Networked Fault Detection for Singular Systems

Yaohui Wu Faculty of Electronic and Information Engineering

Zhejiang Wanli University Ningbo, China

e-mail: yaohuinb@yahoo.com.cn

Zhongzhe Lv Faculty of Electronic and Information Engineering

Zhejiang Wanli University Ningbo, China

e-mail: zhongzhelv@hotmail.com

Abstract—This paper is concerned with the networked fault detection for a class of singular systems. The considered plant has multiple outputs and only one of them can be sent to fault detector at each time instant due to medium access constraint. Moreover, the packet may also be lost during transmission. The whole networked fault detection system is modeled as a stochastic singular switched system, sufficient conditions for the stochastic exponential stability of the system as well as the design method of the fault detector are derived via the Lyapunov function method and the Linear matrix inequality technique (LMI). A numerical example is provided to illustrate the feasibility of the proposed method.

Keywords-Singular system; medium access constraint; missing measurement; fault detection

I. INTRODUCTION Singular systems, also referred to as descriptor systems

are extensively used to represent various engineering systems, such as electrical networks, power systems and so on. Due to the fact that singular systems can preserve the structures of physical systems more accurately by describing the dynamic part, the static part and even the improper part of the system in the same form, they provide a more general representation than the standard state-space systems in the sense of modelling [1]. There have been a number of fundamental results on the stability analysis [1], controller synthesis [2], etc. Fault detection and isolation (FDI), on the other hand, has been an active field of research over the past decades because of the increasing demand for higher performance, higher safety and reliability standards. However, little attention has been paid on the singular systems. Recently, some interesting results on the fault detection for singular system have been presented in [3], the H� filtering based fault detection method has been extended to the singular system.

It should be pointed out that the connection between the physical plant and the fault detector is point to point, this structure has many disadvantages such as high cost, complex installation and low reliability. The development of networked control system has well solved the above problems[4]. The research on the networked control systems has been an active domain for control engineering, communication engineering and computer engineering. Hence, it is not only theoretical but also practical important to investigate the networked fault detection for singular systems. This topic was firstly studied in [5] for the singular

Markov systems with missing measurement. However, an implicit assumption that the nonlinear plant output is transmitted in the same packet was made in [5]. Such an assumption may not hold in real systems. For example, the output may have to be transmitted via different sensors because the sensors are often deployed in a large geographical region and thus can not be grouped into one sensor. On the other hand, only one sensor is allowed to transmit the packets to the FD filter at any transmission instant due to the limited communication bandwidth. Clearly, the results in [5] can not be applied in practice. The main motivation of this paper is to further study the networked fault detection for the singular systems with both medium access constraint and random missing measurement.

This paper is concerned with the networked fault detection for singular systems with communication constraint and random missing measurement. By modelling the networked fault detection problem as a stochastic singular switched system, we use the switched system approach and stochastic analysis method to derive the sufficient conditions that guarantee the exponential stability of the system. Moreover, the parameter gains of the filter can be obtained by solving a set of LMIs. A simulation result is given to show the effectiveness of the proposed design method.

Notations: The notation used throughout the paper is fairly standard. We use 1, , ( ), Tr( )TW W W Wλ− and W to denote, respectively, the transpose, the inverse, the eigenvalues, the trace and the induced norm of any square matrix W . We use 0W > to denote a positive-definite matrix W with min ( )Wλ and max ( )Wλ being the minimum and maximum eigenvalues of W and I to denote the identity matrix with appropriate dimension. Let R n denote the n dimensional Euclidean space. Rm n× is the set of all m n× real matrices. The notation 2[0, )l ∞ refers to the space of square summable infinite vector sequences with the usual norm 2• . The symbol “*” will be used in some matrix expressions to represent the symmetric terms, and “E” denote the mathematical expectation.

