Fault Detection with Sensor Errors

Copyright «:> 2001 IFAC IFAC Conference on New Technologies for Computer Control 19-22 November 2001, Hong Kong

FAULT DETECTION WITH SENSOR ERRORS

Min You

Hong Zhuan Qiu

Hong Yue Zhang

Beijing University ofAeronautics and Astronautics, Beijing 100083, China Email: hQ)'zhanf'1j)DuhUc3 bta net cn

Abstract: A scheme for fault detection with sensor errors is discussed. Sensor output generally contains scale factor error and misalignment error as well as constant bias. In this paper the neural network is applied to compensate the residuals calculated by optimal parity vector method in order to remove the effect of sensor errors. And then compensated residuals are used to detect and isolate fault by analytic method and neural network. Thereby the accuracy of fault detection and isolation is improved. Copyright'O 2001 IFAC

Keywords: fault detection, fault isolation, neural network, sensor systems, sensitivity functions

Consider redundant measurement systems:

I. INTRODUCTION The reliability of inertial navigation system (INS) is most important as a reference system in integrated navigation system. Usually it can be increased by using the redundant and fault-tolerant techniques. Now there are many methods, for example the Generalized Likelihood Test (GLT) and Optimal Parity Vector (OPT), can be used to detect and isolate failure in redundant sensor system. lin and Zhang (1999a, b) presented optimal parity vector without considering sensor errors. But in fact sensor error is affinned to be existing which can restrict the ability of fault detection and isolation. In this paper neural network is applied to compensate the residuals calculated by OPT in order to remove the effect of sensor errors, and then compensated residuals are used to detect and isolate failure by analytic method and neural network. Thereby the accuracy of fault detection and isolation is improved.

y = Hx + g i ¥ i +

I. g ¥ + E j

j

E

(1)

j~i

Where x E 9\ n is the state vector. H E 9\ mxn is the geometry matrix of sensor configuration. yE 9\m is the m dimensional measurement vector. £ is measurement noise which is assumed to be Gaussian white noise with zero mean and covariance matrix :~b(J2I.

¥= (¥P'''' ¥m)'

is the sensor failure

vector. G =(gp· .. ,gm) is failure input matrix and

gi is input vector of i th sensor. E is noise input matrix. A new performance criterion for robust failure detection is proposed (lin and Zhang, 1999a). Due to this performance criterion, the parity equation is most sensitive to specified sensor failure and most insensitive to other sensor failures and system noise.

2. OPTIMAL PARITY VECTOR (OPT)

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In the following a brief introduction of this method is given.

k th normalized residual is maximum. For given false alarm rate a, detection threshold TD is determined by quantity a / 2 for normal distribution.

In order to design optimal parity vector sensitive to specified sensor, consider a performance criterion

If 1G I:::; TD , there is no failure occurred. If 1,; I> TD , fault occurs in the k th sensor and can be isolated simultaneously.

(2)

Where numerator is sensitivity to the fault of i th sensor, and denominator is sensitivity to noise and fault of all other sensors. In addition, constrained

3. APPLICATION OF NEURAL NETWORK TO COMPENSATE RESIDUAL

condition of parity: v' H = 0 is still kept. Parity vector v can be expressed as a linear combination of all columns of V

In practice the true value for the geometry matrix H can not be obtained due to the mounting and scale factor errors of inertial measurement units (Qin and Zhang, 1998). The parity vector v designed as above will be affected by the state and this will decrease the performance of failure detection. When considering sensor errors, measurement systems without fault is:

(3)

v=Ve

Where V is a matrix formed by a set of basis in parity space, C is a weight vector. Substituting (3) into (2), the extremum problem can be rewritten as Sup (u'e)2 le'Be

(4)

Where

c"o

Hn

matrix, Hse E B = V'(EE' +GG' -gjg)V is a Where m-n, matrix with dimension symmetrical u = V'gj E 9\m-n . If B is positive matrix, then

matrix, Hme E

E 9\mxn

9\mxm 9\mxn

sensor bias, and

El

is is

nominal scale

measurement factor

error

is misalignment matrix, b f

is

is Gaussian white noise. Let

1) When e = k · B-Iu, (4) achieves maximum. 2) Optimal parity vector is v = k· VB-Iu . Where k is an arbitrary real number not equal to zero. Suppose v j is an optimal parity vector sensitive to

