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Digital Signal Processing 20 (2010) 276288 Contents lists available at ScienceDirect Digital Signal Processing Mechanical equipment fault diagnosis based on redundant second generation wavelet packet transform Rui Zhou, Wen Bao, Ning Li, Xin Huang, Daren Yu Harbin Institute of Technology, 458#, No. 92, West Da-Zhi Street, Harbin, Heilongjiang, PR China a r t i c l ei n f oa b s t r a c t Article history: Available online 3 May 2009 Keywords: Second generation wavelet packet transform Lifting scheme Feature extraction Fault diagnosis Wavelet transform has been widely used for the vibration signal based mechanical equipment fault diagnosis. However, the decomposition results of the discrete wavelet transform do not possess time invariant property, which may result in the loss of useful information and decrease the classifi cation accuracy of fault diagnosis. To overcome this defi ciency, a novel fault diagnosis method based on the redundant second generation wavelet packet transform is proposed. Firstly, the redundant second generation wavelet packet transform is constructed on the basis of second generation wavelet transform and redundant lifting scheme. Secondly, the vibration signals are decomposed by redundant second generation wavelet packet transform and then the faulty features are extracted from the resultant wavelet packet coeffi cients. Finally, the extracted fault features are given as input to classifi ers for identifi cation. The proposed method is applied for the fault diagnosis of gearbox and gasoline engine valve trains. Test results indicate that a better classifi cation performance can be obtained by using the proposed fault diagnosis method in comparison with using second generation wavelet packet transform based method. 2009 Elsevier Inc. All rights reserved. 1. Introduction Growing demand for high quality production requires that deviation of machine conditions from its normal setting should be identifi ed and fi xed promptly to reduce costly machine downtime and maintain high productivity. As a result, research on effective mechanical equipments health monitoring and diagnosis has been enhanced in recently years 1,2. Since the vibration signal collected from these equipments during operation contains valuable information about the machine or part condition, vibration analysis has been adopted widely as a means for machine failure identifi cation 3. The vibration signal is often a mixture signal which simultaneously contains stationary, non-stationary and noisy components. Therefore, the information for maintenance decisions is not readily available from these vibration data unless the appropriate signal processing techniques are chosen 4. The wavelet transform (WT), as a state-of-the-art tool for signal processing, has focused much attention on both theoretic analysis and engineering applications in many fi elds. WT can be used for multi-scale analysis of a signal through dilation and translation, so it can extract signal features from both time domain and frequency domain effectively. Consequently, WT has been successfully applied for the condition monitoring and fault diagnosis of electromechanical equipment 5. A drawback of WT is that the frequency resolution is rather poor in the high-frequency subband which the faulty characteristics always exist in. The wavelet packet transform (WPT), a generalization of wavelet bases, is alternative bases formed by taking linear combinations of usual wavelet functions 6,7. WPT divides the frequency space into various parts and allows a better * Corresponding authors. Fax: +86 451 86413241. E-mail addresses: hit_zhourui (R. Zhou), yudaren (D.R. Yu). 1051-2004/$ see front matter2009 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2009.04.005 R. Zhou et al. / Digital Signal Processing 20 (2010) 276288277 timefrequency localization of signals. In recent years, WPT has been used as a popular method in the fi eld of condition monitoring and fault diagnosis 812. In application of WT and WPT, it is crucial for selecting a proper wavelet function for a special problem and the engineering experiences show that the wavelet function should be selected according to the fault feature to be detected 13. The second generation wavelet transform (SGWT) is a new wavelet construction method using lifting scheme in the time domain. It abandons the