Intelligent Fault Diagnosis Based on Neurofuzzy Networks for Nonlinear Dynamic Systems

Copyright ~ 2001 IFAC IF AC Conference on New Technologies for Computer Control \9-22 November 2001 , Hong Kong

INTELLIGENT FAULT DIAGNOSIS BASED ON NEUROFUZZY NEWORKS FOR NONLINEAR DYNAMIC SYSTEMS

Wang Y., Chan C.W. and Cheung K.C.

Department of Mechanical Engineering The University of Hong Kong Pokfulam Road, Hong Kong, China Email: [email protected] Fax: (852)28585415

Abstract: In this paper, a new fault diagnosis technique based on B-spline neurofuzzy networks is proposed for nonlinear dynamical systems. The main reasons for using neurofuzzy networks here are that they can approximate non linear functions with arbitrary accuracy, and can be readily trained online. In the proposed scheme, the fault is approximated by the output of a neurofuzzy network, which is to be trained online, and the fault is isolated from the fuzzy rules extracted from the neurofuzzy network. However, as the number of fuzzy rules generated by neurofuzzy networks can be quite large, it is necessary to reduce these rules, which can be a difficult process. A simple approach proposed here is to delete fuzzy rules that have too small a confidence level. It is shown in the simulation example involving a nonlinear single-link robot arm that the proposed procedure can detect and isolate the fault. Keywords: fault diagnosis, neurofuzzy networks, nonlinear observer, online training

1.

INTRODUCTION

Fault diagnosis for nonlinear systems has attracted a lot of attentions recently, since most practical systems are complex and nonlinear. A number of approaches have been proposed (see e.g., the survey papers, Frank (1990), and Isermann (1993), and the book (Chen and Patton, 1999». In comparison with linear systems, fault diagnosis for non linear systems is much more difficult, and a general fault diagnosis scheme is difficult, if not impossible, to derive. Fault diagnosis for a class of nonlinear dynamical systems with the fault a

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nonlinear function of the state and the input of the system is considered in Chan, et al. (1999). It was proposed that the fault is estimated on-line using a neural network, and fault is diagnosed from its estimate. The result in Chan, et al. (1999) is extended to systems with model uncertainties in Wang, et al. (1999). Two groups of neural networks are used, one of which is to approximate the model uncertainties, and the other, to estimate the fault on-line. The main emphasis of most existing fault diagnosis schemes is mainly on fault detection

rather than fault isolation. In this paper, an intelligent fault isolation scheme using B-spline neurofuzzy networks is proposed based on the fault diagnosis schemes proposed in Chan, et al. (1999), and Wang, et al. (1999). As neurofuzzy networks incorporate the qualitative reasoning via fuzzy logic, and the quantitative learning from training data via neural networks, it combines intelligent qualitative and quantitative reasoning (IQ2) within a single framework (Harris et al., 1999). Zhang and Morris (1994) proposed a neurofuzzy network for on-line process fault diagnosis by inserting a fuzzification layer to a conventional feed-forward neural network. In the fault classifier proposed by Ayoubi (1995), a neurofuzzy network is used to model the system, and an initial knowledge base was extracted from a set of training data for fault diagnosis. Benkhedda and Patton (1996) used B-spline neural networks to generate residuals for fault diagnosis, and fuzzy rules are also used to interpret the results obtained from fault diagnosis. In this paper, the fault is estimated online first by a neurofuzzy network, and fuzzy rules are then extracted from the network to isolate the fault. The paper is organized as follows. A brief review of the online fault estimation scheme proposed in Chan et al. (1999) is presented in Section 2. The procedure to extract fuzzy rules from B-spline neurofuzzy networks is described in Section 3, and is illustrated by a simulation example involving a nonlinear single-link robot arm is given in Section 4.