( 7 )

i th sensor fault, corresponding residual is:

Then equation (6) is rewritten as: (5)

( 8 )

In order to determine detection threshold, assume that e is a normal distribution of multiple variables. Thereby residual obtained f.'om (5) is a normal random variable as follows: I)If

y=o,

Due to not knowing He' optimal parity vector can only be designed by nominal measurement matrix H n' Suppose

v j is an optimal parity vector

sensitive to i th sensor fault, corresponding residual is:

Si -NI(O,v~E=E'Vi)

2)If Y;t:O, Si -NI(v~Gy,v~E=E'v;) ,,=v;Hex+v;b+v;e Let normalized residual be r' == ':"

x+ 'r=' l=iv:H e

'i ",vjE -=-E ,Vi

v:b + v:G Y + v:I I 1

E

(With fault)

(9) (10)

I'

From above formula, the residual 'i(i = I, ... ,m) is not only sensitive to its own sensor fault but also to the state of the system and bias error. If the aircraft takes a big maneuver, it is possible that two former terms in equation (9) become so big as to exceed detection threshold and result in false alarm. Also in equation (10) it is possible that two former terms are bigger than the fault term and result in false alarm. To remove the effect of system state and constant bias on the residual, neural network is applied to compensate residual to enhance the ability of failure

I)If Y=O, Si' - NI (0,1) 2)If Y;t:O, Si'-NI(V~GY/~v~ESE'vj,I) So, m normalized residuals 'i' (i = 1, ... , m) sensitive to their own sensor fault are obtained. Then normalized residual with maximal absolute value can be determined. If

(Fault free)

1G 1= max 1'i" 1, it indicates that i

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the residual sensitive to the 1st sensor fault and the 2nd sensor fault respectively. Apparently, system state affects the uncompensated residual seriously. Compensated residual is not affected by system state and an obvious jump occurs only after there is a fault. Maximum normal residual is shown in Fig.3. It indicates that the maneuver can easily make the uncompensated maximal normalized residual exceed detection threshold resulting in a false alarm, and compensated residual is not changed along with system state but is sensitive to the fault. Let false alarm a be 0.01, so detection threshold TD is 2.33 from normal distribution table. The simulation result shows that the first sensor has developed a fault at the time t =1000 . Statistical test has good ability of detecting fault as above. In the follow ing, Kohonen networks (Kohonen, 1988) in FigA is introduced which does not need the statistical property of noise.

detection as follows. Because neural networks have good ability of non linear mapping, the multih .yer perceptron is used to compensate the residual. Taking the residuals without fault as the samples, multilayer perceptron is trained off-line. The compensated residuals are the difference between the output of multilayer perceptron trained and the residuals calculated by optimal parity vector. It is shown in Fig.l where NN is multi layer perceptron and H is real measurement matrix. Compensated residuals are sensitive to their own sensor faults and not affected by system state and bias error. After that statistical test technique as shown in section 11 and Kohonen networks are applied to detect and isolation failure respectively. The former one must have the knowledge of statistical property of noise but the latter one does not. Kohonen networks has a good capability of pattern classification and is an effective technique in fault classification.

residuals 'leCtor

I ••

b

I

z(l ) z(2)

-0.2

E

-0.4

(

200

0

400

600

800

1000

1200

1400

1600

1800

2000

compensatW~~rdua's \ector

0.6 0.4 0.2

I

Fig.l: Compensating residuals by neural network

l

-Z(l )

-

- z(2)

-0.2 OL--200~-400~--'600C":C---"800--1OOO~-1~200--1400':-:--1600~-1~800':-:---:-'2OOO (b)time

4. SIMULATION EXAMPLE

Fig.2 The uncompensated compensated residual

The measurement matrix H of a redundant sensor system consisting of six sensors mounted on dodecahedron is as follows

residual

and

the

Decision func tion for faun detection 1-

-sine cose cose 0 o ]T 0 sine -sine cose cose [ cose cose 0 0 sin e -sine

uncompensated OFd

- compensated DFd

sine

H=

0

5

,

., .\

e

Where =31.72° is the exact mounting angle. Suppose misalignment angle is 20"' , the value of

;.V" I ;, ;

