APPICATION OF DISCRETE WAVEET TRANSFORM TO FAUT DETECTION 1 SEDA POSTACIOĞU KADİR ERKAN 3 EMİNE DOĞRU BOAT 1,,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 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 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 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 []. 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. owever, 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,
to detect them effectively. Mallat, wang, 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 []. 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. 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 highpass 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. shows this procedure, where and are the high-pass and low-pass filters, respectively [1,3]. x(n) d1...... d1 x(n) f d [k] = (t) ψ( t k) dt In this study db wavelet has been used. The filter for decomposition stage are shown in Table 1. () Table 1 Filter for decomposition stage ow pass filter igh pass filter -.19.41.8365.483 -.483.8365 -.4 -.19 The filter for reconstruction stage are shown in Table. Table Filter for reconstruction stage ow pass filter igh pass filter.483.8365.41 -.19 -.194 -.4.8365 -.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]. Decomposition Wavelet Coefficients Reconstruction Fig. Decomposition and reconstruction process N being the total number of samples in x[n].as shown in equations (1)- (), the c[k] are called approximation and the d[k] are called detail. Parameter determines the scale or the frequency range of each wavelet basis function ψ. Parameter k determines the time translations. f c[k] = (t) ϕ(t k) dt (1) 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 []. 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. 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]. 4 3 (a) 1 5 6 7 (b) 8 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 ADS11 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-3 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 ADS11 ADC are used. The power unit includes M57959A Mitsubishi IGBT driver and IXS45N1 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 M57959A Mitsubishi IGBT driver and IXS45N1 IGBT power transistor. Unit names of the system are presented in Table 3 [5,6]. Fig. 3 a) The designed oven b) The control card designed for the oven. Table 3: Units for the system. 1 Oven Termocouple ( sensor) 3 Fan Disturbances (Two holes on 4 the top of the oven) 5 Power supply 6 RS-3 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 1 31 Fault 38 Fault Type Two holes on the top of the oven were open and fan was on during minutes oles were close and fan was on during 6 minutes Fig. 4 represents the PID control for the ISTE disturbance with faults.
1 8 6 4 5 55 5 155 5 55 35 355 45 Time (sec) 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 was taken samples in 593 sec.. Because the temperature 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 reflect edges (abrupt change) and noise. Fig. 5 (a) shows the function and detail produced by the DWT at each level. 1 4 6 8 d1-1 3 4 5 6 d - 5 5 15 5 3 d3-5 5 15 d4-5 1 3 4 5 6 7 8 d5-5 5 1 15 5 3 35 4 a).1 4 6 8 d1 -.1 1 4 6 8 d -1 4 6 8 d3-4 6 8 d4-1 4 6 8 d5-4 6 8 b) Fig. 5 a) Detail produced by decomposition b) Detail produced by reconstruction 455 55 555 As shown in Fig. 5 (a), small 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. After thresholding, the signal has been reconstructed. By upsampling the 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 downsampling. Fig. 5 (b) shows the detail after the thresholding and reconstruction at five levels.. Only the detail that contain the high frequency information are used to find the edges which are faults. Groups of large magnitude detail, called wavelet maxima. Fig. 6 shows the faults as abrupt changes or edges. of the set point is C. Until the temperature gets the set, the edge is detected as a fault for the system as shown in Fig. 6. 1 8 6 4 4 6 8 1.8.6.4. 4 6 8 Sample Fig. 6 Result of the fault detection 4 CONCUSIONS 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 of noisy signals. So the signal has been reconstructed. Groups of large magnitude detail 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. [] i, J.,:3, Wavelet Approach to Edge Detection, Master of Science, The Department of Mathematics and Statistics, Sam ouston State University untsville, Texas. [3] Mallat,S.,:1989, "A Theory for Multiresolution Signal Decomposition: The Wavelet Transform, IEEE
Trans.Pattern Anal. Mach. Intelligence,Vol.11,No.7, pp.674-693. [4] E.-J. Manders and G. Biswas,:3, FDI of abrupt faults with combined statistical detection and estimation and qualitative fault isolation, Washington, DC. [5] Bolat, E.,D., Erkan, K., Postalcıoğlu, S.,:5, Experimental Autotuning PID Control of Using Microcontroller EUROCON 5, Serbia & Montenegro, Belgrade. [6] Bolat, E.,D., Erkan, K., Postalcıoğlu, S.,:5, Implementation of Microcontroller Based Control Using Autotuning PID Methods, IICAI-5, The nd Indian International Conference on Artificial Intelligence, India. [7] Bolat, E.,D., Erkan, K., Postalcıoğlu, S.,: 5, Microcontroller Based Control of Oven Using Different Kinds of Autotuning PID Methods, ecture Notes in Artificial Intelligence 389 pp.195-13. [8] Postalcıoğlu, S., Becerikli, Y., :5, Nonlinear System Modeling Using Wavelet Networks, ecture Notes in Computer Science (NCS), Vol.3497, pp.411-417. [9] Postalcioglu, S., Erkan, K., Bolat, E.,D.,: 5, Comparison of Wavenet and Neuralnet for System Modeling, KES 5, NAI 368, pp..17,