Fourier transform as design tool for wavelets, and wavelets are no longer defi ned as translates and dilates of one fi xed function. Compared with classical WT, SGWT possesses several advantages, including possibility of adaptive and nonlinear design, in-place calculations, irregular samples and integral transform 1416. Recently, the appli- cations of SGWT and second generation wavelet packet transform (SGWPT) to condition monitoring and fault diagnosis of electromechanical equipment deserve more attentions 1720. Unfortunately, SGWT does not have the property of time invariant. Using SGWT, the decomposition results of a delayed signal are not the time-shifted version of those of the input signal, which may lead to the loss of useful faulty information for feature extraction and fault diagnosis. The redundant lifting scheme possesses time invariant property and overcomes the disadvantage of lifting scheme by getting rid of the split step and zero padding of prediction operator and update operator. The approximation and detail signals at all levels are the same length as the input signal in the redundant lifting scheme 2123. In this paper, on the basis of SGWT and redundant lifting scheme, the redundant second generation wavelet packet transform (RSGWPT) is constructed, and then fault diagnoses of mechanical equipments are performed by using the pro- posed RSGWPT. The rest of the paper is organized as follows. In Section 2, the fundamental theory of SGWT and SGWPT is reviewed briefl y. In Section 3, the construction method of RSGWPT is introduced. The fault diagnosis method based on RSGWPT is described in Section 4. In Section 5, the proposed fault diagnosis method is applied to diagnose different states of a gearbox and the valve trains on a gasoline engine. The comparison results with SGWPT based fault diagnosis method are also shown. Finally the conclusions have been drawn in Section 6. 2. Review of second generation wavelet transform 2.1. Second generation wavelet transform Second generation wavelet transform, proposed by Wim Sweldens, is a new wavelet construction method using lifting scheme. It can be seen as an alternate implementation of classical discrete wavelet transform. The main feature of the second generation wavelet transform is that it provides an entirely spatial domain interpretation of the transform, as opposed to the traditional frequency domain based constructions 15. The decomposition stage of SGWT consists of three steps: split, prediction and update. In the split step, an approximate signal alat level l is split into even samples and odd samples. al+1=al(2i),dl+1=al(2i+1)(1) In the prediction step, a prediction operator P is designed and applied on al+1to predict dl+1. The resultant prediction error dl+1 is regarded as the detail coeffi cients of al. dl+1(i) =dl+1(i) M/2 ? r=M/2+1 pral+1(i+r)(2) where pr are coeffi cients of P and M is the length of pr. In the update step, a designed update operator U is applied on dl+1. Adding the result to the even samples, the resultant al+1 is regarded as the approximate coeffi cients of al. al+1(i) =al+1(i)+ N/2 ? j=N/2+1 ujdl+1(i+j1)(3) where uj are coeffi cients of U and N is the length of uj. Iteration of the above three steps on the output a, and then the detail and approximation coeffi cients at different levels are generated. The reconstruction stage of SGWT is a reverse procedure of the decomposition stage, which includes inverse update step, inverse prediction step and merging step. al+1(i) =al+1(i) N/2 ? j=N/2+1 ujdl+1(i+j1) dl+1(i) =dl+1(i)+ M/2 ? r=M/2+1 pral+1(i+r) al(2i) =al+1,al(2i+1) =dl+1 (4) 278R. Zhou et al. / Digital Signal Processing 20 (2010) 276288 Fig. 1. Structure of second generation wavelet transform, both analysis side and synthesis side. The operators P and U are built by means of interpolating subdivision method (ISM) 16. Choosing different P and U is equivalent to choosing different biorthogonal wavelet fi lters 24. Fig. 1 depicts the structure of SGWT. The computational costs of the forward and inverse transform are exactly the same. 2.2. Second generation wavelet packet transform The timefrequency resolution of SGWT varies with the decomposition levels. It gives good time and poor frequency resolution at high frequency subband, and good frequency and poor time resolution at low frequency subband. In order to obtain a higher resolution in the high frequency