2.

fault is much more difficult to estimate, as x(k) is unknown. If there is no fault, ifJ in (1) becomes zero. The fault diagnosis scheme proposed in (Chan, et ai, 1999, Wang, et ai, 1999) is extended here for discrete systems. Let ifJ(x,u) =

[~(x,u) ~(x,u)

... ifJp(x,u)Y

and the estimate of ifJ(.) is denoted by

~(x, u) = ~(x,u) ~(x,u)

...

~p(x,u)r

where " N' denotes the estimate of (i...). The nonlinear observer for system (1) is x(k + 1)

= AX(k) + a(y(k),u(k»

+ B(~(k) - 5(k»

+ K(y(k) - y(k»

(2)

YCk) = Cx(k)

where Ker is the observer gain matrix, which is chosen such that A - KC is stable. The robust vector 8.,.k) is for compensating the estimation error ifJ(x,u)-~(x,u), which will be defined later. Let the nonlinear fault ifJ,(x,u) be modeled by the neurofuzzy network, ~;(k) = W/ f..i;(x(k),u(k» + f.;(k)

(3)

where f..i; is the reception field function of the /h neurofuzzy network with appropriate dimension, and W; is the weight of the network. It is assumed that the optimal weights are located in a convex region, i.e., IIW;IIF 5, W;M , where W;M are preselected. As Wi is unknown, its estimate, W; , is obtained from the training of the network. The

ON-LINE FAULT ESTIMATION

estimate of the fault, ~(x,u) , is given by

Consider the following nonlinear dynamic system,

~;(k) = W/ (k)f..i;(x(k),u(k»

x(k +1) =M..k) +c4y(k),li,k» + qi.,k) + B(i..x(k».zi..k» (1) }(k)=Cx(k)

where xeK'. yeJi and xeEr' are respectively the state, the output and the input of the system. Assume that the measurement of u(k), y(k) are available, but not that of x(k). The system matrix AeJrX", the output matrix Ce1F' and the fault matrix BeJ("-P are assumed known. The known nonlinear term, a : Ji x Er' ~ K', the unknown model uncertainties rp : K ~ K', and the unknown fault ifJ : K' x Er' R+ ~ Jr are assumed to be smooth nonlinear functions.

(4)

From (1), (2), (3) and (4), the dynamical equation for the error is x(k + 1) = Acx(k) + rp(k) + B(i(k) + o(k»

(5)

y(k) = CX(k)

where

W; =W; -W;, x = x-x , y = y- y , and

~(k)-~(k) =w{f..i;(x(k),zi..k»)-W{(k)f..i,{X{k), zi..k) )+q(k) =w{f..i,{x(k),u(k»)-w{f..i,{X{k),zi..k» +w{f..i,{X{k),zi..k)-W{(k)f..i,{X{k),zi..k»+q(k) =W{f..i,{X{k),u(k»+ti{(k)+'1(k)

and Though only the case that a is a function of u(k) and y(k) is considered in this paper, the method can be readily extended to the case that a is a function of x(k) and u(k). Note that (i...) is a function of the input u(k) and the state x(k), the

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tiT;(k) = W;T f..i;(x(k),u(k»-W/ f..i; (x(k), u(k» .

The online training of the network (4) is shown schematically in Fig. 1.

u(k)

to make. Under certain conditions, the qualitative description of the fault can be obtained from the fuzzy rules extracted from the B-spline neurofuzzy network. Consider, for example, the

I--_ _ _ _...... y(k)

membership functions of the fuzzy sets of t/J(k) are chosen to be triangular B-spline functions, as shown in Fig. 2. Then the weight Wj is given by (Brown and Hams, 1994),

1(k)

F.uh Decision Logic

q

Wo I

Fig. I Online training of the neurofuzzy network approximating the fault

i = "cicr ~ I C

(9)

i=\

where

c/

is defined as the confidence of the /h

aJ

An on-line trammg algorithm for W; of the network (4) is given below (Wang, 2001).

fuzzy set, is the center of the /h B-spline fuzzy set, and q is the number of fuzzy sets. In

