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APPLICATION OF DISCRETE WAVELET TRANSFORM TO FAULT DETECTION 1 SEDA POSTALCIOĞLU 2 KADİR ERKAN 3 EMİNE DOĞRU BOLAT 1,2,3 Department of Electronics and Computer Education, University of Kocaeli Türkiye Abstract. The aim of this paper is to explain the application of discrete Wavelet transform (DWT) to fault detection. Edges which can be occurring by faults in signal can be mathematically defined as local singularities. The Wavelet transform characterizes the local regularity of signals by decomposing signals. In this study, ISTE disturbance PID (Proportional Integral Derivative) has been used. This method has been applied to the experiment set. Two faults were given to the experiment set while working. Faults have been found using Wavelet transform. Only the detail coefficients that contain the high frequency information are used to find the edges which are faults. After decomposition stage, wavelet denoising method has been applied because of the environmental noise. So the signal has been reconstructed. Groups of large magnitude detail coefficients shows the edges which occur by faults. Key-Words: -Wavelet, fault detection, discrete Wavelet transform, PID. 1 Introduction Wavelet is a waveform of limited duration that has an average value of zero. In Fig. 1, we compare wavelet with sine wave, which are the basis functions of Fourier analysis. Sinusoids do not have limited duration. They extend from minus to plus infinity. Fourier analysis consists of breaking up a signal into sine and cosine waves of various frequencies. Similarly, wavelet analysis consists of breaking up of a signal into shifted and scaled versions of the original or mother wavelet. a) b) Fig. 1 a) Sine wave b) Wavelet STFT was able to analyze either high frequency components using narrow Windows, or low frequency components using wide windows, but not both. Therefore came up with the ingenious idea of using a different window function for analyzing different frequency bands. Furthermore, Windows were all generated by dilation or compression of a prototype Gaussian. These window functions had compact support both in time and in frequency [1]. Wavelet analysis is a powerful tool for time- frequency analysis. Wavelets show local characteristics which is the main property in both space and spatial frequency [8,9]. Fourier analysis is also a good tool for frequency analysis, but it can only provide global frequency information, which is independent of time with Fourier analysis. It is impossible to describe the local properties of functions in terms of their spectral properties [2]. Edges which can be occurring by faults in signal can be mathematically defined as local singularities. Until recently, the Fourier transforms was the main mathematical tool for analyzing local singularities. However, the Fourier transform is global and not well adapted to local singularities. It is hard to find the location and spatial distribution of singularities with Fourier transforms. Wavelet analysis is a local analysis; it is suitable for time-frequency analysis, which is necessary for singularity detection. The Wavelet transform has been found significant mathematical tool to analyze the singularities including the edges, and further, Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp149-153)
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Page 1: APPLICATION OF DISCRETE WAVELET TRANSFORM TO FAULT DETECTION€¦ ·  · 2006-09-29APPLICATION OF DISCRETE WAVELET TRANSFORM TO ... was run through DWT using Daubechies wavelet of

APPLICATION OF DISCRETE WAVELET TRANSFORM TO FAULT DETECTION

1SEDA POSTALCIOĞLU 2KADİR ERKAN 3EMİNE DOĞRU BOLAT

1,2,3Department of Electronics and Computer Education, University of Kocaeli

Türkiye

Abstract. The aim of this paper is to explain the application of discrete Wavelet transform (DWT) to fault detection. Edges which can be occurring by faults in signal can be mathematically defined as local singularities. The Wavelet transform characterizes the local regularity of signals by decomposing signals. In this study, ISTE disturbance PID (Proportional Integral Derivative) has been used. This method has been applied to the experiment set. Two faults were given to the experiment set while working. Faults have been found using Wavelet transform. Only the detail coefficients that contain the high frequency information are used to find the edges which are faults. After decomposition stage, wavelet denoising method has been applied because of the environmental noise. So the signal has been reconstructed. Groups of large magnitude detail coefficients shows the edges which occur by faults.

Key-Words: -Wavelet, fault detection, discrete Wavelet transform, PID.