subband, SGWPT has been constructed and hence the detail coeffi cients at each level are further decomposed to obtain their approximation and detail components 25,26. The decomposition and reconstruction stages of SGWPT are described as below. In the decomposition stage, Xl,kis split into even samples Xl,keand odd samples Xl,ko, Xl,ke=Xl,k(2i),Xl,ko=Xl,k(2i+1)(5) where Xl,k represents the coeffi cients of the kth node at level l. Then calculate each subband coeffi cients at level l+1. Xl+1,2=Xl,1oP(Xl,1e) Xl+1,1=Xl,1e+U(Xl+1,2) . . . Xl+1,2l+1=Xl,2loP(Xl,2le) Xl+1,2l+11=Xl,2le+U(Xl+1,2l+1) (6) In the reconstruction stage, the subband coeffi cients to be reconstructed are reserved, and then other subband coeffi cients are set to be zeroes. Finally, the reconstructed results are obtained by the following formula. Xl,2le=Xl+1,2l+11U(Xl+1,2l+1) Xl,2lo=Xl+1,2l+1+P(Xl,2le) Xl,2l(2i) =Xl,2le Xl,2l(2i+1) =Xl,2lo . . . Xl,1e=Xl+1,1U(Xl+1,2) Xl,1o=Xl+1,2+P(Xl,1e) Xl,1(2i) =Xl,1e Xl,1(2i+1) =Xl,1o (7) Overall, the decomposition and reconstruction stages of SGWPT are shown in Figs. 2 and 3. R. Zhou et al. / Digital Signal Processing 20 (2010) 276288279 Fig. 2. The decomposition stage of SGWPT. Fig. 3. The reconstruction stage of SGWPT. 3. Redundant second generation wavelet packet transform 3.1. Redundant lifting scheme In the redundant lifting scheme, the split step is discarded. Assuming Pland Ulrepresent the prediction and update operators of the redundant lifting scheme at level l, the coeffi cients of Pland Ulare obtained by padding Prand Ujof initial operator P and U with zeroes 21. pli=p0 0,0,.,0 ? ? ? 2l1 ,p0 1,0,.,0 ? ? ? 2l1 ,p0 2,.,p 0 M20,.,0 ? ? ? 2l1 ,p0 M1 (8) ulj=u0 0,0,.,0 ? ? ? 2l1 ,u0 1,0,.,0 ? ? ? 2l1 ,u0 2,.,u 0 N20,.,0 ? ? ? 2l1 ,u0 N1 (9) The decomposition results of an approximation signal alat level l via redundant lifting scheme are expressed by following equations. 280R. Zhou et al. / Digital Signal Processing 20 (2010) 276288 Fig. 4. The forward and inverse transform of redundant lifting scheme. ?d l+1=alPl+1al al+1=al+Ul+1dl+1 (10) where al+1and dl+1 are approximation coeffi cients and detail coeffi cients of alat level l+1. The reconstruction procedure of redundant lifting scheme is directly achieved from its forward transform, which is expressed as below. al= 1 2 ?a l+1Ul+1dl+1+dl+1+Pl+1 ?a l+1Ul+1dl+1 ? (11) The forward and inverse transform of redundant lifting scheme is shown in Fig. 4. 3.2. Construction of RSGWPT With the redundant lifting scheme and SGWPT, the RSGWPT is easily to be constructed. The prediction step and update step of RSGWPT at level l are performed by using Pland Ul, which are expressed as follows. Xl+1,2=Xl,1Pl+1(Xl,1) Xl+1,1=Xl,1+Ul+1(Xl+1,2) . . . Xl+1,2l+1=Xl,2lPl+1(Xl,2l) Xl+1,2l+11=Xl,2l+Ul+1(Xl+1,2l+1) (12) The reconstruction stage of RSGWPT can be obtained from its decomposition stage and expressed by following equations. Xl,2l= 1 2 ?X l+1,2l+11U l+1(X l+1,2l+1)+Xl+1,2l+1+P l+1?X l+1,2l+11U l+1(X l+1,2l+1) ? . . . Xl,1= 1 2 ?X l+1,1Ul+1(Xl+1,2)+Xl+1,2+Pl+1 ?X l+1,1Ul+1(Xl+1,2) ? (13) The forward and inverse transform of RSGWPT are shown in Figs. 5 and 6. Owing to without the split operation in the decomposition stage of RSGWPT, the approximation and detail coeffi cients at all levels have the same length as that of the input signal. Consequently, the decomposition results of RSGWPT possess time invariant property and keep the information of the raw signal perfectly. 4. The proposed fault diagnosis method Fig. 7 shows the fl ow diagram of the proposed fault diagnosis method. The vibration signals are acquired from the monitoring mechanical equipments. After A/D conversion, the sampled vibration data are decomposed by RSGWPT. In the feature extraction stage, nine statistical characteristics (i.e. peak value, mean, standard deviation, root mean square, shape factor, skewness, kurtosis, crest factor and pulse index) are calculated from each of the resultant subband wavelet packet coeffi cients. R. Zhou et al. / Digital Signal Processing 20 (2010) 276288281 Fig. 5. The decomposition stage of RSGWPT. Fig. 6. The reconstruction stage of RSGWPT. For classifi cation, three classical machine learning algorithms including C4.5 decision tree (C4.5), radial basis function neural network (RBFNN) and support vector machine (SVM) are employed. In the following, the principles and some basic mathematical expressions