FAULT DETECTION ALGORITHM USING DCS METHOD

COMBINED WITH FILTERS BANK DERIVED FROM THE

WAVELET TRANSFORM

Oussama Mustapha

1,2

, Mohamad Khalil

2,3

, Ghaleb Hoblos

4

, Houcine Chafouk

4

and Dimitri Lefebvre

1

1

University Le Havre, GREAH, Le Havre, France

2

Lebanese University, Faculty of Engineering, Section I- El Arz Street, El Kobbe, Lebanon

3

Islamic University of Lebanon, Faculty of engineering, Biomedical Department, Khaldé, Lebanon

4

ESIGELEC, IRSEEM, Saint Etienne de Rouvray, France

Keywords: Signal, Filters Bank, DCS, Fault, detection, wavelet transform.

Abstract: The aim of this paper is to detect the faults in industrial systems, such as electrical machines and drives,

through on-line monitoring. The faults that are concerned correspond to changes in frequency components

of the signal. Thus, early fault detection, which reduces the possibility of catastrophic damage, is possible

by detecting the changes of characteristic features of the signal. This approach combines the Filters Bank

technique, for extracting frequency and energy characteristic features, and the Dynamic Cumulative Sum

method (DCS), which is a recursive calculation of the logarithm of the likelihood ratio between two local

hypotheses. The main contribution is to derive the filters coefficients from the wavelet in order to use the

filters bank as a wavelet transform. The advantage of our approach is that the filters bank can be hardware

implemented and can be used for online detection.

1 INTRODUCTION

The fault detection and diagnosis are of particular

importance in industry. In fact, the early fault

detection in industrial machines can reduce the

personal damages and economical losses. Many

researchers have performed fault detection by using

mechanical conditions such as vibration analysis.

Recently the current or voltage signature analysis is

used for the detection of electromechanical faults,

such as a broken bar in electrical drives (Sottile and

Kohler, 1993; Schoen et al., 1995; Kliman et al.,

1996). Other researchers use the AI tools

(Awadallah and Morcos, 2003) and frequency

methods (Benbouzid, et al., 1999).The aim of this

paper is to propose a method for the on-line

detection of changes in the electric current feeding

an induction motor due to a mechanical fault. The

method is based on a filters bank, whose coefficients

are derived from the wavelet, to decompose the

signal in order to explore their frequency and energy

components of the signal. Then, the Dynamic

Cumulative Sum method is applied to the filtered

signals in order to detect any change in the signal.

The filters bank is derived from the wavelet

transform, by using the Prony method, so the

wavelet characteristics are approximately conserved

and this allows both filtering and reconstruction of

the signal. The main contributions are to derive the

filters and to evaluate the error between filters bank

and wavelet transform. This study continues our

investigation concerning fault detection by means of

wavelet transform and filters bank (

Mustapha et al.,

2006a, 2006b). Extraction and detection will be

applied on simulated and real signals. The real

signals are issued from long duration experiments,

with GREAH, on asynchronous machines of 4kW.

These signals are recorded when the machine is

properly

operating and then when a bar of the same

machine is broken.This paper is decomposed as

follows. First we present the wavelet transform

(WT) and the filters bank technique. In section 3 we

detail the derivation of filters from a WT. In section

4, the Cumulative Sum and the Dynamic Cumulative

Sum methods are presented. In section 5, some

results are discussed. Then, the choice of the suitable

filter are discussed in section 6.

226

Mustapha O., Khalil M., Hoblos G., Chafouk H. and Lefebvre D. (2007).

FAULT DETECTION ALGORITHM USING DCS METHOD COMBINED WITH FILTERS BANK DERIVED FROM THE WAVELET TRANSFORM.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 226-231

DOI: 10.5220/0001640802260231

Copyright

c

SciTePress

2 WT AND FILTERS BANK

The Fourier analysis is the most well known

mathematical tool used for transforming the signal

from time domain to frequency domain. But it has

an important drawback represented by the loss of

time information when transforming the signal to the

frequency domain. To preserve the temporal aspect

of the signals when transforming them to frequency

domain, one solution is to use is the WT (

Truchetet,

1998

) which analyzes non-stationary signals by

mapping them into time-scale and time-frequency

representation. The Wavelet Transform is similar to

the Short Time Fourier Transform but provides, in

addition, a multi-resolution analysis with dilated and

shifted windows. The multi-resolution analysis

consists of decomposing the signal x(t) using the

wavelet

)(t

ψ

and its scale function )(t

φ

(Frandrin,

1993; Krim, 1995):

(t)dtx(t)ψ=b)(a,T

ab

ψ

x

∫

+∞

∞−

,

ab

1t-b

ψ (t)= ψ()

a

a

(1)

where a and b are respectively the dilation and

translation parameters. The filter associated with the

scale function

)(t

φ

is a low pass filter and the filter

associated to the wavelet

)(t

ψ

is a band pass filter.