II. PROBLEM FORMULATION Consider the following discrete-time singular system:

2012 Third International Conference on Intelligent Systems Modelling and Simulation

978-0-7695-4668-1/12 $26.00 © 2012 IEEE

DOI 10.1109/ISMS.2012.68

565

2012 Third International Conference on Intelligent Systems Modelling and Simulation

978-0-7695-4668-1/12 $26.00 © 2012 IEEE

DOI 10.1109/ISMS.2012.68

567

1 2

1 2

( 1) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )x k Ax k Bu k E d k E f ky k Cx k Du k F d k F f k

+ = + + +�� = + + +�

(1)

where ( ) R nx k ∈ is the state,

1( ) [ ( ) ( )]my k y k y k= � Rm∈ is the plant output vector, ( )iy k is a scalar for all {1,2,..., }i M m∈ = .

( ) R pd k ∈ is the unknown disturbance, ( ) R su k ∈ is the

control input, ( ) R qf k ∈ is the fault signal to be detected, which are assumed to belong to 2[0, )l ∞ .

1 2 1, , , , , ,A B E E C D F and 2F are some constant matrices, E is the singular matrix with rank( )E r n= < .

An FD system consists of a residual generator and evaluation stage including an evaluation function and a threshold. To generate a residual signal, we consider the following FD filter [6]:

ˆ ˆ( 1) ( ) ( )ˆ( ) ( ) ( )f f

f f

x k A x k B y kr k C x k D y k

+ = +��� = +��

�� (2)

where, ( ) R nfx k ∈ is the state of the filter, ( ) Rmy k ∈�

is the input signal of the filter and ( ) R qr k ∈ is the residual

signal. , ,f f fA B C and fD are filter gains to be determined. Due to the networked connection between the sensors

and the FD filter, ( ) ( )y k y k≠� . By using the communication sequence ( )kρ to describe the medium access constraint phenomenon,we have ( ) ( )i iy k y k=�

when ( )k iρ = ,and ( ) 0, \jy k j M i= ∀ ∈� . We can use the

matrices ( ) 1 2{ , ,..., }k mρΠ ∈ Π Π Π to describe the whole process. where diag{ ( 1), ( 2),..., ( )}i i i i mδ δ δΠ ∈ − − − is

a diagonal matrix and0, 0

( )1, 0t

tt

δ≠�

= � =�. Due the

measurement signal may also be lost due the sensor’s temporal failure or unreliable channel link. The binary valuable ( )kα is introduced to describe the random measurement missing process, where ( ) 1kα = packet is transmitted successfully and ( ) 0kα = means packet is lost. Combining the medium access constraint and random missing measurement, we have ( )( ) ( ) ( )ky k k y kρα= Π� , where, E{ ( ) 1}kα α= = is the packet successful transmission rate.

For the purpose of fault detection, it is not necessary to estimate the fault ( )f k . Sometimes one is more interested in the fault signal of a certain frequency interval, which can be formulated as the weighted fault ˆ ( ) ( ) ( )ff z W z f z=

with fW being a given stable weighting matrix. A minimal

realization of ˆ ( ) ( ) ( )ff z W z f z= is supposed to be

( 1) ( ) ( )ˆ ( ) ( ) ( )

w w

w w

x k A x k B f k

f k C x k D f k

+ = +���

= +�� (3)

where ( ) R nx k ∈ is the state of the weighted fault, and ˆ ( ) R qf k ∈ is the weighted fault. , ,w w wA B C and wD are

the known constant matrices. In this paper, we aim to minimize the error between the residual signal ( )r k and the weighted fault signal ˆ ( )f k in an H� framework.

Denote ˆ( ) [ ( ) ( ) ( )]T T T Tk x k x k x kξ = ,

( ) [ ( ) ( ) ( )]T T T Tw k u k d k f k= and

( ) ( ) ( )e k r k f k= − , for each communication sequence ( )k iρ = , we have

( )( )

( )( )

