An Approach to Fault Detection Using Nonlinear Modelling and Estimation

Copyright © II'AC 12th Triennial World Congress . Sydney, Australia, 1993

AN APPROACH TO FAULT DETECTION USING NONLINEAR MODELLING AND ESTIMATION 11. Wang* and S. Daley** +f)eparlm£nl 0/ Paper Science, UMIST Manchesler, M60 IQD, UK ++I:nxineerinx Research Cenlre, European (;a.' Turbines LLd., Leicc.l'ler, LEII 311L, UK

Abstrad. In this paper, a novel approach to the modelling of dynamic systems is presented and its application to fault detection. diagnosis and prediction is studied. It is shown that any dynamic system can be described by two models; the first is a dynamic model for the physical parameters and the second is the ordinary system inpuHlutput dynamic model. The parameters of the laller model are often nonlinear functions of the physical parameters and therefore this can he regarded as an output equation for the first model. Using this representation it is shown how a statc estimator can he constructed to provide estimates of the physical parameters and so assess system health . More general models arc also considered. and it is shown that extended Kalman filters and artificial neural networks can he used to estimate fault location and magnitude. A simulation example is included to illustrate the proposed approach.

Key Wurd.s. Fault det eLlion and diagnosis; fault prediLlion; state estimate; Kalman filter; artificial neural networks.

I. INTRODUCTION

parameters. p . Even for identification based approaches where 8 can he estimated and tracked (Isermann, 1992, Wang and Daley, 1993), the il1'/erse of 4>(.) is difficult to calculate due to it'> complexity. Some efforts have been made to solve this problem, such as the influence matrix method (Ono, el aI, 1987) which assumes valid linearization for equation (4), but no general solution has heen found . In this paper an allempt is made to obtain a general solution to this problem by considering different dynamic models for the fault behaviour. This enables the status of the physical parameters to be estimated directly and obviates the need to calculate the inverse of 4>(.) . Moreover. in addition to fault detection and diagnosis, fault prediction is also possible. 2. MODELLING OF FAULT BEHAVIOUR

Due to an increasing demand for reliahility in engineering systems many advanced methods have been developed for fault detection and diagnosis (FDI) in recent years (Pallon el aI, 19R9). Analytical redundancy approaches have heen widely applied to systems where signal relationships can he expressed mathematically with either known or unknown parameter values. In most cases, faults cause unexpected changes in physical system parameters, p and analytical redundancy FDI methods can he descrihed as follows. Let the dynamic system model he represented hy x(k

+ I) =J(x(k),u(k),8)

y(k) = g (x(k), u(k), 8)

(I) (2)

It is assumed that the single-input single-output healthy system can he described by

wherex(k) E R" is the state, u(k) E R'" is the measured input and y(k) E RI is the measured output, 8 E Rq arc model parameters which generally depend upon the physical parameter vector p in a complex nonlinear manner, J( ... ) and g( ... ) are functions which can he linear, nonlinear, known or unknown. Then using the information provided hy equations (I) and (2), a detection signal (parity check) of the form ECk)

= 0(y(k),u(k) , 8)

(5)

where {y (k)} and {u (k)} are the measured output and input sequences, q - t is a unit backward shift operator and cl is the known pure time delay. A(q - t) and 8(q - l) are known polynomials of the following form

(3) A(qt)

= 1+ i i

is obtained. where 0( ... ) is a function which can he either linear or nonlinear. Using equation (3). fault detection can be carried out hy checking whether ECk) exceeds a preset limit.

~

a,q -'

(6)

I

(7)

However. since the model parameters 8 can he expressed as

where a, and bj form the system model parameter vector, i.e.,

(4)

(8)

where 4>(.) is generally a complex nonlinear function, difficulties can arise in diagnosing the fault. For diagnosis. not only do the location and size of changes in the model parameters. 8. need to he determined. hut also the location and size of changes in the physical

As discussed in section 1, system faults arc manifested as unexpected changes in physical parameters (Isermann. 19R4) and the purpose ofFDI is to estimate the location and size of the fault. Let the physical parameters be characterised by 527

p = [(1-!;)PI,(l- f~P2' ... ,(1- fh)Ph)

(9)

where p,(i = 1,2, ... , h) is the nominal value of ith physical parameter and /" (i = 1,2, ... , h) represents the health status of the ith physical parameter defined as

f

=

:t

(18)

~l1(k - 1)

(19)

( ) - 1 + 1I11(k - 1)11 2

where ~ is a positive number. After a fault occurs the second term of equation (11) vanishes therefore, in the main, the physical parameters of a real system are described by equation (10). If the variation of w(k) is neglected, then the state estimator, (17)-(19), can be shown to converge if the following conditions are satisfied.