1 Introduction Wavelet is a waveform of limited duration that has an average value of zero. In Fig. 1, we compare wavelet with sine wave, which are the basis functions of Fourier analysis. Sinusoids do not have limited duration. They extend from minus to plus infinity. Fourier analysis consists of breaking up a signal into sine and cosine waves of various frequencies. Similarly, wavelet analysis consists of breaking up of a signal into shifted and scaled versions of the original or mother wavelet.

a)

b)

Fig. 1 a) Sine wave b) Wavelet

STFT was able to analyze either high frequency components using narrow Windows, or low frequency components using wide windows, but not both. Therefore came up with the ingenious idea of using a different window function for analyzing different frequency bands. Furthermore, Windows were all generated by dilation or compression of a prototype Gaussian. These window functions had compact support both in time and in frequency [1]. Wavelet analysis is a powerful tool for time- frequency analysis. Wavelets show local characteristics which is the main property in both space and spatial frequency [8,9]. Fourier analysis is also a good tool for frequency analysis, but it can only provide global frequency information, which is independent of time with Fourier analysis. It is impossible to describe the local properties of functions in terms of their spectral properties [2]. Edges which can be occurring by faults in signal can be mathematically defined as local singularities. Until recently, the Fourier transforms was the main mathematical tool for analyzing local singularities. However, the Fourier transform is global and not well adapted to local singularities. It is hard to find the location and spatial distribution of singularities with Fourier transforms. Wavelet analysis is a local analysis; it is suitable for time-frequency analysis, which is necessary for singularity detection. The Wavelet transform has been found significant mathematical tool to analyze the singularities including the edges, and further,

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp149-153)

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to detect them effectively. Mallat, Hwang, and Zhong proved that the maxima of the Wavelet transform can detect the location of the irregular structures as faults. The Wavelet transform characterizes the local regularity of signals by decomposing signals into elementary building blocks that are well localized both in space and frequency. This not only explains the underlying mechanism of classical edge detectors, but also indicates a way of constructing optimal edge detectors under specific working conditions when system has a fault [2]. The aim of this paper is to explain the working mechanism of fault detection using Wavelet transforms. Faults in the system cause as certain changes in the response of measured signals, changes in time response and in frequency response. These changes would result in transient behavior of system variables and transient analysis becomes critical for fast and accurate fault detection. Wavelet analysis is capable of detecting the change or transition in the signal. For this reason wavelet analysis has been used. 2 Discrete Wavelet Transform Mallat showed that Multiresolution can then be used to obtain the Discrete Wavelet Transform (DWT) of a discrete signal by iteratively applying low-pass and high-pass filters and subsequently down sampling them by two. Decomposing stage for discrete signal using a series of low-pass and high-pass filters for computing signal’s DWT. Quadrature mirror filters (QMF) and sub-band filtering were developed by A. Croisier, D. Esteban and C. Galand around 1976. Fig. 2 shows this procedure, where H and L are the high-pass and low-pass filters, respectively [1,3].

x(n)

H

L

H

L

d1

Reconstruction

...

x(n)

H

L

H

L

d1

Wavelet CoefficientsDecomposition

...

Fig. 2 Decomposition and reconstruction process

N being the total number of samples in x[n] .As shown in equations (1)- (2), the c[k] are called approximation coefficients and the dj[k] are called detail coefficients. Parameter j determines the scale or the frequency range of each wavelet basis functionψ . Parameter k determines the time translations.

∫∞

∞−

−ϕ= dt)kt()t(f]k[c

(1)

∫∞

∞−

−ψ= dt)kt2(2)t(f]k[d j2j

j

(2)

In this study db2 wavelet has been used. The filter coefficients for decomposition stage are shown in Table 1.

Table 1 Filter coefficients for decomposition stage

Low pass

filter

coefficients

-0.129 0.2241 0.8365 0.4830

High pass

filter

coefficients

-0.483 0.8365 -0.224 -0.129

The filter coefficients for reconstruction stage are shown in Table 2.

Table 2 Filter coefficients for reconstruction stage

Low pass filter

coefficients 0.4830 0.8365 0.2241 -0.129

High pass filter

coefficients -.1294 -0.224 0.8365 -0.483

The low-pass and high-pass decomposition filters together with their associated reconstruction filters form a system of what is called quadrature mirror filters. Decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. The signal can also be reconstructed from a wavelet representation with a similar pyramidal algorithm [3].