for such algorithms are briefl y reviewed. The C4.5 decision tree classifi er is a type of induction algorithm 27. An example of a decision tree is shown in Fig. 8. A test node represents the selected predictor, which is used to divide the samples into subsets. Each branch descending from that node corresponds to one of the possible values for this predictor. Finally, the leaf node provides the classifi cation of samples in the subsets. Generally speaking, a decision tree is constructed from a set of samples by using the divide-and- conquer strategy; that is, a best predictor is selected in each test node to split samples into the smaller subsets. The quality of predictors is evaluated by using the information gain ratio 28. When the dataset is divided into several subsets by the predictor, the information gain ratio is used to measure the reduction of the uncertainty of samples associated with this process. This means that the samples with the same classifi cation are sorted into the same subset as much as possible. For a continuous predictor P, the threshold t in P?t should be found to maximize the information gain ratio 29. The samples are sorted on their values of predictor P to give ordered distinct values v1,v2,.,vN. Every pair of adjacent values suggests a potential threshold t= (vi+vi+1)/2. When the best predictor and its threshold are determined, the predictor is used to test these samples at the root node of the tree. A descendant of the root node is then created according to the threshold of this predictor, and the training samples are sorted into an appropriate descendant node. C4.5 employs the top-down and recursive splitting technique to produce the subtree. When samples in the subset have the same classifi cation or all possible tests have the same class distribution, the leaf node is generated. The classifi cation of the samples in the leaf node is the same as the most frequent classifi cation in this leaf node. 282R. Zhou et al. / Digital Signal Processing 20 (2010) 276288 Fig. 7. Flow chart of fault diagnostic procedure. Fig. 8. Example of a decision free. The RBFNN has a feed forward architecture with an input layer, a hidden layer and an output layer as shown in Fig. 9. The radial basis functions are embedded into a two-layer feed forward neural network. Such a network is characterized by a set of inputs and a set of outputs. In between the inputs and outputs there is a layer of processing units called hidden units. Each of them implements a radial basis function 30. The input layer of this network has NIunits for a NIdimensional input vector. The input units are fully connected to the NHhidden layer units, which are in turn fully connected to the NC output layer units, where NCis the number of output classes. The activation functions of the hidden layer were chosen to be Gaussians, and are characterized by their mean vectorsi, and covariance matrices Ci=2 i I, i=1,2,.,NH. Then the activation function of the ith hidden unit for an input vector xjis given by gi(xj) =exp ?x ji?2 22 i ? (14) Theiand2 i are calculated by using suitable clustering algorithm. The hidden layer units are fully connected to the NCoutput layer through weights wik. The output units are linear, and the response of the kth output for an input xjis given by yk(xj) = NH ? i=0 wikgi(xj),k=1,2,.,NC(15) Training the RBF involves two stages. First, the basis functions must be established using an algorithm to cluster data in the training set. Typical ways to do this include Kohohen self-organizing maps, K-Means clustering, decision trees, genetic algorithms or orthogonal least squares and Max-Min algorithms 31,32. Next, it is necessary to fi x the weights linking the hidden and the output layers. If neurons in the output layer contain linear activation functions, these weights can be R. Zhou et al. / Digital Signal Processing 20 (2010) 276288283 Fig. 9. Architecture of a radial basis function neural network. Fig. 10. (A) A separating plane with small margin; (B) A separating plane with larger margin. calculated directly using matrix inversion and matrix multiplication. Because of the direct calculation of weights in an RBF, it is usually much quicker to train than an equivalent multilayer perceptron training algorithm. The SVM is a statistic machine learning technique that has been widely applied in the pattern recognition area 3336. Let(xi,yi),i=1,.,Nbe a t