1 1

2 2

1 1

2 2

( 1) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

i i

i i

i i

i i

E k A k B w k

k A k B w k

e k C k D w k

k C k D w k

ξ ξ

α ξ

ξ

α ξ

� + = +�� + +��

= +��

+ +��

�� �

� ��

� �

� ��

(4)

where

0 00 00 0

IE E

I

� �� = � � �

� , 1

0 00 00

W

i

f i f

AA A

B C Bα

� �� = � � �

� ,

1 1 2

1 2

0 0 W

i

f i f i f i

BB B E E

B D B F B Fα α α

� �� = � � РРР�

� ,

2

0 0 00 0 00 0

i

f i

AB C

� �� = � � �

� ,

2

1 2

0 0 00 0 0i

f i f i f i

BB D B F B F

� �� = � � РРР�

� ,

1 [ ],i W f i fC C D C Cα= − Π�

2 [0 0],i f iC D C= Π�

1 1 2[ ]i f i f i f i WD D D D F D F Dα α α= Π Π Π −� ,

2 1 2[ ]i f i f i f iD D D D F D F= Π Π Π� , ( ) ( )k kα α α= −� .

Since system (4) is a stochastic singular switched system, we need the following definitions before the main results.

Definition 1. The stochastic singular switched system

1 2( 1) ( ) ( ) ( )i iE k A k k A kξ ξ α ξ+ = +� �� � is said to be

566568

stochastically regular if for each ,i M∈

( )1 2det E{ }i izE A Aα− −� �� � is not identically zero. Definition 2. The stochastic singular switched system

1 2( 1) ( ) ( ) ( )i iE k A k k A kξ ξ α ξ+ = +� �� � is said to be stochastically casual if, for each ,i M∈

( )( )1 2deg det E{ } rank( )i izE A A Eα− − =� �� �� .

Definition 3. For any given communication sequence ( )kρ and initial condition 0( )kξ , system (4) with ( ) 0w t = is stochastically exponentially stable if there exist

scalars 0K > and 0 1β< < ,

{ } 0( )0 0E ( ) ( ) ,k kk K k k kξ β ξ−< ∀ ≥ hold.

Definition 4 [7]. For any 0k k> , and any switching signal ( )ρ τ , 0k kτ≤ ≤ , let Nσ denote the number of switching ( )ρ τ over 0( , )k k . If 0 0( ) aN N k k Tσ ≤ + − holds for 0aT > and 0 0N ≥ , then aT is called average dwell time and 0N the chatter bound. As commonly used in the literature, we choose 0 0N = .

According to the model based fault detection theory, the fault detection can be transformed into an H� filtering problem [8]. Thus, system (4) is also required to have an

desired H� performance level, i.e., 0

E ( ) ( )T

se s e s

∞

=

� �� � ��

2

0( ) ( )T

sw s w sγ

∞

=

≤ � .

After designing the residual generator, we then introduce the following residual evaluation function:

( ( )) ( ) ( )k L

TL

s kJ r k r k r k

+

=

� �= � �� (5)

where L is the length of evaluation time. The occurrence of the fault can be observed by comparing the ( ( ))LJ r k and the threshold thJ according to the following logic:

( ) fault alarm( ) no fault

L th

L th

J r JJ r J

> � �

≤ � (6)

where { }2 2, , 0sup E ( ( ))th L

d l u l fJ J r k

∈ ∈ == .

III. PROBLEM FORMULATION Theorem 3.1. For any given scalars

0 1, 1, 0σ μ γ< < > > and communication sequence ( )kρ , system (4) is said to be stochastically exponentially stable and achieves a prescribed H� performance level γ if there exist matrices 0,i iP Q> , iQ , and matrix ( ) ( )R n n n n rR + + × −∈ with full column rank satisfying 0TE R =� , such that the following inequality(7-9) hold:

1 2 3 4

* 0 0 00* * 0 0

* * * 0* * * *

i i i i i i i

i

i

P PP

PI

I

θ θΩ Ψ Ψ Ψ Ψ� �� −� � <−� −� � − �

(7)

i jP Pμ≤ (8) * ln lna aT T μ σ> = − (9)

where

1 1 12

,*

T T T T Ti i i i i i i

iE PE Q R A A RQ Q R B

Iσ

γ� �− + +

Ω = � − �

� �� � �

1 1 1 2 2 2[ ], [ ]i i i i i iA B A BΨ = Ψ =� �� � ,

3 1 1 4 2 2[ ], [ ]i i i i i iC D C DΨ = Ψ =� �� �

4 2 2[ ], (1 )i i iC D θ α αΨ = = −� � . Proof: By Schur complement, (7) guarantees that