3 Fault Detection Using Discrete Wavelet Transform Edges which can be occurring by faults in signal can be mathematically defined as local singularities. Singularities can be characterized easily as discontinuities where the gradient approaches infinity. Edge detection is an important task in image and signal processing. It is a main tool in pattern recognition, image segmentation, and scene analysis [2]. Faults in the system cause as certain changes in the response of measured signals, changes in time response and in frequency response. These changes would result in transient behavior of system variables and transient analysis becomes critical for fast and accurate

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp149-153)

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fault detection. Wavelet analysis is capable of detecting the change or transition in the signal. For this reason wavelet analysis has been used. The purpose of this example is to show how analysis by wavelets can detect the precise instant when a signal changes or taking place a fault in the system [3]. The DWT is a linear transform that is very suitable to represent the non-stationary events in signals. The DWT has good localization properties of high frequency components [4].

3.1 Experimental Results In this study, ISTE disturbance PID has been used. This method has been applied to the experiment set which is an FODPT (First Order Plus Dead Time) system. To implement the temperature control, a digital signal processing card and an oven are designed for temperature control. To be able to control this system a digital signal processing card is designed. PIC17C44 is used as microcontroller and ADS1212 is used as A/D converter. Experiments have been realized on this oven [7]. Fig. 3 (a) and (b) present the oven and control system card respectively. As shown in Fig. 3 (b) digital signal processing unit is designed and a power unit including an IGBT and an IGBT driver is produced. This power unit uses PWM (Pulse Width Modulation) technique. Since the control method is wanted to be flexible, it is achieved by using a computer. The digital signal processing unit gets the temperature data from the experiment set by using a thermocouple temperature sensor and makes the data appropriate for the computer. Then, this unit transmits the data to the computer by using an RS-232 protocol. The computer produces control data by using the control method. Afterwards, this control data is transmitted to the digital signal processing unit again. This unit derives a PWM signal from the control data. And, this PWM signal is applied to the power unit. Finally, the PWM signal determines the energy level of the heater. So, the control is achieved by applying necessary amount of energy to the heater. In the digital signal processing unit PIC17C44 microcontroller and ADS1212 ADC are used. The power unit includes M57959AL Mitsubishi IGBT driver and IXSH45N120 IGBT power transistor. And this PWM signal is applied to the power unit. Finally, the PWM signal determines the energy level of the oven. So, the control is achieved by applying necessary amount of energy to the oven. The power unit includes M57959AL Mitsubishi IGBT driver and IXSH45N120 IGBT power transistor. Unit names of the system are presented in Table 3 [5,6].

1

2

3

4

5

6

7

8

(a) (b)

Fig. 3 a) The designed oven b) The control card designed for the oven.

Table 3: Units for the system. 1 Oven

2 Termocouple (Temperature sensor)

3 Fan

4 Disturbances (Two holes on the top of the oven)

5 Power supply 6 RS-232 connection 7 DSP unit (controller card) 8 Power block

Two faults were given to the system while working. So PID parameters and faults have been examined in these conditions. In this study, the main purpose is to detect the faults in the system. Faults are given in Table 5.

Table 4. Fault Types

Fault Start Time (sec)

Fault Type

Fault 1 3120

Two holes on the top of the oven were open and fan was on during 2 minutes

Fault 2 3802 Holes were close and fan was on during 6 minutes

Fig. 4 represents the PID control for the ISTE disturbance with faults.

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp149-153)

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0

20

4060

80

100

120

5505

1005

1505

2005

2505

3005

3505

4005

4505

5005

5505

Time (sec)

Temperature

Fig. 4 PID control for the ISTE disturbance with faults

Faults in the system cause as certain changes in the response of measured signals, changes in time response and in frequency response. These changes would result in transient behavior of system variables and transient analysis becomes critical for fast and accurate fault detection. For fault detection with DWT the temperature value was taken 1200 samples in 5930 sec.. Because the temperature value is not change quickly. Firstly the signal was run through DWT using Daubechies wavelet of length 4 for decomposition. Decomposition is realized by down-sampling. Down –sampling by a factor of two means every other term is removed. When the faults occur detail coefficient show abrupt changes. The detail coefficients reflect edges (abrupt change) and noise. Fig. 5 (a) shows the function and detail coefficients produced by the DWT at each level.

