An intelligent fault diagnosis method based on wavelet packer analysis and hybrid support vector machines

Expert Systems with Applications 36 (2009) 12131–12136

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

An intelligent fault diagnosis method based on wavelet packer analysis and hybrid support vector machines Guang-Ming Xian *, Bi-Qing Zeng Computer Engineering Department of Nanhai Campus, South China Normal University, Guangdong, Foshan 528225, China

a r t i c l e

i n f o

Keywords: Wavelet packet analysis Hybrid support vector machines Multi-fault diagnosis 1-v-r approach

a b s t r a c t In this paper, a new intelligent method for the fault diagnosis of the rotating machinery is proposed based on wavelet packet analysis (WPA) and hybrid support machine (hybrid SVM). In fault diagnosis for mechanical systems, information about stability and mutability can be further acquired through WPA from original signal. The faulty vibration signals obtained from a rotating machinery are decomposed by WPA via Dmeyer wavelet. A new multi-class fault diagnosis algorithm based on 1-v-r SVM approach is proposed and applied to rotating machinery. The extracted features are applied to hybrid SVM for estimating fault type. Compared to conventional back-propagation network (BPN), the superiority of the hybrid SVM method is shown in the success of fault diagnosis. The test results of hybrid SVM demonstrate that the applying of energy criterion to vibration signals after WPA is a very powerful and reliable method and hence estimating fault type on rotating machinery accurately and quickly. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The efficient and accurate fault diagnosis is important for improving reliability and performance in a mechanical system (Chen, Chua, & Lim, 2008). Most of the rotating machinery such as internal combustion engines, fans and motors may develop faults. These faults may cause the machine to break down and decrease its level of performance (Lei, He, Zi, & Hua, 2007). Traditionally, the condition of rotating machinery can be monitored by measuring he respective vibration signal or sound emission signal (Wu et al., in press). Many methods such as Time Synchronous Average (TSA), Fast Fourier Transform (FFT)-based spectrum analysis and short-time Fourier Transform (STFT) have been applied in fault diagnosis and condition monitoring of mechanical system. The above methods analyze the signal in frequency domain with low resolution, which is not suitable for non-stationary vibration signal (Chen et al., 2008). Wavelet packet analysis (WPA) is the typical signal processing method for mechanical fault diagnosis. WPA can multi-decompose the signal into the different frequencies to obtain the localized impulse signals. The energy of the WPA coefficients is used for fault detection. Hasan Ocaka developed a new scheme based on wavelet packet decomposition and hidden Markov modeling (HMM) for tracking the severity of bearing faults (Ocaka, Loparob, & Discenzoc, 2007). A fault diagnosis system (Wu & Liu, 2009) is proposed for internal combustion engines using WPA and artificial * Corresponding author. Tel.: +86 757 83125963; fax: +86 757 86687309. E-mail address: [email protected] (G.-M. Xian). 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.03.063

neural network (ANN) techniques. The experimental results showed the proposed system achieved an average classification accuracy of over 95% for various engine working conditions. In Zarei and Poshtan (2007), bearing defect is detected using the stator current analysis via Meyer wavelet in the wavelet packet structure, with energy comparison as the fault index. The advantage of this method is in the detection of incipient faults. Compared to conventional methods, the superiority of the proposed method is shown in the success of fault detection. Among the various methods for condition monitoring of machinery, ANN have become in the recent decades the outstanding method exploiting their non-linear pattern classification properties, offering advantages for automatic detection and identification of gearbox failure conditions, whereas they do not require an in-depth knowledge of the behaviour of the system (Rafiee, Arvani, Harifi, & Sadeghi, 2007). These methods are based on an empirical risk minimization principle and have some disadvantages such as local optimal solution, low convergence rate, obvious ‘‘over- fitting” and especially poor generalization when the number of fault samples is limited (Yuan & Chu, 2006). Support vector machines (SVM) is a very effective method for general purpose pattern recognition based on structural risk minimization principles. This characteristic is very important in fault diagnostics under the condition that the fault samples are few (Guo-hua, Yong-zhong, Yu, & Guang-huang, 2007). The main difference between ANNs and SVMs is in their risk minimisation (Gunn, 1998). In the case of SVMs, structural risk minimisation principle is used to minimise an upper bound based on an expected risk. Whereas in ANNs, traditional empirical risk minimisation is used

