2005 International Conference on Control and Automation (ICCA2005)June 27-29, 2005, Budapest, Hungary WE-2.8

Fault Detection Based on Robust III states Observer on Networked ControlSystems

Yong Bao, Qiu-qiu Dai, Ying-liu Cui and Hui-zhong Wu

Abstract-The buffer is added into NetworkedControl Systems and random time-delay is changedinto constant delay. An approach to design a robustHe* states observer based on H., control theory onNetworked Control Systems (NCS) is proposed in thispaper. The theorem on existing the observer and howto get the gain matrix is presented and proved indetail. Robust fault detection based on robust H,o.states observer is achieved. Finally, the simulationresults on an example demonstrate that the method iseffective and robust.

Key words: Networked Control Systems; Robust H.States Observer; Time-delay; Fault Detection

I. INTRODUCTION

Networked Control Systems are the real-timenetworked closed-loop control systems, through whichthe information transmits. Recently, the NCS becomes avery hot problem and has some important applications inthe control area.

Because the study of NCS started in recent years, theviewpoint dwells on modeling, attemperment, stabilityanalysis, estimate and control of the network time-delay,etc. Little attention has been paid to states observer, andespecially in the area of the fault detection and diagnosethere has few study work.

In the paper [2], a states observer of NCS wasdesigned with the time-delay compensation method. Itprovided the structure of the states observer and provedits apex under any condition without consideration of thesystem noise and more disturbances, so its practicabilityis weak. In the paper [3], the author designed a statesobserver of NCS with a buffer, but in theory, it did notsolve the robust problem of the states observer. In the

Bao Yong is with the Department of Computer Science, NanjingUniversity of science and Technology ,and the Department of ComputerScience Engineering ,NanJing Audit University, Nanjing 210000,China

Dai Qiu-qiu is with JingLing Institute of Science and Technology,Nanjing 210001,China

Cui Ying-liu is with the Department of Computer ScienceEngineering ,NanJing Audit University, Nanjing 210000,China

Wu Hui-zhong is with the Department of Computer Science, NanjingUniversity of science and Technology, Nanjing 210014,China

paper [4], depended on the design method of the paper[2], the states observer achieved the fault insulation(FDI).But it is very difficult to search the disturbing Q matrixas in the text, especially in the actual NCS where themodel was not uncertainty. In the paper [5],[6], Kalmanfilter was introduced to get states estimation which wasapplied to states feedback control of the system. But itwill need great calculation times and has poor robust.

In this paper, the buffer is added in the NCS andrandom time-delay is changed into constant delay. Anapproach to design a robust H. states observer basedon Ho control theory on Networked Control Systems(NCS) is deduced with the usage of time-delaycompensation. We get the theorem on existing theobserver and the robust of fault detection based on H

0.

Finally, the simulation results demonstrate that themethod is effective and robust.

II. PROBLEM FORMULATIONThe Fig. 1 is the structure ofNCS, we suppose the linear

time invariant system and the random variable of thenetwork-induced delay r include sensor-to-controllerTSC and controller-to-actuator delay ?7ca.Considering the influence of the interference in real

systems, the model of the control process after dispersioncan be shown as follows [2]

{x(k + 1) = Ax(k) + Bu(k) + D1v(k)y(k) = Cx(k) + D2v(k)

(1)

Where x(t) E Rn,t) E R r y(t) E Rq are the input

and output vector of the system states, v(k) is therandom interference added outside and its norm satisfiesthe IIv(k)IL, <Vmax. A, B, C, D ,D2 are the known

Fig. 1. The structure of the network system

12370-7803-9137-3/05/$20.00 © 2005 IEEE

suitable dimension matrix.

In the Fig. 1, we suppose:

1. A is nonsingular and (A, C) can beobserved,

2. The node of the sensor is driven by time, which issampled with known fixed period T. Thecontroller and processor are driven by event.

3. There don't exist time disturbance and datapackage lost, when the signal transfers in thesystem.

4. The large time-delay value of the system isknown.

When the buffer is added into Networked ControlSystems and random time-delay is changed into constantdelay, the Fig. 1 becomes the Fig. 2. Under this conditionNCS only have output time-delay and the number of itssampling periods are known integer r .

The approach to design a robust H. states observerbased on time-delay compensation is used when the faultappears in the systems. And w(k) is considered at thesame time. Then we have the states observer function:

ikk1)AStk)+Bik)+K(C4k _r) _-CA-ri(k)r (2)

+ CAr'Ai'-B(k-i) +D2i4(k))i I

Where, Cx(k - r) + D2 w(k) = y(k) is the outputinformation at the k time arriving at the observer,x(k) is the states estimate; K is the gain matrix.

Known from the system, we have

r

x(k-r) = A-rx(k) -ZAI-'Bt(k-i) +ArD(k) (3)i1

From (2) and (3), we have the states estimate error andthe output residual equation

{e(k + 1) = (A - KCA-r)e(k) + D w(k)e(k) = He(k)

(4)

Where H E Rqxn is the known residual output weightmatrix:

Dk = D1(I - KC) + KD2

Lemma I[21 In the system (1), if (A, C) can be

observed and the matrix A is nonsingular, (A,CA r)can be observed at the random time-delay r .