1 1 1 1 0T T T T Ti i i i i i i iA PA E PE Q R A A RQσ− + + <� � � �� � . Using

similar proof method in [5], it is easy to show system (4) is stochastic regular and casual. The proof is omitted here. We now consider the exponential stability of system (4) with

( ) 0w k = . To do this, construct the Lyapunov function as ( ) ( ) ( )T Ti iV k k E PE kξ ξ= � � (10)

Then, { }

{ }E ( 1) ( )

E ( 1) ( 1) ( ) ( )i i

T T T Ti i

V k V k

k E PE k k E PE k

σ

ξ ξ σξ ξ

+ −

= + + −� � � �

(11) Noticing 0TE R =� , we get for any matrix iQ that

{ }2E ( 1) ( ) 0T Tik E RQ kξ ξ+ =� (12)

Thus { }

21 1 2 2

1 1

E ( 1) ( )

( ) ( )

( ) ( )

i i

T T T Ti i i i i i i

T T T Ti i i i

V k V k

k A PA A PA E PE k

k Q R A A RQ k

σ

ξ θ σ ξ

ξ ξ

+ −

� �= + − �� �+ + �

� � � � � �

� � (13)

It follows from (7) that { }E ( 1) ( ) 0i iV k V kσ+ − < . For the switching signal ( )kρ , let 0 1 , 0lk k k k l< < < ≥� denote the switching points over the time interval 0[ , )k k , we have

( ) ( )E{ ( )} E{ ( )}l

l

k kk k lV k V kρ ρσ −≤ (14)

Combing (8) and (14), we have

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1

0 0

0

0

0

( ) ( )

( )

( 1)

( )( ) 0

1 ( )( ) 0

E{ ( )} E{ ( )}

E{ ( )}

E{ ( )}

E{ ( )}

( ) ( )

l

l

l

l

l

l

a

a

k kk k l

k kk l

k kk l

k k k k Tk

T k kk

V k V k

V k

V k

V k

V k

ρ ρ

ρ

ρ

ρ

ρ

σ

σ μ

σ μ

σ μ

σμ

−

−

−

−−

− −

−

≤

≤

=

≤ ⋅ ⋅ ⋅ ≤

≤

0

0

1 ( )( ) 0( ) ( )aT k kkV kρσμ −≤ (15)

Since system (4) is regular and casual, we choose non-singular matrices G� , H� and H

� such that

0 0, ,

0 0TI

GEH G RHI

−� � � �= =� � � �

�� �� � �

1 1 21

3 4

0, ,

0T i ii

i ii i

P PAGAH G PG

P PI− − � �� �

= = � � � �

� ��� � � ��� �

1

2

T T ii

i

QH Q H

Q− � �

= � �

���� .

Let 1 1

2

( )( ) ( )

( )k

k H kk

ηη ξ

η−� �

= =� �

� , the stability of system

(4) is equivalent to the following system

( )1 1 2 1( 1) ( )i ik A A kη α η+ = +� �� (16) According to the Lyapunov functional (10), we have

1 1 1( ) ( ) ( )Ti iV k k P kη η= � , which implies that

2min 1 1( ) min { } ( )i ii M

V k P kλ η∈

≥ � . On the other hand, there

exist two scalars 1β and 2β such that

0

0

0

21 1 ( )

1( ) 0

212 0

E{ ( ) } E{ ( )}

( ) ( )

( ) ( )

a

a

k

T k kk

T k k

k V k

V k

k

ρ

ρ

β η

σμ

σμ β ξ

−

−

≤

≤

≤

(17)

i.e.,

02 22

1 01

E{ ( ) } ( )k kk kβη β ξβ

−≤ (18)

where 1 aTβ σμ= . It follows from the condition (9) that 1β < , this means that 1( )kη is exponentially stable. Thus,

system (4) is exponentially stable. We now consider the H� performance.

