Proceedings of the American Control Conference Philadelphia, Pennsylvania June 1998

Fault detection for time-delay systems: a ]parity space approach F. KRATZ, W. NUNINGER, S. PLOIX Centre de Recherche en Automatique de Nancy - CNRS UPRES A 7039 Institut National Polytechnique de Lorraine 2, avenue de la Fort de Haye - 54516 Vandoeuvre Cedex - FRANCE Tel: 33-(0)3-83-59-56-35 Fax: 33-(0)3-83-59-56-44 . , E-mail: {fkratz, wnuninge, sp1oix)Qensem.u-nancy.fr Abstract- In this paper, it is shown how a residual generator can be synthesized for time-delay systems using parity relations. The goal of this study is to investigate the application of a modern symbolic computation like MAPLE to the development and the design of fault detection systems for time-delay systems. In the literature, the problem brougth up by time-delay systems has been solved in two different ways at least: first; retarded functional differential equations based upon the Razumikhin theorem [l];well the secund method is based upon the use of the delay operator V and the properties of the ring R[v] [2]. Taking into account these properties, a FDI scheme based on the parity space is proposed. Finally, the applicability of the presented scheme is illustrated by a numerical example. Keywordstems.

failure detection, parity space, time-delay sys-

in [12] for linear time invariant systems by using symbolic computation programs. A simple and systematic design method is presented. In addition, a numerical example is given to illustrate the design method. 11. MATHEMATICAL PRELIMINARIES A. A ring model for tame-delays systema Consider a linear discrete time-delay system described by: z(kf1) = CA;z(k-i)+ i=O

y(k) I. INTRODUCTION

2B j u ( k - j ) j=O

(1)

= C Z ( k )+ d ( k )

where x(k+l) E R" and x(k-i) E R" are the state vecN the last two decades, a large variety of methods have tors, u(k) E RP and u(k-j) E RP are thle control vectors, been proposed to solve the failure detection and isola- y(k) E Rm is the output vector, d(k) E: Rm is the meation (FDI) problem ([3], [4], [5] and [SI). In one way or surement noise vector (zero-mean white noise process with another, the procedure of FDI essentially consists in two constant covariance matrix V). Matrices; Ai (i=O, . . . , v), stages. The first stage is the generation of signals, called Bj (j=O, . . . , s) and C are constant with appropriate diresiduals, which values are nominally zero or close to zero mensions. Integers v>O and s>O respectively denote the when no failure occurs and differ significantly from zero number of time delays in the state and control vectors. otherwise. The second stage consists in the residuals eval- Using the delay operator v defined by \7f(k) = f(k-1) (for any continuous function f ) , system (1) cim be modelled by uation to make the appropriate decisions. Different methods to generate residuals have been pro- the equation: posed ([7], [8] and [9]). These methods use the direct relationships between the inputs and the outputs of the system z(k+1) = A(v)z(k)+B(v)u(k) (2) (transfer function approach) [lo], the reconstruction error or the redundancy relations generated as a linear combination of the system inputfoutput data over a finite horizon As only commensurate delays are comidered, A(v)and (parity space technique) ([ll], [12] and [13]). B ( v ) are matrices over R[v] (the ring of polynomials in In general, most of these studies concentrate on linear v with coefficients in R) and can be developed as: systems [14], [15], and some others also consider bilinear and nonlinear systems. Only few papers deal with the FDI problem in time-delay systems [16] and [17]. The FDI problem of linear systems with time delays has become of great interest in recent years, time delays are inherent in many Note that, any matrix M(.) over a ring R[.] can be real physical systems (i.e. mechanical and chemical pro- decomposed as: cesses, power and water distribution networks, air pollution systems etc.). This paper is concerned with the FDI problem for M (.) = MO +MI (.) + . . . + A,fK ( . ) K discrete-time systems with delays in both the state and where K E N and Mi (i=O, . . . , K) are real matrices. control vectors. The purpose is to extend the result given 2009 0-7803-4530-4/98 $10.00 0 1998 AACC

I

In the sequel the following notations are used. First

R ( v ) stands for the field of rational functions with

CO-

where the matrix W is chosen such that the residual r(k) is independent of the system operating state x(k-L), thus

efficients in R, Rp"q(v)for the field of rational matrices of dimension pxq, and R"[v]for the module of vectors with n polynomial coordinates.