III. AN APPROACH TO DESIGN A ROBUST H ,STATES OBSERVER

Define 1 States observer (2) was defined as the normalstates robust H.0 observer in the network controlsystem (1). When the system has no fault, there existsobserver gain matrix K, which makes the estimate errorequation (upper the (4)) of the states approach to stability.As to the constanty > 0, the Ho, norm of the transferfunction, which is the residual output (down the (4))£(k) relative to interference w(k), satisfies

IGr(z)II. < (5)

Where G, (z) = H(zI, - (A - KcA -r )) IDk isthe transfer function of the residual output £(k)relative to the interference w(k),

z = ejo0 E[0,2z],

111llo Jimax [( ) (.)].In order to achieve the robust HO states observer,

the solution of the observer gain matrix K and itstheorem, we firstly introduce two lemmas.

Lemma 2171 Supposing A,C,C-' +DA-'Bfinite-dimension nonsingular square matrix, we have

is

(A +BCD)-' = A -A-B(C-' +DA-'B)DA-' (6)

Lemma 3(8] (Schur mend theorem) If A,B,C arefinite-dimension constant matrix, thenA AT ,0 <B = BT, we have

A+CTB-C<0<C B] <Oo 4C A < (7)

Fig. 2. The buffer added into NCS

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Theorem 2121 For the observer equation (4), if(A,CA r) can be observed, then existing symmetry

positive-definite matrix P and a given r > 0. Then

and defining AV(k) = V(k + 1) - V(k)

We have

LP 0 lFAk Dk1[P 01Ak Dk]

L0 y2I jLH 0 J LO IjLH O J

(8)

Where Ak = A - KCA r, thus the H,, norm of the

transfer function, which is the residual output 6(k)relative to interference w(k) , satisfies the formula (6).

proof: Because the matrix inequation (8) is strict tenable,there always exists a constant a,O < a < 1, then wehave

LP 0 ]Ak Dk ][P 0A1 Dk]

L0 (1-a)y2Ij LH 0 [O IIH 0j (9)

At the same time, V(k) = eT (k)Pe(k) is a

Lyapunov function. As interference not considered,we can define the Lyapunov function

V(k) = eT (k)Pe(k) according to the output errorequation (12) of the observer, then

k = [IIe(k)112 - (1- a)y2||w(k)||2 - AV(k)] - AV(k)e(k) H(F [P kl [A, )

L[ (k)j_L[ 0J Jj (I(la) y2ILDkj)

e(k) Vk

X [ew( )-AV(k)

k k k k

i4(k)j H 0 IIH 0oj[o (1-a)y2Ij

e(k) Vk

L (k J

From the inequation (1 1), we can get

ik <-AV(k)

That is

IIe(k)|1 < (1- a)y Iw(k)|| - AV(k) (13)

AV(k) = V(k + 1) - V(k) = eT (k + 1)Pe(k + 1)- eT (k)Pe(k) = (Ake(k))T P(Ake(k) - (10)

eT (k)Pe(k) = e (k)(A PAk - P)e(k)From (8), we can obtain P > A PAk.

Then we have

AV(k) <0

That is to say the observer output approaches stability.

Considering the interference, for the residual channelse(k) -4 w(k), then

AV(k) = e (k)(AkPAk- P)e(k) +w (k)D[PAke(k)+ e (k)A[PDk (k) + wI (k)PDk (k)

(11)

To prove y2Ijw(k)jI2 is the upper boundary of

I||(k)112 , considering

Jk = klc(k)ll2 (1- a)y2 IIw(k)ll2 (12)

If we sum up the inequation (13) from k = 0 tok = n, then we define the zero initializtion condition as

n

LAV(k) = V(n + 1)k=O

At the same time we notice the w(k - r) is aboundary-function, then it follows that

n n

ZIV(k)112 < (1- a)72 ||k(k)B|2 - V(n + 1)k=O k=O

< (I _o)y2 |lw(k)||2 - V(n + 1)(14)

The above formula must be valid for arbitrary n,when n - oo, lim e(n) =0, then we have

nf->

(15)

That is

(16)

Theorem 1 indicates the existence of the robust HOstates observer. But we have to find the method and the

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JIG,-,,, (z)JI' < y' =:> JIG, (z)II. < y2

II.c(k)ll' < (I a),V 2llw(k)112 < y'llw(k)ll22 2 2

formula to solve the observer gain matrix.

Theorem 2. Supposing the given constant y > 0, whenthe following optimize problem has the best solution

*kY~A, where * is the block matrix got from the

symmetry property of the matrix.

miny (17)Ak

° -r2I * *s.t. < O (18)

PA, PDK k * *

T Ht g -Im

Then the observer gain matrix can be written as

K=(A-A*)(CA-r)- (19)

proof based on the lemma 2,inequation (18) equal to

[ [~0 -yI] [A °]T [° ]< (20)P1 [Ak H -P 0]]

Inequation (20) is a linear matrix inequation. By usingthe mincx of LMI toolkits to seek the minimum value ofr, e.g. a convex optimization problem of linear object

function, we can gain the leasty and the method to

solve the observer gain matrix under the condition of thecorresponding optimal H,.