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to minimise the error in training of data. The difference in risk minimisation leads to a better generalization performance for SVMs than ANNs (Kim, Pang, Je, Kim, & Bang, 2003). Samantaray, Dash, and Panda (2007) focus on the development of an advanced signal classifier for small reciprocating refrigerator compressors using noise and vibration signals. Three classifiers, self-organising feature map, learning vector quantisation and SVM are applied in training and testing for feature extraction and the classification accuracies of the techniques are compared to determine the optimum fault classifier. The classification technique selected for detecting faulty reciprocating refrigerator compressors involves artificial neural networks and SVMs. The results confirm that the classification technique can differentiate faulty compressors from healthy ones and with high flexibility and reliability. Abbasiona, Rafsanjani, Farshidianfar, and Irani (2007) provides a procedure for fault classification of rolling bearings, using SVM classifier. Vibration data from bearings were denoised using discrete Meyer wavelet. The results that they achieved from wavelet analysis and SVM are fully in agreement with empirical result. SVM theory is proposed for binary classification. Many approaches have been proposed to extend the binary SVM to multiclass problems. The common scheme is that a multi-class SVM is designed to deal with the problem as a collection of two-classifications that can be solved by binary SVM. In Guo-hua et al. (2007), a hybrid SVM scheme is proposed for multi-fault classification. This hybrid scheme integrates two SVM strategies, 1-v-1 (one versus one) and 1-v-r (one versus rest), respectively adopted at different classification levels, parallel classification and serial classification levels. In this paper, the object of the fault diagnosis is the rotating machinery. For designing WPA and hybrid SVM based fault design system, the features of the rotating machinery were used for training and testing of hybrid SVM after preprocessing. A serial classification method based on 1-v-r SVMs strategy is proposed in our research.

pffiffiffi X 2 h0k /ð2t  kÞ;

/ðtÞ ¼

ð2Þ

k

pffiffiffi X h1k /ð2t  kÞ: 2

wðtÞ ¼

ð3Þ

k

When n = 2; 3; . . . the function can be defined by the following recursive relationships:

w2n ðtÞ ¼

pffiffiffi X 2 h0k wn ð2t  kÞ;

ð4Þ

k2Z

pffiffiffi X h1k wn ð2t  kÞ; w2nþ1 ðtÞ ¼ 2

ð5Þ

k2Z

where h0k and h1k are the quadrature mirror filter (QMF) associated with the predefined scaling function and mother wavelet function. The wavelet packet coefficients, wnj;k are defined as

wnj;k ¼

Z

UðtÞwnj;k ðtÞ dt:

ð6Þ

The symbol W 03 presents the symbol for a subspace that stands for the third resolution and the 0th subspace. In the experimental study, the signals will be broken up to four resolutions. As a result, four resolutions will produce sixteen subspaces and the frequency intervals of each subspace can be computed by Hu, Wang, and Ren (2005):

 n1 2jþ1

Sf ;

 S ; f jþ1

n 2

n ¼ 1; 2; . . . ; 16

ð7Þ

where Sf is sampling frequency. In this research Sf = 2000 Hz, f00 is  i S the original signal with the frequency interval 0; 2f ¼ ð0; 1000, f10 with the frequency interval (0, 500], f20 with the frequency interval (0, 250], f30 with the frequency interval (0, 125], f37 with the frequencies interval (875, 1000]. The wavelet packet analysis is used regarding the data preprocessing for fault diagnosis. 2.2. Feature extraction of fault conditions using Shannon entropy and wavelet selection

