊤

Since N (M) 6= 0, the second-order observers ex-

ist for the system. Accordingly, β 6= 0 exists and is

obtained by taking any column of N (M ). Matrices

and b

are then obtained based on (45), where



0 0.2339 0.3898



⊤



0 0.3509 −0.8187



⊤

From (35) and (36), L

and L

are obtained



0 −1.4034 −2.3390

0 −1.4034 3.2747





−4.2103 0

4.2103 0



AFaultDetectionSchemeforTime-delaySystemsusingMinimum-orderFunctionalObservers

It follows (17) and P = I

, L =





and it is

easy to verify condition (10) that

LE =



9.3562 −5.1459

−4.6781 0.4678



6= 0.

Thus, all the conditions for Theorem 3 are satis-

ﬁed and hence a second-order observer and ﬁrst-order

residual generator exist and can be constructed to de-

tect faults in the system by determining all other un-

known parameters, where

N =



−3 0

0 −5



, T =



1 1



G =



14.5021 0.4678 −0.9356

−13.5665 3.2747 −5.6137





4.2103 1.4034 −1.8712

−4.2103 1.4034 0.9356



H =



−2.3390 0.9356

3.2747 −4.6781



F =



0 2.8069 −0.9356



Figure 2 indicates that the residual generator can

detect the faults f

(t) and f

(t) in the system. It is

clear in this example that the design of residual gen-

erator is very easy and systematic. Furthermore, the

order of the functional observer is very low compar-

ing to conventional FD schemes using full-order or

reduced-order state observers. This example thus fur-

ther highlights the attractiveness of the FD scheme

proposed.

Time(sec)

0 5 10 15 20 25 30 35 40 45 50

-30

-20

-10

Fault f

(t)

Fault f

(t)

Input u(t)

Residual r(t)

Figure 2: Residual generator based on second-order fuc-

tional observers detects faults in the system.

Example 3: This example is given to demonstrate

Case 2 (Section 4), where 1 ≤ p ≤

. Let us con-

sider a system which has matrices A, B, and E as same

as in Example 2, however, matrices A

and C given as







−1 0 0 0 0

2 −1 0 0 0

1 −1 0 0 0

−1 0 0 0 0

0 1 0 0 0







, C =







1 0

0 1

0 0







⊤

Thus, for this example, we have n = 5, p = 2, m =

2 and l = 2. Since C =





, implies P = I

and

according to (18) and (19), sub-matrices A

, A

d11

, A

d12

, A

d21

and A

d22

are obtained, where











−1 0

0 1 −2

0 −5

3 4 0

1 1 −8 3 0

−4 0

2 −6 0

0 0 0 1 −1









d11

d12

d21

d22









−1 0

0 0 0

2 −1 0 0 0

1 −1 0 0 0

−1 0

0 0 0

0 1 0 0 0







Let us assign q = 2, the poles of N to be at s

−7, s

= −9, Q = I

and TQ =



1 1



. It follows

the same line as in the calculations of Example 2, the

matrix M in (45) is obtained as

M =







0 0 0 0 0 0 −16 −8 14

0 0 0 0 0 0 −30 −78 0

0 0 0 0 0 0 −20 10 6

0 0 0 0 0 0 8 −40 0







⊤

Since rank(M) = 3 < qp = 4, the condition (49)

holds and N (M) 6= 0 exists, where

N (M) =



0.3152 0.1686 −0.7354 −0.5757



⊤

Accordingly, b

and b

are obtained, where

= [

0.3152 0.1686

]

⊤

, b

= [

−0.7354 −0.5757

]

⊤

From (35) and (36), L

and L

are obtained



13.2368 7.0828

−17.6490 −13.8173





−10.1017 −15.6751 4.4123

10.1017 15.6751 −4.4123



It follows (17) and P = I

, L =





and it is

easy to verify that

LE =



41.8003 22.4096

−39.4781 −42.3809



6= 0.

Thus, all the conditions for Theorem 3 are satis-

ﬁed and hence a second-order observer and a residual

generator exist and can be constructed to detect faults

in the system by determining all other unknown pa-

rameters, where

N =



−7 0

0 −9



, T =



1 1



G =



132.019 4.064

−193.791 −45.168





6.502 7.431

−15.559 −0.697



H =



3.135 21.597

−7.547 −28.331



, F =



4.412

6.734



⊤

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

Time(sec)

0 5 10 15 20 25 30 35 40 45 50

-25

-20

-15

-10

-5

Fault f

(t)

Fault f

(t)

Input u(t)

Residual r(t)

Figure 3: Residual generator based on second-order func-

tional observers detects faults in the system.

Figure 3 indicates that a residual generator based

on a second-order observer can effectively detects the

faults f

(t) and f

(t) in the system. It clearly illus-

trates the Remark 2.

6 CONCLUSION

This paper has proposed a new fault detection scheme

using minimum-order functional observers to con-

struct residual generators to timely trigger actuator

faults in time-delay systems. The proposed approach

is based on solving a generalized Sylvester matrix

equation via a parametric approach. Existence con-

ditions and systematic procedures for designing the

proposed fault detection scheme have been presented.

The lowest possible order and the simplicity of the

approach are the hallmark of the proposed novel fault

detection scheme. Three examples have been con-

structed to prove the theory of the scheme.

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AFaultDetectionSchemeforTime-delaySystemsusingMinimum-orderFunctionalObservers