MV;(k) =r[J{(X(k),z.(k»Y;(k +1)-BfV;(k)] i =1, .. ·,p(6)

Fig. 2, q = 5, and c;

8;(k)

= -7]; · sign(y;(k+l»

i

= 1, .. . ,p

y. Note that y = MY, where

cl Cl

0

Or. Note as

where p; is the B-spline basis function. The mapping from the confidence level of the fuzzy sets to the weights and the inverse mapping from the weight to confidence level of the fuzzy sets are given respectively by (9) and (10). Hence for a given weight, the confidence of the fuzzy sets, Cj

M is chosen

can be evaluated from (10). For example, t/J(k) shown in Fig. 2 can be interpreted as follows: (1) If x(k) is B{ and u(k) is B~, then ~(k) B~ (Rule confidence

=

IS

cl)

(2) If x(k) is B{ and u(k) is B~, then ~(k) is B~+\ (Rule confidence =

With certain conditions, fuzzy rules can be extracted from the B-spline neural networks. This is a useful feature in isolating faults (Harris, et aI., 1999), as discussed in the following section.

cl)

As t/J(k) is an estimate of the fault, a qualitative knowledge of the fault can be obtained from these fuzzy rules. The isolation of fault using this approach depends on that t/J(k) is a sufficiently good approximation of the fault. This approach is used to isolate the fault in a nonlinear single-link robot arm, as presented in the next section.

FUZZY RULES EXTRACTIONS WITH NEUROFUZZY NETWORK

As discussed in Section 2, the fault
Wj,

(10)

satisfying BTp = MC, and P is a positive definite matrix given by (A - Kcl P(A - KC) - P < O. It is shown in Wang (2001) that the training algorithm given by (6) and (7) is stable, and the estimated errors, though bounded from above, do not necessarily approach zero asymptotically. Further, the upper bound of the estimated errors depends on K. Consequently, the upper bound of the estimate errors can be reduced further by a suitable choice of K.

3.

r

that c( can also be expressed in terms of follows,

(7)

where sign(.) is the sign function, 7]j is the bounded value of (iJj, r j , the size of the training step, is a matrix chosen to ensure the training of the network is sufficiently fast, Yj is the i'h entry of

=

(8)

Though the fault can be detected from ~(k) given by (8), isolating the fault from t/J(k) is not straightforward. If a-priori knowledge of the system is available, then it is possible to isolate the fault, otherwise further assumptions may have

Output universe i(k) Fig. 2 Confidence of fuzzy sets for weight Wi

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4.

nine weights, {Wlo ... W9}, are obtained as given below.

SIMULATION EXAMPLE

Consider a single-link robot arm given by the following equation (Marino and Tomei, 1995), ..

W=[0..1834 5.3359 2.1448 -4.3694 0..6949

3.3201 - 1.1928 - 4.9298 - 0..6431 f

1

Mj3+-mglsinj3=u; y=j3 (11) 2 where 13 is the angle, u, the input torque, M, the moments of inertia, g, the gravitation acceleration and m and I are respectively the mass and the length of the link. Let

Memberships of the fuzzy sets for u(k) and xI (k) are given by B-spline functions, as shown in Fig. 4. Fuzzy sets for the output rjJ(k) are chosen to be ''Negative (N)", "Positive (P)", "Big (B)" and "Small (S)" respectively. Fuzzy rules are extracted as follows . For example, for the 7th weight, W7 = -1.1928, and the confidence level

m = 1 kg, 1= 101, M = 0.5kg and g = 9 .8m1s 2 and

the state be defmed as, x = [XI x 2 f = [13 then the state space mode of the system is

jJf,

of

[;:J y =

=[~

~

I;:J +[~]

(12)

Assuming that the model uncertainties and the noise are bounded by [0.01 0.02f, and the input is given by u = sin2t + cos5t. Consider a fault occurring at t = 5s that reduces M by 40%, which arises from a sudden loss of the load. Then M is given by