0 200 400 600 800 1000 12000

100

200

0 100 200 300 400 500 600-1

0

1

0 50 100 150 200 250 300-2

0

2

0 50 100 150-5

0

5

0 10 20 30 40 50 60 70 80-20

0

20

0 5 10 15 20 25 30 35 40-50

0

50

d1

d2

d3

d4

d5

Tem

pera

ture

va

lue

a)

0 200 400 600 800 1000 12000

100

200

0 200 400 600 800 1000 1200-0.1

0

0.1

0 200 400 600 800 1000 1200-1

0

1

0 200 400 600 800 1000 1200-2

0

2

0 200 400 600 800 1000 1200-10

0

10

0 200 400 600 800 1000 1200-20

0

20

Tem

pera

ture

va

lue

d1

d2

d3

d4

d5

b)

Fig. 5 a) Detail coefficients produced by decomposition b) Detail coefficients produced by reconstruction As shown in Fig. 5 (a), small coefficients are likely to represent noise and should be removed using wavelet shrinkage or thresholding. Soft thresholding has been applied in this study for the detail coefficients. After thresholding, the signal has been reconstructed. By up-

sampling the coefficients and reversing the filtering process, the signal is reconstructed. Up-sampling is accomplished by adding zeros between every term, thus taking the place of the coefficient removed by down-sampling. Fig. 5 (b) shows the detail coefficients after the thresholding and reconstruction at five levels.. Only the detail coefficients that contain the high frequency information are used to find the edges which are faults. Groups of large magnitude detail coefficients, called wavelet maxima. Fig. 6 shows the faults as abrupt changes or edges. Temperature value of the set point is 100 0C. Until the temperature gets the set value, the edge is detected as a fault for the system as shown in Fig. 6.

0 200 400 600 800 1000 120020

40

60

80

100

120

0 200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

Sample

Tempe

rature

value

Fig. 6 Result of the fault detection

4 CONCLUSIONS

Faults cause as certain changes in time response and in frequency response in the system. These changes would result in transient behavior of system variables and transient analysis becomes critical for fast and accurate fault detection. For this reason Wavelet transform has been used. The Wavelet transform has been found important mathematical tools to analyze the singularities including the edges, and further, to detect them effectively. ISTE disturbance PID has been used for temperature control. These methods have been applied to the experiment set. Two faults were given the system while working. Because of the environmental noise, wavelet denoising method has been applied to the detailed wavelet coefficients of noisy signals. So the signal has been reconstructed. Groups of large magnitude detail coefficients shows the edges which occur by faults. In this study, the main purpose is to detect the faults using DWT.

References [1] Polikar,R.,:1999, “The Story of Wavelets”, IMACS/IEEE CSCC'99 Proceedings, pp. 5481-5486. [2] Li, J.,:2003, “Wavelet Approach to Edge Detection”, Master of Science, The Department of Mathematics and Statistics, Sam Houston State University Huntsville, Texas. [3] Mallat,S.,:1989, "A Theory for Multiresolution Signal Decomposition: The Wavelet Transform”, IEEE

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp149-153)

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Trans.Pattern Anal. Mach. Intelligence,Vol.11,No.7, pp.674-693. [4] E.-J. Manders and G. Biswas,:2003, “FDI of abrupt faults with combined statistical detection and estimation and qualitative fault isolation”, Washington, DC. [5] Bolat, E.,D., Erkan, K., Postalcıoğlu, S.,:2005, “Experimental Autotuning PID Control of Temperature Using Microcontroller” EUROCON 2005, Serbia & Montenegro, Belgrade. [6] Bolat, E.,D., Erkan, K., Postalcıoğlu, S.,:2005, “Implementation of Microcontroller Based Temperature Control Using Autotuning PID Methods”, IICAI-05, The 2nd Indian International Conference on Artificial Intelligence, India. [7] Bolat, E.,D., Erkan, K., Postalcıoğlu, S.,: 2005, “Microcontroller Based Temperature Control of Oven Using Different Kinds of Autotuning PID Methods”, Lecture Notes in Artificial Intelligence 3809 pp.1295-1300. [8] Postalcıoğlu, S., Becerikli, Y., :2005, “Nonlinear System Modeling Using Wavelet Networks”, Lecture Notes in Computer Science (LNCS), Vol.3497, pp.411-417. [9] Postalcioglu, S., Erkan, K., Bolat, E.,D.,: 2005, “Comparison of Wavenet and Neuralnet for System Modeling”, KES 2005, LNAI 3682, pp. 100.107, 2

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp149-153)