B. Observability condition of s y s t e m (1) The system (2) is Observable (i.e. Observable Over the field R(v))if the associated observability matrix satisfies [HI:

(4)

W G ( v )= 0

(8)

An appropriate value for s can be found by the designer bv a systematic increase of L. Of Darticular interest the parity relations for which ones the length (L+1) of the data window is minimal (i.e. L is the observability indice). Now, using the definition of W, it is clear that r(k) is zero mean when the system is working correctly. On the contrary in the presence of a fault in the system this residual may becomes non-zero.

IV. ILLUSTRATIVE EXAMPLE Consider a time delay system of the form (l),

[

111. PARITY SPACE RESIDUAL GENERATION FOR FAULT

Ao=

DETECTION

The task for FDI is t o design a residual signal which is zero in the fault free case and non-zero when a fault occurs in the monitored system. The redundancy is constructed by collecting sensor outputs over a time interval. The combination of eq. (2) from time k-L to time k yields to the following scheme (no fault case):

where:

AB=

[

0.607 0.239 0 0 0.368 0 0 0 0.5

0 -:.5

0 yl

According t o [12], a residual signal (parity signal) can be defined as:

?-

(IC) = W [Y ( I C )

-

(v)U @)I

[E];

C=13

,so

2w

250

3M

150

ua

a0

JM

ua

7 -

O 5

(6)

1.20

50

0

Equation (5) can be simplified as:

(0) U (IC) + D ( k )

0 0 1 ; B= 0

0.25 0 -0.2

The approach presented in section 2 can be used to transform the above system into the form (2). Then the projection matrix W (8) and an associated residual (7) can be designed so that the faults can be detected. The computations in equation (8) t o provide a parity residual (7) are carried out using Maple VR4 (see appendix). The result leads t o a residual vector with 6 degree of freedom, represented by -t[i] in the result. To investigate the applicability of the approach it is assumed that the output measurements are corrupted by random noise. In addition, 2 biases failure are simulated: one of magnitude 10 occuring in sensor 3 from the time 101 t o 150 and the other one of magnitude 1 occuring in actuator from the time 201 t o 250.

-yl

y (IC) = G (0) (IC - L ) +

1

; Ai =A2 = O

~ 0

' IO

"

Im

' 150

"

mo

' 2M

Fig. 1. Measurements and control variable.

Figures 1 and 2 give the simulation results without faults. Figures 3 and 4 give the simulation results with sensor and actuator failures. The close and accurate detection for the ( 7 ) sensor and actuator faults is obvious.

2010

Fig. 2. Residuals.

Fig. 3. Measurements and control variable.

V. CONCLUSIONS A simple approach of parity space residual design for time delay systems has been presented. First, an algebraic approach is used t o model time-delay systems with point delays. Moreover it is shown that realistic problems can be quickly and easily solved by using computer algebra. The proposed method was illustrated by designing a parity residual for actuator and sensor failures. Simulation results demonstrated the usefulness of such FDI scheme. Of course, for more complex systems, further studies in this direction are required.

REFERENCES [l] Hale J., Theory of Functional Differential Equations, SpringerVerlag, 1977.