\

~

detection tI1reshotd

2

scale factor is 10-4, bias error is 0.1° / h, the covariance of measurement noise is 0.01° / h . Simulation runs 2000 s(>conds. The aircraft trajectory is consisted of sinusoidal maneuver along three orthogonal axes with angular rate OS / s . At t =500, swift sinusoidal maneuver with angular rate

o~~~"~'·~~-~-~~-~-~~-~ 0200

400

600

800

1000

1200

1400

1600

18002000

time(sec)

Fig.3 The uncompensated detection function and the compensated detection function

2° / s is added on z axis. Suppose the first sensor develops a fault from t = 1000 . The magnitude of

For six sensors of redundant system, seven output neurons are required in Kohonen networks (Jin, 1998). The six neurons are chosen such that each one of them represents the fault occurring at each sensor, and the seventh one is for the normal operation of the

fault is 0.3° / h . Uncompensated residual is shown in fig.2 (a). Compensated residual by multi layer perceptron is shown in fig.2 (b). Where z(l) and z(2) represent

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system. As shown in FigA, the input of Kohonen networks (KN) is the residual which has been compensated by the multilayer perceptron. The weighting vector is w i = (w jl , w j 2" " W j6)" Where

the all other time periods.

s·

W ij

is the weighting coefficient linking the j th

input s~ and the ith output

Zj'Z, =wjS .

The

competition rule in the KN is to determine the winning output neuron by comparing the distance between the weighting vector and the input before updating the corresponding weighting vector,. The Euclidean distance is defined as the input intensity of the i th output neuron of the K.N: 00

(11 )

200

400

600

800

1000 1200 lime{sec)

1400

1600

1S00 2000

Fig.5 Comparison with the input intensities of all neurons

5. CONCLUSIONS The residual generated using optimal parity vector method with sensor errors is not only sensitive to specified sensor fault but also to the state of the system and bias error. To solve this problem a approach applying neural network to compensating the residual is proposed. Compensated residual is sensitive to specified sensor fault and without the effect of sensor errors. Then statistic test and Kohonen networks are used to detect the fault. The effectiveness of two techniques in fault detection is illustrated by an example of a redundant sensor system consisting of six sensors.

Fig.4 Construction ofkohonen networks The output neuron with the smallest Euclidean distance is the winner and its output is set to one, whilst all outputs of other neurons are set to zero. Let the i th output neuron be the winner, the weighting vector Wj is adjusted by the following learning algorithm

REFERENCES

w7ew = (1_1])w~ld +1] ~ W

new

)

= wo}1d

J'

= 1,... 7,J' ... l' .,J.

Jin H. and Zhang H.Y.(1999a). Comparison of two techniques used to FDI of redundant system. Proceeding of the first Conference on technique process and security in China, Beijing. Jin H. and Zhang H.Y.(l999b). Optimal parity vector sensitive to designated sensor fault, IEEE Trans.on Aerospace and Electronic Systems, Vo1.35, NoA, pp 1122-1128. Qing Y.Y. and Zhang H.Y.(1998). Principle of Kalman filter and integrated navigation, Northwestern Polytechnical University Publishing House. Hecht-Nielsen R.(1990).Neurocomputing. Macmillan College Publishing Company, ppl0l-l12. Jin H.(1998). Study on accuracy and fault-tolerant performance of navigation system. Doctor dissertation, Beijing University of Aeronautics and Astronautics. Kohonen T.(1988). Self-organization and associative memory, 2nd ED., Springerverlag Berlin.

(12)

Where 1] is the learning rate. Suppose the first sensor develops a fault between t = 1000 and t =1400, whilst the other sensors are operating normally. The magnitude of fault is 3° / h . The input intensities of all seven neurons are shown in Fig.5.For t between 1000 and 1400, the input intensity of the first neuron is the smallest one indicating that a fault has developed in first sensor. At all other time periods, the input intensity of the seventh neuron is the smallest one indicating that the system is operating normally. Further more, from the competition rule ofKN, the output of the first neuron is 1 for t between 1000 and 4000, whilst the output of the seventh neuron is 1 in all other time periods. The outputs of other neurons are zero all the time. Consequently, it can deduce that the first sensor has developed a fault during the time period from 1000 to 4000, and the system operates normally in

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