0; otherwise

Without loss of generality, it is assumed throughout that only abrupt changes in parameters occur. Since the nominal values of physical parameters are constant, dynamic modelling can only be introduced for the fault behaviour or health status, f. The following principles can be used to establish representative models. Principle 1: When no fault occurs, all the physical parameters are constant and are equal to their nominal value, in this case only small variations of physical parameters will present. Therefore F(k + 1) = F(k) + w(k)

(17)

i(k) = y(k) -l1 T (k -1)cjl(F(k}) L k _

{O; when the ith physical parameter is normal}

,

F(k + 1) = F(k) + L(k)i(k)

(i) for any two vectors, f!.., pE Rh, the nonlinear function cjl(.) satisfies ~ "1011 f!.. -

11 cjl(f!..) - cjl(p)1I 0< "1111 f!.. - pll \

xh

plI

~ (cjl(f!..) - cjl(p}) (f!.. -

(20)

p/ (21)

(10)

(ii)

where F(k) = [J;(k)'/2(k), "''/h(k»)T '" 0 and w(k) is a time-varying signal of very small magnitude.

~

Principle 2: When a fault for the ith physical parameter has occurred, its value becomes a constant which is different from its nominal value, therefore, equation (10) still holds except that the ilh component of F(k) is a non-zero number. As a result, there must be an impulsive change in the mean value of w(k) at the time of the fault. Using these two principles, the dynamic model for fault behaviour can be expressed as F(k + 1) = F(k) + ao(k) + w(k)

(11 )

a= diag(al'~' ... , a h )

(12)

E

2'1,) (0, 10

(iii) 1I11(k - 1)11

(22) :t

0

FDI for system (5) can now be carried out by using this state estimator to provide F(k) and checking each element against a pre-selccted threshold. If any element exceeds the threshold, then the magnitude of the fault can be determined from

Pi = (1 -l,(k»Pi

(23)

wherej,(k) is the ith component of F(k).

4. A GENERAL MODEL AND FAULT

PREDICTION Principle 2 in section 2 isa simplification of the situation following the occurrence of a fault. In many cases, faults in one physical element will induce faults elsewhere also the magnitude of the fault may continue to grow. Therefore a more general model for fault behaviour takes the form

and ki(i = 1,2, .. . , h) is the sample time when a fault in the ith physical element occurs, 0(.) is an impulse function which is equal to 1.0 when k =ki and is equal to 0.0 otherwise, ai(i = 1, 2, ... , h) is the corresponding magnitude of the change in mean value. Both ki and a i are unknown. Denote

F(k + l) = GF(k)+aO(k)+w(k)

l1(k -1) = [y(k -1),y(k -2), .. . ,y(k -11) u(k-d) , u(k-d-l), ... ,u(k - d - m)(

where G E Rh x h is assumed to be a known matrix. The variation term w(k) is now considered to be a white noise sequence with zero mean and a known small variance matrix, Q. and another white noise sequence, v(k), can be added to the output equation (16) to represent mea<;urement noise. As a result, equation (16) becomes

(14)

then by definition (8) the system input-output equation (5) can be re-written as T (15) y(k) = e l1(k -1)

y(k)

Using definition (9), it can be seen that y(k)

=l1(k -

1/cjl(F(k»

(24)

= l1T (k -

1)cjl(F(k» + v(k)

(25)

where {v(k)} is assumed to be a white noise sequence with zero mean and a known small variance, R. Similar to section 2. model (24) is taken as a state space equation and model (25) as a nonlinear output equation. Denote

(16)

Since vectorl1(k -1) is mea<;urable, equation (ll) and equation (16) can therefore be regarded as a state space equation and a nonlinear output equation, respectively. At this stage, faults can be detected and diagnosed if an estimate of F (k) can be made via equations (11) and

y~

_I

= {y (k -

1), Y (k - 2), ... , y (O)}

(26)

Then the problem of fault detection and diagnosis is tmnsferred to the problem offinding the estimate of F(k) using the measurement Y~ _ I' In the sense of minimum variance estimation, the calculation

(\6).