Denote 2( ) ( ) ( ) ( ) ( )T Tk e k e k w k w kγΓ = − . By the

same function (10) and { }2E ( 1) ( ) 0T Tik E RQ kξ ξ+ =� , it is

easy to obtain that { }E ( 1) ( ) ( ) ( ) ( )T T

i i iV k V k k k kα η η+ − + Γ = Δ (19)

where ( ) [ ( ) ( )]T T Tk k w kη ξ= ,

2 21 1 2 2 3 3 4 4 .T T T T

i i i i i i i i i i i iP Pθ θΔ = Ω + Ψ Ψ + Ψ Ψ + Ψ Ψ + Ψ Ψ By Schur complement, (7) guarantees

{ }E ( 1) ( ) ( ) 0i iV k V k kσ+ − + Γ < , which implies

0 0

0

0

( )

( , )( ) 0

1( , ) 1

E{ ( )}

( )

E{ ( )}

k

k k N k kk

kN s k k s

s k

V k

V k

s

ρ

ρσ μ

μ σ

−

−− −

=

≤

− Γ� (20)

Considering the zero initial condition, we have

0

1 ( ) 1 ( ) 0k k ss ke sλ σ− − −

=Γ <� , where ln (ln ) aTλ σ μ= + .

Further, 1 ( ) 1

1 0( ) 0k k s

k se sλ σ∞ − − −

= =Γ <� � is also true.

Summing both sides from 1k = to k = ∞ and change the

order of summation, we have ( ) 10 1

( ) k ss k s

s eλ σ∞ ∞ − −= = +

Γ� � 1

0( ) 0

1 s

e se

λ

λσ −

∞

== Γ <

− � . Thus, { }0E ( ) ( )T

se s e s∞

=�

20

( ) ( )Tsv s v sγ ∞

=≤ � . This completes the proof.

We have presented a sufficient condition for the existence of the fault detection filter in theorem 3.1. However, the condition is not in the LMI form, which can not provide the filter gains directly. We now give the detailed design method in the following theorem.

Theorem 3.2. For any given scalars 0 1, 1, 0σ μ γ< < > > and communication sequence ( )kρ , system (4) is said to be stochastically exponentially stable and achieves a prescribed H� performance level γ if there exist matrices 0,i iP Q> , and any matrix ( )

1 Rn n rR × −∈ with

full column rank satisfying 1 0TE R = , such that (8), (9) and

1 2 3 4

* 0 0 00* * 0 0

* * * 0* * * *

i i i i iT

i i iT

i i i

P T TP T T

II

Ξ ϒ ϒ ϒ ϒ� �� − −� � <− −� −� � − �

(21)

hold, then the filter gains can be determined by

3T

f FA T A−= , 3T

f FB T B−= , fi FiC C= , fi FiD D= ;