M- {

Mo 0.6Mo

t

< 5s

2

x2 (k)

(13)

j(k)=rf XI (k)]

1X2(k)

where rjJ(k) is the on-line estimate of the fault obtained from the neurofuzzy network (8), and

A=

[6

0.~05l

is chosen to be

B = [0.g05

J.

are

c~ = 0.8012

and

** If xI(k) is NB and u(k) is NB, then the fault is S (Rule confidence is 0.9694)

(2)

* If xI(k) is NB and u(k) is NB, then the fault is PB (Rule confidence is 0.0306)

(3)

* If xI(k)

is NB and u(k) is S, then the fault is S (Rule confidence is 0.1107)

(4)

If XI (k) is NB and u( k) is S, then the fault is PB (Rule confidence is 0.8893)

(5)

**

(6)

* If xI(k) is NB and u(k) is PB, then the fault is PB (Rule confidence is 0.3575)

(7)

* If xI(k)

is S and u(k) is NB, then the fault is S (Rule confidence is 0.2718)

(8)

If xI(k) is Sand u(k) is NB, then the fault is NB (Rule confidence is 0.7282)

(9)

** If xI(k) is Sand u(k) is S, then the fault is S (Rule confidence is 0.8842)

(10)

* If xI(k)

is S and u(k) is S, then the fault is PB (Rule confidence is 0.1158)

(11)

**

M

+K6(k)-j(k))

sets

(1)

t > 5s

where Mo denotes the value of M under "healthy" condition. The discrete state-space equation of the robot is as follows, ~I (k+ 1)]={~1 (k)] + Bu(k)-o.5mgsinxl (k) + B#...k)

[x (k+l)

fuzzy

d = 0.1988 respectively. Considering the combination of the fuzzy rules, 18 rules are obtained, as presented below.

u - 0.5:;1 sin XI

[1 0{;:J

the

C = [1 0] and K

[fi~]'

As shown in Fig. 3, the estimate of the fault by the B-spline neurofuzzy network is reasonably accurate. Assuming the location of the fault is not known, and a qualitative knowledge of the fault is to be obtained from the fuzzy rules extracted from the neurofuzzy network, as discussed in the previous section. Suppose the fuzzy rules are extracted from the network at t = 15s. A total of

If xI(k) is NB and u(k) is PB, then the fault is S (Rule confidence is 0.6425)

If xI(k) is Sand u(k) is PB, then the fault is S (Rule confidence is 0.4466)

(12) If xI(k) is S and u(k) is PB, then the fault is PB (Rule confidence is 0.5534)

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**

(13)

If XI (k) is PB and u(k) is NB, then the fault is S (Rule confidence is 0.8012)

(14)

* If

xI(k) is PB and u(k) is NB, then the fault is NB (Rule confidence is 0.1988)

network. Although only a relatively simple nonlinear system involving a single-link robot arm is considered in the simulation example, it is demonstrated that the proposed procedure is able to detect and isolate the fault in the system. The proposed technique will form a basis for further development in fault isolation by extracting fuzzy rules from neurofuzzy networks.

(15) * If xI(k) is PB and u(k) is S, then the fault is S (Rule confidence is 0.1784) (16) If xI(k) is PB and u(k) is S, then the fault is NB (Rule confidence is 0.8216) (17) ** If xI(k) is PB and u(k) is PB, then the fault is S (Rule confidence is 0.8928) (18) * If xI(k) is PB and u(k) is PB, then the fault is NB (Rule confidence is 0.1072)

REFERENCES Now the fuzzy rules are reduced by deleting rules that have a small confidence level, as marked by "*" . For the ten rules remaining, the rules marked by "**,, are also deleted, since only faults that are either positive big or negative big are considered. After deleting these rules, only four rules remain.