2. Principle of wavelet packet analysis and feature extraction 2.1. Wavelet packet analisis Both of WPA and discrete wavelet transform (DWT) have the framework of multi-resolution analysis (MRA). The main difference in the two techniques is the WPA can simultaneously break up detail and approximation versions, but DWT only breaks up as an approximation version. Hence, the WPA have the same frequency bandwidths in each resolution and DWT does not have this property. The mode of decomposition does not increase or lose the information within the original signals. Therefore, the signal with great quantity of middle and high frequency signals can offer superior time–frequency analysis. The WPA suits signal processing, especially nonstationary signals because the same frequency bandwidths can provide good resolution regardless of high and low frequencies. The theory of WPA can be defined as below (Li, Song, & Li, 2004; Ortiz & Syrmos, 2006; Wu & Liu, 2009; Xu & Li, 2007; Yen & Lin, 2000). The WPA is a generalization of the wavelet transform and the wavelet packet function is also a time–frequency function, it can be described as j=2 j w0n j;k ðtÞ ¼ 2 wð2 t  kÞ;

j; k 2 Z

ð1Þ

where the integers j and k are the index scale and translation operations. The index n is an operation modulation parameter or oscillation parameter. The the scaling function /(t) and mother wavelet functions w(t) can be defined as:

A high-frequency-resolution wavelet is obtain to study the frequency characteristic of a signal. Shannon wavelet has the most resolution theoretically (Mallat, 1998) among orthogonal wavelets. Sharp edges of these filters make them non-causal, therefore their approximation is utilized in practical application. In the field of signal processing, entropy is a common idea used (Zhang, Walter, Miao, & Lee, 1995). Wavelet packet decomposition is applied to the fault signal using wavelet packet filters w with the ‘‘Shannon entropy” and is defined as below

En ¼

15 X

  n2 wn2 j;k log wj;k ;

j; k 2 Z

ð8Þ

n¼0

where wnj;k is the coefficients of the subspace after wavelet packet decomposition and n = 1, 2, . . . , 15 (Avic & Akpolat, 2006). For containing massive noise, the low frequencies in Shannon entropy of E0 will not used. Hence, after being normalized the feature vector T composed of En can be expressed as

T ¼ ½E1 =E; E1 =E2 ; . . . ; E15 =E:

ð9Þ

For identifing the different faults of rotating machinery by support vector macine, the feature vector T will be used in fault classification. Meyer wavelet is an approximation of Shannon wavelet. This wavelet is a frequency bandlimited function whose Fourier transform is smooth, unlike that of the Shannon wavelet, and cause a faster decay of wavelet coefficient in the time domain. However, the time decay of this wavelet in time domain is high. It is faster than Shannon wavelet, but the supporting area in time domain is

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The variables ni are slack variables which are needed to allow misclassifications in the set of inequalities. We can introduce a set of Lagrange multipliers ai ; bi to slove the problem. The Lagrange multipliers ai are then determined by means of the following optimization problem (dual problem):

Minimize Q ðaÞ ¼

l X

ai 

i¼1

Subject to

l X

l X l 1X ai aj yi yj xTi xj ; 2 i¼1 i¼1

ð17Þ

ai yi ¼ 0;

ð18Þ

i¼1

0 6 ai 6 C;

i ¼ 1; . . . ; l

ð19Þ

If 0 6 ai 6 C, the corresponding data points are called support vectors (SVs). By introducing kernel function Kðx; xi Þ, SVM maps the input vector into a higher dimensional feature to solve the nonlinear case. Kðx; xi Þ is the innerproduct kernel performing the nonlinear mapping into feature space

Kðx; xi Þ ¼ Kðxi ; xÞ ¼ uðxÞT uðxi Þ: Fig. 1. Discrete Meyer wavelet and scaling functions.