The following formulas can be used to calculate

detail and approximation coefficients (

Truchetet,

1998):

∫

+∞

∞−

= dtttxmna

nmx

)()(),(

φ

(2)

∫

+∞

∞−

= dtttxmnd

nmx

)()(),(

ψ

(3)

where m and n are integers.

In this way, the relevant events, to be detected, can

be shown as details on specific scale levels. In

Discrete Wavelet Transform (DWT), the multi-

resolution analysis uses a scaling function and a

wavelet to perform successive decomposition of the

signal into approximations and details (figure 1: a

and b).

Figure 1: (a) multi-resolution analysis: Successive

decomposition into approximations and details (b) multi-

resolution analysis of the original signal into an

approximation and three detail levels.

At each time t, the signal is first decomposed by

using an N-channels band-pass filters bank whose

central frequency moves from lowest frequency f

1

up

to the highest frequency f

N

. Each component m

∈

{1,

…, N} is the result of filtering the original signal x

by a band-pass filter centered on f

m

. The frequency

response of the filters bank is shown in (figure 2).

H(jf) in d

B

f

f

1

f

m

f

N

Figure 2: Response curves of the filters bank.

For each component m, the sample y

(m)

(t), is on-line

computed according to the original signal x(t) and

using the parameters a

i

(m)

and b

j

(m)

of the

corresponding band-pass filter according to (4):

)4()()(.)(

1

)(

0

)()(

∑∑

==

−−−=

p

i

m

i

q

j

mm

ityajtxbjty

where x is the original signal, f

s

is the sampling

frequency of the original signal x, f

N

must satisfy the

condition f

N

≤ f

s

/2, N is the number of channels

used, p and q are the orders of the filter at level m.

The choice of the filters bank depends on the

original signal and its frequency band. The number

of filters N depends on the details that we have to

extract from the signal and on the events that must

be distinguished. In our case we will use N =3

filters.

The procedure of decomposing x(t) into signals

y

(m)

(t), m=1…N, allows us to explore all frequency

components of the signal. y

(1)

(t) gives the low

x

(t

)

sc ale

w

a

v

e

l

e

t

approx

[v ]

1

1

detail

[w ]

1

1

scale

w

a

v

e

l

e

t

app rox .....

[v ]

2

2

detail

[w ]

2

2

scale

w

a

v

e

l

e

t

approx . ....

[v]

j

j

detail

[w]

j

j

(a )

1

(d )

1

(a )

0

(a )

2

(d )

2

(a )

3

(d )

3

FAULT DETECTION ALGORITHM USING DCS METHOD COMBINED WITH FILTERS BANK DERIVED FROM

THE WAVELET TRANSFORM

227

frequency components and y

(N)

(t) gives the high

frequency ones. Therefore, the points of change of

each component give information about the

frequency and energy contents and will be used to

detect any changes in frequency and energy in the

original signal.

3 PRONY'S METHOD

In the present work, the main objective is to derive

the filters coefficients of a filters bank from a

wavelet in order to use the filters bank as a WT. The

filters bank is derived from the WT, by using the

Prony's method, so the wavelet characteristics are

approximately conserved and this allows both

filtering and reconstruction of the signal.

For a given wavelet, we can use the approximation

coefficients of the wavelet function ψ(t) to extract

the coefficients a

i

and b

i

in order to design an IIR

filter that behaves as the wavelet. The extraction of

the filter coefficients can be done by using the

Prony's method. The main advantage of the wavelet-

derived filter is that it can be used instead of the

wavelet and can be hardware implemented in order

to be used for online signal filtration. Figure 3 shows

the response curves (h

wav

) of the wavelet function

'db3' and the response curves (h

filt)

of the derived

filter.

Figure 3: response curves of the wavelet function 'db3' and

of the derived filters bank (the filter's order is 30).

Prony's method is an algorithm that can be used to

find an IIR filter with a prescribed time domain

impulse response. According to the time domain

impulse response h

wav

of the wavelet function )(t

ψ

,

the numerator order p and the denominator order q

of the desired filter, Prony's method is used to

compute the filter's coefficients a

i

and b

j

, i=1…p and

j=1…q. If the length of h is less than the largest

order (p or q), h is padded with zeros. It is

fundamentally based on signal approximation with a

linear combination of adjustable exponentials.

The impulse matching problem for modeling an

entire causal signal x(t), t=0,1,…,∞, produces an

infinite number of equations. The problem is to find

the parameters a

i

and b

i

such that the equation (x) is

satisfied:

)5(

0

0

0

1

1

0)0()1()2(

0....).........0()1(

0.................0)0(

1

0

?

~~

2

~

1

?