3. STATE ESTIMATION Assume that the number of model parameters is the same as that of physical parameters, then for the system represented by equations (11) and (16). the following state estimator is constructed,

F(k) = E {F(k) I Yk }

528

(27)

must be made, where E {.I.} denotes conditional expectation. Similarly, the I-step ahead prediction of the fault can be achieved by evaluating i(k+llk)=E{F(k+l)IYk }

(28)

network (ANN) techniques. Define C(F(k» and $(F(k» as the approximations. then using the B-splines type of ANN (Brown and Harris, 1991), the following expressions can be obtained N

A

y(k) = r{(k -1) [9 H+ CHF(k)] + v(k) aq,(F)

CH = (

i

)T

Denote w(k) = y(k)-r{(k -1)9 11 , then C(F(k»

(31 )

L. vB .(F(k» j=1 } }

(32) T

T

y(k)

where P(O I 0) = P(O) = crI and F(O) = 0 arc initial conditions and cr is a small positive number indicating that the calculations (32)-(35) start before a fault has occurred.

G(O)

A

ad

Using (32)-(35), I-step-ahead fault prediction can be directly formulated as

<1>(0)

= r{(k N

A

(42)

1) ( $(0) + ;; IF = 0 F(k») + V(k)

(43)

(44)

_

= L.

w,B,(O)

(45)

i =I

N

aF IF =o= A

a$

(36)

_

aB i

'~I 1'.', aF

IF =o

(46)

N _

= j L.= 1 v }B/O) N

aF IF =0= j ~I

This result is important since fault prediction. rather than fault detection, means that an advance alarm can be given and early preventative action can he taken.

_

Vj

aB j aF Ip =0

(47)

(48)

can be readily calculated. Denote T aq, C(k)=ll (k-1)aF Ip =o

5. NONLINEAR MODEL AND ARTIFICIAL NEURAL NETWORKS APPLICATION

(49)

then from equation (44) the following result can be obtained

A direct extension of model (24) to an even more general representation can be made by replacing the term G F(k) with a nonlinear function. G(F(k». and the model for fault elements can thus be expressed by

w(k) = y(k) -ll T (k -1)$(0) = =C(k)F(k)+v(k)

(37)

(50)

It can be seen that equations (43) and (50) are linear, time-varying state space equations, and a similar approach to that used in section 3 can be used to construct an extended Kalman filter to generate F(k) and F(k + II k).

while the output equation y(k) = r{(k -l)(F(k» + v(k)

= (F(k»

t. aF Ip =o!,(k)+G(O)+UO(k)+w(k) (ac

F(k+1)=

(33)

P(k+1Ik)=GP(klk)G +Q (34) P(k + II k + 1) = (/ -L(k + 1)C(k + l»)P(k + 1 I kX35)

F(k + 1) = G(F(k» + uO(k) + w(k)

(41)

Since B,(.) and Bl) are differentiable, the same first order approximations in the neighbourhood of healthy physical parameters as in section 4 can be made for both C(.) and $(.). Therefore equations (37) and (38) can be approximated by

L(k+1)=P(k+1Ik)C (k+1)

F(k+llk)=G1F(k)

L. v B(F(k» j =1 } }

(F(k» =

F(k + 1) = GF(k) +L(k + 1)(w(k + 1)- C(k + 1)GF(k))

l

= G(F(k»

=1

N _

A

T

(40)

= ~ w,Bi(F(k» i

is a linear output equation. Let C (k) = r{ (k - 1)CI/. then the following extended Kalman filter (Meditch, 1969) can be constructed to generate F(k).

(C(k+1)P(k+llk)C (k+1)+Rf

(39)

W,Bi(F(k»

where W i and Vj are the weights and B,(.) and Bl) are known basis functions with their orders larger than 2. The training of the weights can be performed in different ways (Brown and Harris, 1991; Wangand Harris, 1992) and discussion of this is omitted here due to the space limitation. However, it can be assumed that wi and Vj are trained weights such that

(30)

w(k) = r{(k - l)CIIF(k) + v(k)

=I

N

A

q,(F(k» =

(29)

aF IF =o

L.

G(F(k» =

Evaluating the right-hand sides of equations (27)-(28) is difficult because of the nonlinearity in q,(.). However, under the assumption that q,(.) is linearizablc in the neighbourhood of the healthy physical parameter vector PH and that 9H = q,(PI/) is known, the nonlinear output equation (25) can be approximated by

(38)

is kept unchanged. Unlike sections 2 and 4. it is assumed that both G (.) and <1>(,) are unknown. In this cw;e, the estimation and the prediction in a minimum variance sense can still be given hy equations (27) and (28), hut to calculate F(k) and F(k + I I k), nonlinear functions. G(.) and <1>(.). need to he estimated first. If it is assumed that training patterns for physical parameter hehavour are availahle and are denoted hy F](k). then estimates of G(.) and <1>(,) can he ohtained using artificial neural