Therefore, the dual optimization problem can be given by

not limited and an approximation of it is used as ‘‘Discrete Meyer”. Fig. 1 shows the Discrete Meyer wavelet and the scaling function (Zarei & Poshtan, 2007). 3. Principle of hybrid SVM 3.1. Basic theory of SVMs The SVM introduced by Vapnik is a learning method on the foundation of statistical learning theory (Abbasiona et al., 2007). SVM can creates a line or a hyperplane between two sets of data for classification. We assume that a training set of N data points is given by

ð10Þ

where xi 2 Rn , and corresponding binary class labels yi 2 f1; þ1g. The SVM classifier, according to Vapnik’s original is to find an optimal hyperplane formulation satisfies the following conditions

wT uðxi Þ þ b P 1 if yi ¼ þ1; wT uðxi Þ þ b 6 1 if yi ¼ 1

ð11Þ

If the inequality in Eq. (11) holds for all training data, it will be a linearity separable case (Abbasiona et al., 2007, Huang & Liu, 2002; Vapnik, 1995, 1998). The non-linear function uðÞ maps the input space to a high (possibly infinite) dimensional feature space. Therefore, in the linearly separable case, for finding the optimal hyperplane, one can solve the following constrained optimization problem:

1 T w w; 2 T Subject to yi ðw ðxi Þ þ bÞ P 1;

Minimize UðwÞ ¼

ð12Þ i ¼ 1; . . . ; l:

ð13Þ

If inequality in Eq. (11) does not hold for some data points in S, SVMs become linearly not separable. We can solve the following constrained optimization problem to find an optimal hyperplane for a linearly not separable case

min w;b;n

l X 1 T ni ; w wþC 2 i¼1

ð14Þ

Subject to yi ðwT uðxi Þ þ bÞ P 1  ni ;

ð15Þ

ni P 0;

ð16Þ

i ¼ 1; . . . ; l

Minimize Q ðaÞ ¼

l X i¼1

ai 

l X l 1X ai aj yi yj Kðx; xi Þ: 2 i¼1 i¼1

ð21Þ

The kernel Kðx; xi Þ is required to satisfy the Mercer’s theorem (Huang & Liu, 2002). Unseen data are classified by kernel functions as below

x2

S ¼ fxi ; yi gli¼1 ;

ð20Þ



positive type; if mðxÞ > 0; ngative type;

ð22Þ

if mðxÞ > 0:

where the decision function can be expressed as

mðxi Þ ¼ yi

l X

! yj aj Kðxi ; xj Þ þ b ;

j¼1

f ðxÞ ¼ sign

l X

ð23Þ

!

ai yi Kðx; xi Þ  b :

ð24Þ

i¼1

For the kernel function Kðx; xi Þ (Parikh & Das, 2007), we have the following choices: Radial-basis function: Kðx; xi Þ ¼ expðgkx  xi k2 Þ; Polynomial: Kðx; xi Þ ¼ ðcxT xi þ rÞd ; c > 0 Sigmoid: Kðx; xi Þ ¼ tanhðcxT xi þ rÞ

g>0

3.2. Serial classification level based on 1-v-r SVMs strategy Support vector machines are essentially binary classifiers. To improve their applicability, several methods have been suggested for extending SVMs for multi-classification, including one-versusone (1-v-1), one-versus-rest (1-v-r) (Guo-hua et al., 2007; Lingras & Butz, 2005, 2007) and DAGSVM. In this article, we extend the interval set formulation of SVMs to fault diagnosis of roller bearing that involve four fault types that are separated using the 1-v-r approach. Hybrid SVM using 1-v-r SVM strategy is designed to realize multi-fault diagnosis for roller bearing of rotating machinery (shown in Fig. 2). Larger values indicate that the input samples are far from the classification hyperplane and the classification results are more reliable. There are four fault type to be classified: fault type I (tooth root crack), fault type I (fatigue wear), fault type III (surface pitting) and fault type IV (surface scrape). SVM1 is trained to separate the fault type I from the other three fault types. When the feature input is fault type I, the output of

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SVM1 F(x)=-1

Fault type (I,II,III,IV)

N

N

SVM1 F(x)=-1

Y

Fault type I

N

SVM1 F(x)=-1

Y

Y

Fault type II

Fault type IV

Fault type III

Fig. 2. Scheme of gear multi-fault classification based on 1-v-r SVM strategy.