2

1

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

⎥

⎦

⎤

⎢

⎣

⎡

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

#

#

#

#%##

"

q

p

b

b

b

b

aXx

X

a

a

a

xxx

xx

x

where X

1

is the top part of matrix X,

~

a is a p

dimentional vector of parameters {a

i

}, b is a q+1

dimentional vector of parameters {b

j

},

~

x is the first

column of bottom part of matrix X, and

~

2

X

is the p

last columns of matrix X bottom part.

The equation

~~~?

2

0xXa

+

=

contains an infinite

number of equations to be solved for

~

a

. This linear

equation is usually over determined and no exact

solution exists. This means that since the vector

~

x

can only be approximated by the columns of

matrix

~

2

X

, it's necessary to choose

~

a

to minimize

the equation error defined by the equation (6):

∑∑∑

∞

+==

∞

+=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

−+===

1

2

11

2

)6()()()(

qt

p

i

i

qt

T

itxatxteee

ε

with

~~~

2

exXa=+

. The error is minimized by using

partial differentiation with respect to parameters

{a

i

}:

~

2

~~

2

~

2

~

xXaXX

TT

−=

(7)

In order that the orthogonality condition

0

2

~

min

=Xe

T

is satisfied, (7) provides a solution for the optimum

vector

~

a

, which can then be used to find the

solution to vector b by simple matrix multiplication

in (5).

4 CUMSUM AND DCS

The Cumulative Sum algorithm (CUMSUM)

algorithm is based on a recursive calculation of the

logarithm of the likelihood ratios. This method can

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

228

t

M

be considered as a sequence of repeated tests around

the point of change t

M

(figure 4) (Nikiforov, 1986;

Basseville and Nikiforov, 1993

). For the seek of

simplicity x(t) will be referred as x

t

in the following.

Let x

1

,x

2

,x

3

,…,x

t

be a sequence of observations. Let

us assume that the distribution of the process X

depends on parameter

0

θ

until time t

M

and depends

on parameter

1

θ

after the time t

M

. At each time t we

compute the sum of logarithms of the likelihood

ratios as follows:

)8(

),...,/(

),...,/(

1

11

11

1

)(

),(

1

0

1

∑∑

=

−

−

=

==

t

i

tt

tt

t

i

i

m

mt

xxxf

xxxf

LnsS

θ

θ

The importance of this sum comes from the fact that

its sign changes after the point of change. The

detectability (

Basseville and Nikiforov, 1993) is due to

the fact that the expectation

0][

0

<

i

sE

θ

and

0][

1

>

i

sE

θ

. We, then, calculate the following

detection function

),(

1

1

),(

1

)(

min

mi

ti

mt

t

m

SSg

≤≤

−= . This

function compares, at any time t, the difference

between the value of the sum of the logarithm of the

likelihood ratio and its minimal current value. The

instant at which the procedure is stopped is t

a

= min

{t : g

(m)

t

≥

h}, where h is the detection threshold (i.e

the change can be detected when the detection

function reaches a predetermined threshold h). The

point of change can be defined as follows t

M

= max

{t : g

(m)

t

= 0}.

Figure 4: CUNSUM algorithm (a) Signal (b) CUNSUM

(c) Detection function.

At any time t and for the observation

vector

),...,(

1 tt

xxXX == , suppose that the

distribution of the process X depends on

parameter

θ

. A change can affect the frequency

distribution of the signal. The Dynamic Cumulative

Sum method (DCS) is a repetitive sequence around

the point of change t

M

. It is based on the local

cumulative sum of the likelihood ratios between two

local segments estimated at the current time t. These

two dynamic segments

)(t

a

S

(« after t ») and

)(t

b

S

(« before t ») are estimated by using two windows of

width W (figure 5) before and after the time instant t

as follows:

*

}1,...,{i ;:

)(

−−= tWtxS

i

t

b

follows a probability

density function

(

)

i

xf

0

θ

*

},...,1{i ;:

)(

WttxS

i

t

a

++=

follows a probability

density function

(

)

i

xf

1

θ

Figure 5: DCS algorithm (a) Signal; (b) Dynamic

cumulative sum Cumulative sum; (c) Detection function.