Sontag E.D., Linear systems over commutative rings: A survey, Ricerche Automat., (7), pp. 1-34, 1976. Basseville, M., Detecting changes in signals and systems - a survey, Automatica, (24), pp. 309-324, 1988. Frank, P.M., Fault diagnosis in dynamic syst,ems using analytical and knowledge-based redundancy - a survey and some new results, Automatica, (26) 6, pp. 459-474, 1990. Gertler, J., Analytical redundancy in fault detection and isolation, proc. of the IFAC/IMACS symp. Safeprocesis, pp. 9-22, 1991. Ragot, J. and Maquin, D., Failure detection, identification and accommodation based on a new accommodation filter, Int. J. Systems Sciences, (24) 6, pp. 1215-1220, 19913. Willsky, A.S., A survey of design methods for failure detection in dynamic systems, Automatica, (12), pp. 606-611, 1976. Patton, R.J., Frank, P.M. and Clark, R.N., (Eds), Fault diagnosis in dynamic systems, Prentice Hall, 1989. Kratz, F., Maquin, D. and Ragot, J., Robusst generation of analytical redundancy equations and applications to diagnosis, proc. of the 31st IEEE CDC, DD. 2838-2841. 1992. [lo] Ding, X. and Frank, P:M., Fault detection via factorization approach, Syst. Control Lett., (14), pp. 431-136, 1990. [11] Potter, D. and Suman, M.C., Thresholdless I-edundancymanagement with arrays of skewed instruments, Electronic flxght control systems, Agardo-graph 224, pp. 2115-2121, 1977. [12] Chow, E.Y. and Willsky, A.S., Analytical redundancy and the design of robust failure detection systems, LEEE Tkans. on Aut. Contr., (29) 7, pp. 603-614, 1984. [13] Kratz, F., Bousghiri, S. and Mourot, G., A finite memory observer approach to the design of fault detection algorithms, proc. of the 1994 ACC, p. 3574-3576, 1994. [14] Frank, P.M., Enhancement of robustness in observed-based fault detection, Int. J . of Control, (59) 4, pp. 955-981, 1994 [15] Saif, M. and Guan, Y., A new approach to robust fault detection and identification, IEEE Bans. on Aerospace and Electronic Systems., (29) 3, pp. 685-695, 1993. [16] Yang, H. and Saif, M., Observer design anf fault diagnosis for retarded dynamical systems, proc. of the 35th IEEE CDC, p. 1149-1154, 1996. [17] Trinh, H. and Aldeen, M., A memoryless reduced-order state observer for discrete-time systems with multiple delays, proc. of the 1997 ECC, TU-A F5 (286), 1997. [18] Sename, 0. and Lafay, J.F., coefficient assignement for timedelays systems, proc. of the 1997 ECC, WE-M D2 (75), 1997.

APPENDIX

> with(lina1g): Warning, new definition for norm Warning, new definition for trace

> Al:=matrix(3,3,[0.607,0.239,0,0,0.368,0,0,0,0.5]): > A2:=matrix(3,3,[0,0.25,0,-0.5,0,0,0,-0.2,0]): > A:=matadd( A1 ,A2,1,zd) : > C:=matrix(3,3, [I,O,O,O, 1,O,O,O,l]): > G :=st ack(C ,multiply (C,A) ,multiply( C,multiply(A,A) ) ) : > zer :=vector (3,O): w:=linsolve(transpose(G) ,zer) : > B :=matrix( 3,1, [1,O,O]) : > H:=stack(augment (zer,zer,zer),augment (multiply( C ,B). zer,zer) ,augment(multiply(C,multiply(A,13)),multiply(C,B),:

> YL:=matrix(9,l,[ylk2,y2k2,y3~2,ylld,y2ld,y3ld,~

Y2-hY3-kI): > UL: =matrix( 3,1,[uk2,ukl,uk]): > P :=matadd(YL,multiply (H,UL) ,1,-1): > R:=multiply (w,P); R := [(-.607 -t[l] + .5 zd-t[2] (-.368449 .1195zd .125(~~~)) -t[4] - .1 (z") -t[6] + .4875 zd-t[5])ylk2 ((-.239 - .25zd)4[1] - .368-t[2] + .2zd-t[3] (-.233025 .24375zd)-t[4] (.1195zd .125(zZd)- .135424) A[5] .1736 zd-t[6])y2k2 (-.5-t[3] - .25-t[6])513A2 -t[l](ylAl - uk2) -t[2]y2Al + -t[3]y3Al

+

+

Fig. - 4. Residuals.

+ +

+

+ + +

+

+ +

+

-t[4](yl-S-.607uk2-uS1)+Jt[5](y2-lc+.5~~Ulc2)+-t[6]y3-S] 201 1