To summarise. the fault detection, diagnosis and prediction of system (37) and (38) can be carried out by first; using the available training pattern, {FT(k)}, to evaluate the weights. wi and Vj and then; the extended Kalman filter together the measured w(k) to generate F(k) and F(k + I I k). If there are components in F(k) or in F(k + 1 I k) which exceed the pre-specified threshold then faults have occurred or are going to occur. 529

It can then be seen that the fault has been successfully detected and located. The estimate of the size of the fault is given by

6. SIMULATION RESULTS The simulation is carried out for the following system y(k) = sin(3 .14PI)y(k -1) + (p;)u(k -1) + v(k)

(51)

(56)

where y(k) and u(k) are measurable output and input sequences, v(k) is white noise with zero-mean and 0.01 variance, PI and P2 are physical parameters whose abrupt changes are caused by faults in the system. The nominal values are given by

The small error between and /2 is due to the accuracy of the lincarization discussed in section 3. Nevertheless, direct estimation of the location and size of the fault is obtained without solving the inverse of the nonlinear function q, = (sin(3.14pl) p;+0.2(

PI

J2

(52)

= 1.5P2 = 0.56

7. CONCLUDING REMARKS

It ha'> been shown in this paper that fault detection, diagnosis and prediction can be simultaneously performed for dynamic systems if a representative model for fault behaviour can be established. This model can be taken as a state space equation and the input/output model of the systems can be regarded as a nonlinear output equation. A new state estimator has been constructed for deterministic systems and its convergence ha'> been proved under certain conditions. More general models. both linear and nonlinear, are also proposed for fault behaviour and the extended Kalman filter is shown to be capable of both generating the fault behaviour estimates and predicting faults . For the nonlinear model, artificial neural networks are applied to approximate the fault behavour before the extended Kalman filter is used. A simulation example has demonstrated the potential of the approach.

and the fault vector is defined as F(k)=

cri /l

(53)

which satisfies F(k) = F(k -1) + w(k)

(54)

where w(k) is white noise with zero-mean and 0.001 variance. Using the method discussed in section 3, a simulation is carried out with 0" = 1.5 and u(k)

= 0.01 sin(k)+ 1.0

(55)

A fault is created by changing the value of P2 to 0.392 when the sample number reaches 200, this is equi valent to a change in the value of /2 from zero to 0.3. The estimation results arc given in Fig. 1 and the corresponding residual signal is shown in Fig. 2.

8. REFERENCES Brown. M. and C.l. Harris, (1991) , "A nonlinear adaptive controller: a comparision between fuzzy logic and neuracontrol". IMA . 1. Maths . COlltr, ltif. , Vol. 8 , pp.239-266.

O. ~

12

Freyermuth. B. and R. Isermann. (1991) , "Model based incipient fault diagnosis of industrial robots via parameter estimation and feature classification", Proc. of. Europeall COlltrol COIl/ercll ce. pp.1918-1922 .

0.2 ~

i

-3

0.15

.! ."

.!!

~

01

Isermann , R., (1984). "Process fault detection based on modeling and estimation methods--a survey", Awomatim. Vo1.20. pp.387-404.

01 0.05

fI

o ... r . ·0.05

o

"J'

50

'. '"

Meditch. l. S .. (1969) . "Stochastic Optimal Linear Estimlltioll alld COlltrol", McGraw-Hill.

. 100

150

200

250

300

350

400

450

500

Ono.T .. T. Kumamaru. A. Maeda. S. Sagara, K. Kumamaru . (1987) , "Influence matrix approach to fault diagnosis of para meters in dynamic systems" , IEEE tnws . lndustrial Electron, VoI.IE-34, pp. 285-291.

Sample Number

Fig. I. Behaviour of estimated fault vector 0. 05.--~-~-~~-~-~~-~-~--,

Patton. R.1 .. P.M. Frank and R.N Clark (1989). "Fault diaRnosis in DYllamic Systems : Theory and Applicatioll". Prentice Hall International. Wang. H. and S. Dalcy, (1993), "Fault detection for unknown systems with unknown inputs and its application to hydraulic turbine monitoring", Int. 1. Control, Vo1.57. pp.247-260

~ · 0 05

...

in

] j

Wang, H. and C. l . Harris, (1992), "Modelling and control of a family of nonlinear systems via neural networks", Rcport 'I NO IIn , University of Southampton .

-0.1

·0.1 5

·0.2

o

..

~~-~- . - ~--

50

100

150

200

250

300

350

400

,150

500

Sample Number

Fig. 2. Behaviour of residual

530