where, 1 2 111

3 12

,*i i i

i ii i

Ξ Ξ ϒ� � � �Ξ = ϒ =� � Ξ ϒ � �

,

212

22

ii

i

ϒ� �ϒ = � ϒ �

, 313

32

ii

i

ϒ� �ϒ = � ϒ �

, 414

42

ii

i

ϒ� �ϒ = � ϒ �

,

568570

1 2 1 3

4 21 5 3

2

6

* ,

* *

Ti i i i

T Ti i T T

i i iT Ti

i

P P E Q R A PE P E Q R A

P A RQA RQ

P

σ σ σσ

σ

σ

� �− − + −�

− +� Ξ = − +� +� � − �

1 1 1 1 1 1 1 2

2 2 1 2 1 1 2 1 2

3 1 3 1 1 3 1 2

T T Ti i i

T T Ti i i i

T T Ti i i

Q R B Q R E Q R EQ R B Q R E Q R EQ R B Q R E Q R E

� �� Π= � � �

,

2 2 23 diag{ , , }i I I Iγ γ γΞ = − − − ,

1 2 3

11 4 5 6 ,0

W i W i W iT T T T T T T

i i i i F i i FT TF F

A T A T A TA T A T C B A T C B

A Aα α

� �� ϒ = + Π + Π� � �

4 5 6

12 1 4 1 5 1 1 6 1

2 2 5 3 2 61 2 4

2 2

,

T T T T T T Ti i i F i i F

T T T T T T Ti i i i F i i F

T T T TW i i W i iT T

W i i T T T Ti F i F

B T B T D B B T D BE T E T F B E T F B

B T E T B T E TB T E T

F B F B

α αα α

α α

� ��

+ Π + Π� � ϒ = + Π + Π� � + +

+� + Π + Π� �

21

0 0 000 0 0

T T T Ti i F i FC B C Bθ

� �� ϒ = Π Π� � �

,

22 1 1

2 2

000

T T T Ti F i F

T T T Ti i F i F

T T T Ti F i F

D B D BF B F BF B F B

θ� �Π Π� ϒ = Π Π� � Π Π �

,

31 [ ]Ti W F i FC D C Cαϒ = Π ,

32 1 2[ ]Ti F i F i F iD D D F D Fα α αϒ = Π Π Π ,

41 [0 0]Ti F iD Cθϒ = Π ,

42 1 2[ ]Ti F i F i F iD D D F D Fθ θ θϒ = Π Π Π ,

1 2 3

4 5

6

** *

i i i

i i i

i

P P PP P P

P

� �� = � � �

, 1 2 3

4 5 6

7 70

i i i

i i i i

T T TT T T T

T T

� �� = � � �

,

1 2 3[ ]T T T Ti i i iQ Q Q Q= , 1[0 0]T TR R= .

Proof. Construct the matrices 1 2 3

4 5

6

** *

i i i

i i i

i

P P PP P P

P

� �� = � � �

,

1 2 3

4 5 6

7 70

i i i

i i i i

T T TT T T T

T T

� �� = � � �

, 1 2 3[ ]T T T Ti i i iQ Q Q Q= and

1[0 0]T TR R= . System (9) can be described as (21). This completes the proof.

IV. SIMULATION Consider the singular system (1), the parameters are

given as 1 0 0.6 0.4 0.8

, ,0 0 0.2 0.5 0.1

E A B� � � � � �

= = =� � � − � � �,

1 2

0.5 1.2 1.3 1.0, ,

0.2 1.0 1.0 2.0E E C

� � � � � �= = =� � � � � �

,

1 2

1.2 0.8 1.2, ,

1.0 1.0 0.8D F F

� � � � � �= = =� � � � � �

.

Suppose the measurement signals are transmitted via two sensors, the data successful transmission rate is 80% . The fault weighting system is in the form of (3) and the matrices are chosen as 0.5, 0.5, 1.0W W WA B C= = = and 0WD = . We first choose 0.95σ = , 1.1μ = , 1 [0 1]TR = , it is

obtained that * 1.858aT = . By using the Matlab LMI toolbox

[8], the optimal H� performance level * 0.578γ = , the corresponding filter matrices are

0.116 0.309 0.432 0.379, ,

0.098 0.611 0.040 0.064f fA B− −� � � �

= =� � − � �[ 0.013 0.016], [0.054 0.044]f fC D= − = .

In the simulation setup, we choose ( ) 0.1sin( )d k k= , control signal ( )u k as a step signal, the fault signal are assumed to occur from 11k = to 20k = with amplitude 1. The initial condition of system (1) is chosen to be

(0) [0.5 0.2]Tx = , and the communication sequence ( )kρ is chosen to be a periodical signal as 1,1,2,2,1,1� .

We also select 30L = . Figure 1 shows the residual evaluation function. From which, we can see that

1.910thJ = and15

0( ) ( ) 1.961 1.910T

sr k r k

=� � = > �� ,

which means that the fault can be determined after four time step.

Figure 1. Evaluation function

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V. CONCLUSION We have investigated the networked fault detection for

singular systems with both medium access constraint and random missing measurement. By using the switched system theory and the stochastic analysis method, the sufficient condition for the existence of the fault detection filter has been presented. Moreover, the filter gains can be determined by solving a set of LMIs. The effectiveness of the proposed design method has been illustrated by a numerical example.

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singular systems: An equivalent characterization,” IEEE Trans. Autom. Control, vol. 49, no. 4, pp. 568–574, 2004.

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