The parameters

)(

^

t

b

θ

of the segment

)(t

b

S

, are

estimated using W points before the time instant t

and the parameters

)(

^

t

a

θ

of the segment

)(t

a

S

, are

estimated using W points after the time instant t. At a

time t, the DCS is defined as the sum of the

logarithm of likelihood ratios from the beginning of

the signal up to the time t:

)9(

)(

)(

),(

1

^

1

)(

)(

)(

)()(

^

^

∑∑

==

==

t

i

i

t

i

i

i

i

i

t

b

t

a

m

s

xf

xf

LnSSDCS

b

a

θ

θ

(

Khalil, 1999) proves that the DCS function reaches

its maximum at the point of change t

M

. The detection

function used to estimate the point of change is

(

)()

( ) ( ) () () () ()

1

max , , (10)

mmtttt

t

ab ab

it

gDCSSSDCSSS

≤≤

⎡⎤

=−

⎣⎦

The instant at which the procedure is stopped is

t

a

= inf {t : g

(m)

t

≥

h}, where h is the detection

threshold. The point of change is estimated as

t

M

= max {t>1 : g

(m)

t

= 0}. The DCS is a method that

t+W

t-W

t

FAULT DETECTION ALGORITHM USING DCS METHOD COMBINED WITH FILTERS BANK DERIVED FROM

THE WAVELET TRANSFORM

229

can be used when the parameters of the signal are

unknown.

5 RESULTS

The algorithm is first applied to simulated signals

and then to real signals (figure 6). The simulated

signal is generated by concatenating two random

signals of different variances (σ

0

=1 et σ

1

=3), and

two sinusoidal signals of different frequencies

(f

0

=150Hz et f

1

=600Hz). Real and simulated signals

are decomposed into 3 scales before applying the

DCS method. These scales are computed by using

the ARMA coefficients calculated by Prony’s

method and corresponding to the ‘db3’ wavelet. The

coefficients of the derived filter of order 5 from the

wavelet 'db3' for scale level 3 are detailed in the next

table:

Table 1: Derived filters coefficients.

ai 1.000 -1.247 0.527 -0.165 0.604 -0.409

bi 0 0.0110 0.0130 0.0144 0.0203 0.0252

The results lead us to determine the point of change

of statistical parameters of these signals.

Figure 6: Detection of a real signal.

Table 2: Comparison of the points of change.

Expected

Time of

change

1

st

comp.

2

nd

comp.

3

rd

comp.

1

st

simulated signal

1000 1006 1005 1002

2

nd

simulated signal

2000 2012 2003 2001

Real signal

4000 4107 4097 4098

Note that the third component, which is filtered by a

highest central frequency band-pass filter, presents

the closest point of change to the real one as shown

in the table 2.

6 FILTER'S ORDER

In the wavelet theory, the choice of the wavelet is a

critical problem. To extract the specific events in a

signal, the choice of the wavelet is important to be

adapted to the event to be detected. Many

researchers have performed the detection by using

the wavelet in different domains of application: in

image edge detection (Mallat, 2000), for

compression (Benbouzid et al.,1999), for signal de-

noising in speech processing (Misiti et al.). In

biomedical applications, the quadratic spline wavelet

is used by Li (Li et al.,1995) and the complex

wavelet is used by Shenhadji (Shenhadji et al.,1995)

to process the ECG signal.

In our work, filters derived from many wavelets

such as the Daubechies, the coiflet and the symlet

wavelets are tested and according to the results

obtained in figure 7, the filter derived from the

wavelet 'db3' at level 3 has been used because it

presents the minimum error and then it is chosen.

Note that the error is defined as follows:

)11()(

1

2

∑

=

−=

k

i

filtwav

hherror

Where, h

wav

and h

filt

are the impulse responses of the

wavelet and the derived filter respectively.

Figure 7: Error due to the use of different types of

wavelets.

The orders of the filter (p and q) are very important

parameters and can affect the error due to the

application of Prony's method to extract the filter

coefficients from the wavelet. As shown in figure 8,

we can see that if the order of the filter becomes 30

and above, the error due to the derivation becomes

negligible for filter derived from db3.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

230

Figure 8: Error due to the order of the filter derived from

different types of wavelets (scale 2 and p=q=30).

7 CONCLUSIONS

This article has proposed a method to detect the

point of change of statistical parameters in signals

issued from industrial machines. This method uses a

band-pass filters bank, derived from a wavelet

transform, to decompose the signal and the DCS

algorithm to characterize and classify the

parameters

of a signal in order to detect any variation of the

statistical parameters due to any change in frequency

and energy. The main contribution of the work is to

find a filters bank that approximates a wavelet. The

filters bank derivation is done by using the Prony's

method. After the calculation of the resulting error,

between the derived filters bank and the

correspondent wavelet, the wavelet 'db3' has been

selected. In order to reduce the error due to the order

of the derived filter, the order is taken to be beyond

30. This on-line algorithm is developed and tested

and it gives good results for the detection of changes

in the signals. It is necessary to test the algorithm

with other types of wavelets, to explain the error

depending on the scale levels, and to implement the

whole algorithm in a DSP. The detectability of DCS

must be proved after decomposing the signal,

especially after using the ARMA decomposition.

Another perspective is to complete the filters design

by determining the optimum orders p and q.

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FAULT DETECTION ALGORITHM USING DCS METHOD COMBINED WITH FILTERS BANK DERIVED FROM

THE WAVELET TRANSFORM

231