Because A* is the best solution of the theorem 2 andthe restrict condition of the theorem 2 is the conclusionof the theorem 1, its object function is the best solutionof the performance index of the observer H., A* isthe solution when the observer (6) satisfies the best indexof the robust H.. Combining the formula (12), we have

A;=A-KCA-' (21)

From t(20) it is easy to see the gain matrix K of theshort delay NCS robust HO states observer is deducedby (2 1).

IV. FAULT DETECTION

condition (5), that is the output signal £(k) whichrelative to w(k) has robust HO norm.

When it exits fault, we may know from the aboveanalysis, residual of observer output exceeds thelimitation of A (detection threshold). Theoretically, wemay choose the fault detection threshold 2 = rIlw(k)j12,but IIw(k)112 must be confirmed concretely when the

statistic characteristics of the noise and the interferenceare known. In practice, there has some difficult to get theexplicit value. So we can select the detection threshold2l= XWmax

Therefore, the fault detection logic is

normalfault

(22)

Furthermore, combining the solution of the theorem 2,we can select the optimization threshold A* = 7 Wmto assure the robust fault detection and improve thesensitivity of the fault detection of the system, which isthe accuracy to detect the primary fault and light fault.

V. A SIMULATION EXAMPLE

Considering the following NCS

F0.5 0.011 [I

x(k + 1) = 01 -°x(k) +Iu(k) +

L L

0.1 0 p1.51

0 0.1w(k) + 08]f(k)

y(k)=L I x(k)+ [0.I 0.15]w(k)

(23)

Where

H=c;y=0.8;L=[0.4 0.3]T;U=1;Q=I2.2;r=2Thus the observer gain matrix K = [0.7252 0.6533]

can be obtained from the theorem 3. It supposes thesystem occurs a structural parameter step fault while therecurrence steps number is k = 1400 . Then the

optimize detection threshold A* =0.125. Figure 3-6indicates the simulation results driven by random noise(energy equals to 0.2, sampling time is 0.1 s). Thesimulation steps are 2000.

Observer has parameterized gain matrix (19) and makethe estimate error and detection output approach stabilitywhen there is no fault. The w(k) satisfies the

1240

11,6(k)ll, < A

> A

* ~~~~~

I-.r

11I %

0.2

0.1

0

-0. 1

02, *10.1

-0-

0 500 1000 1500 20(

Fig. 3.The state of xl and its estimate

*Sl. olm ISM X

Fig. 4. The state of x2 and its estimate

U 3:

0:2

Fig. 5.Fault-free detection residual

o i

II) I"

30

Fig. 6. Fault detection residual

Fig. 3 and Fig. 4 demonstrate the practical state traceof the system (real line) and the estimate result of theobserver (dashed) under the practical condition. We cansee that the output of the observer well traces the changeof the practical state. Fig. 5 is the detection residualoutput while the system is well. Thus the detectionresidual output of the observer is all in the range of thethreshold at the condition of fault-free. Fig. 6 indicatesthe detection residual output of the observer while thestructural parameter step fault takes place. Ifk = 1426,it will be II6(k)112 > 2 . So it achieves the faultdetection.

The effect of the network random delay is alwaysdifficult to solve in the on-line states estimate and faultdetection of the NCS. In this paper, we present the designtheory and method of the robust H. states observer

about NCS based on the Hc, control theory. Thesimulation results demonstrate that the method iseffective and robust. Comparing with the current statesestimate and the method of the fault detection, themethod in this paper has smaller calculation, strongrobust and sensitivity. The method can largely improvethe accuracy of the NCS on-line fault detection andachieve good time value.

References

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[2] YU Zhixun, JIANG Ping, CHEN Huitang, et al.Design of States Observer for Network ControlSystems with Transfer Delay[J]. Information andControl, 2000,29(2): 125-130.

[3] YU Zhixun, CHEN Huitang, Wang Yuejuan. Controlof Network System with Random CommunicationDelay and Noise Disturbance[J]. Control AndDecision, 2000,15(5):518-522

[4] ZHENG Ying, FANG Huajing, WANG Hua, et al.Observer-based FDI Design of Networked ControlSystem with Output Delay[J]. Control Theory andApplication, 2003,20(5):653-656

[5] J.Nilssion, B.Bernhardsson, B.Wittenmark. StochasticAnalysis and Control of Real-time Systems withRandom Time Delays[J]. Automatica,1998,34(1):57-64.

[6] Q., Mills, D,L. Jitter-based delay-boundary Predictionof wide-area Networks[J]. IEEE/ACM Transactionson Netwoking, 2001,9(5): 578-59

[7] ZHOU Kemin, Doyle J C, Glover K. Robust andOptimal Control [M]. Beijing, published by Nationaldefence industry, 2002, 7:23-24.

[8] Yuli. Robust Control-method to extract linear matrixinequation [M]. Beijing, published by Tsinghuauniversity press, 2002, 12:8-9.

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VI. CONCLUSION0.3

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