Ayoubi, M. (1997). Neuro-fuzzy systems for diagnosis, Fuzzy sets and systems, 89, pp. 289307. Benkhedda H. and Patton, R. 1. (1996). Fault diagnosis using quantitative knowledge integration, Proceedings of UKA cc Int. Con! on Control, pp. 849-854. Brown, M. and Harris, e. J. (1994). Neurofuzzy adaptive modelling and control, Prentice Hall. Frank, P.M. (1990). Fault diagnosis in dynamical systems using analytical and knowledge based redundancy - a survey and new results, Automatica, 26, 459-474. Harris, CJ., Wu, Z.Q. and Gan, Q. (1999). Neurofuzzy state estimators and their applications, IFAC Annual Reviews in Control, 23, pp. 149-158. Isermann, R. (1993). Fault diagnosis of machines via parameter estimation and knowledge processing - tutorial paper, Automatica, 29(4), pp. 815-836. Chan, C.W., Cheung, K.C., Wang, Y and Chan, W.C. (1999). On-line fault detection and isolation of nonlinear Systems, Proceedings of ACC, San Diego, CA, pp. 3980-3984. Chen, J. and Patton, R.J. (1999). Robust Modelbased Fault Diagnosis for Dynamical Systems, Kluwer Academic Publishers, USA. Wang, Y (2001). On-line fault diagnosis of nonlinear dynamical systems using recurrent neural networks, Chapter, 5, Ph.D. thesis, The University of Hong Kong. Wang, Y , Chan, e.W., Cheung, K.C. and Chan, W.e. (1999). Fault estimation for a class of nonlinear dynamical systems, Proceedings of 38'h CDC, Phoenix, AZ, USA, pp. 3128-3129. Wang Y , Cheung, K.C. and Chan, e.W. (2000). On-line fault estimation for discrete dynamical with unknown nonlinearities. Proc. UKACCCONTROL 2000, Cambridge, UK. Zhang, J. and Morris, AJ. (1994). On-line process fault diagnosis using fuzzy neural networks, lEE Intelligent Sys. Engineering, 3(1), pp. 3747.

(i) If xI(k) is NB and u(k) is S, then the fault is PB (Rule confidence is 0.8893) (ii) If xI(k) is S and u(k) is NB, then the fault is NB (Rule confidence is 0.7282) (iii) If xI(k) is Sand u(k) is PB, then the fault is PB (Rule confidence is 0.5534) (iv) If XI(k) is PB and u(k) is S, then the fault is NB (Rule confidence is 0.8216) From rules (ii) and (iii), it is concluded that when u(k) is positive big (or negative big), the fault is also positive big (or negative big). From rules (i) and (iv), when XI (k) is positive big (or negative big), the fault is also negative big (or positive big), giving a direct relationship between u(k) and the fault, but an inverse relationship between xI(k) and the fault. From (13), xI(k) varies between -

7!

2

and

7!

2

for the given input u,

implying that sin(x) is directly related to x. Consequently, the fault can only occur in M or in u, as the chance of a fault in m or I is very small, and also that sensor fault is not being considered here. Consequently, it is concluded that the fault occurs in M, and it is a decrease in M .

5.

CONCLUSIONS

In this paper, an intelligent fault diagnosis scheme based on B-spline neural networks is proposed for nonlinear dynamical systems. The proposed scheme involves estimating first the fault using neurofuzzy networks, from which fault is detected. The isolation of the fault is achieved by interpreting fuzzy rules extracted from the

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6.---------------.----------------.---------------. Solid line: fault function Dashed line: output of the neurofuzzy network

,

\,

,,

/ ,, I \

,

I

I

:, I,

,, ,I, ,, I, ,,

, ,,,

,,

,,, ,

,

,,

\

\

I

\,

\

-4

-6~--------------~--------------~--------------~

o

5

Time (s)

10

15

Fig. 3 Fault function and its on-line estimate by the B-spline neural network

Fuzzy sets fer

o Fuzzy sets fer u(k)

Fig. 4 Fuzzy interpretation for the fault diagnosis scheme

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«.k)