SVM1 is set to 1 and the classification process is over; otherwise the output is set to +1 and fault diagnosis from will be transferred to SVM2. SVM2 is trained to classify the fault type II and the other two fault types. When the feature input is fault type II, the output of SVM2 is set to 1 and the classification process is over; otherwise +1 and fault diagnosis will be transferred into SVMs3. SVM3 is trained to classify fault type III and fault type IV. When the feature input is fault type III, the output of SVM3 is set to 1; otherwise +1 indicates fault type IV. Vibration signals were acquired from reciprocating compressors consisting of healthy and faulty compressors. Vibration data were measured by an accelerometer located on the compressor. The maximum acquisition frequency was 10 kHz and the number of sampled data was 8000. The signals obtained are processed further for machine condition diagnosis as explained in the in Fig. 3. The experimental equipment is shown in Fig. 4. The fault diagnostic system includes two main parts: fault feature extract and fault classification. WPA was used as a feature extractor which gives distinguishable characteristic features about the signals. After wavelet packet decomposition, energy and Shannon type entropy criterion have been applied to WPA detail. The feature values obtained by these analytical methods include the amplitudes of the specific frequency (gear rotating frequency, gear mesh frequency, harmonic frequencies, etc.) in Power Spectrum, the specific rotating period in Cestrum and the components energy values from Wavelet packet decomposition. Therefore, dimensions of the input patterns can be reduced and useful information can be extracted for the training of hybrid SVM. The wavelet packet analysis was performed with four-level decomposition and Dmeyer was selected as mother wavelet. All of these values will be transferred into serial classification level SVMs as inputs (Guo-hua et al., 2007; Zarei & Poshtan, 2007). The training of SVMs is carried out using the Platt’s SMO algorithm (Platt et al., 1998).

Vibration signal

Feature extraction by WPA

Classification by Hybrid SVM

Roller bearing

Electric motor

Fig. 3. Flow chart of diagnostic procedure.

Reciprocating compressor

Fig. 4. The experimental equipment.

4. Experimental result 4.1. Veracity of the hybrid SVM and selection of kernel function The training experiments were conducted on a small data set (60 vibration signal samples, 15 signals for each of four gear states). Using these samples, the SVMs at a serial classification level based on 1-v-r SVMs strategy were trained. In order to test the performance of the hybrid SVM scheme, 60 testing samples were re-chosen. The Gaussian RBF kernel has been used for training and testing of the SVM and the values of the parameters gamma (g) and regularization parameter (C) have been chosen as 0.5 and 2:34  107 , respectively. The classification results of this test experiment are shown in Table 1 and prove the veracity and reliability of the hybrid SVM. Upon testing over 10,000 test cases, an overall fault recognition accuracy of 92.85% has been obtained by the proposed algorithm. The details of the accuracies obtained for different types of faults are shown in Fig. 5. From Fig. 5, it is clear that different SVMs have different accuracy for different fault style. For surface pitting fault, the performance of the proposed algorithm is little inferior. Analysed results show that the proposed method is effective to detect the machinery fault. In the results shown above, as mentioned earlier, the RBF kernel has been used for the SVM. Studies have also been made to investigate the performance of the SVM for polynomial kernel and sigmoid kernels. During this investigation, the degree of polynomial ðcÞ has been varied from 1 to 15 in step of 1. It had been found that,

Table 1 Serial classification level SVMs classification results. Testing data

Fault type

Roller bearing

1–15 16–30 31–45 46–60

SVM values

Classification results

SVMs1

SVMs

SVMs3

1 1 1 1

– 1 1 1

– – 1 1

Tooth root crack Fatigue wear Surface pitting Surface scrape

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94

100

92

Accuracy(%)

Accuracy (%)

80

60

40

90

88

g

86 20

0.3 0.5 0.7 0.9

84

0

82

Tooth root crack

Fatigue wear

Surface pitting

0

Surface scrape

7

1x10

7

2x10

with polynomial kernel, the maximum classification accuracy attained was 90.47%. Whereas, using the Sigmoid kernel the maximum accuracy obtained was only 86.62%. Hence, RBF kernel had been found to be the most suitable one for this application.

To show the efficiency of the selected features and the classifier algorithm, a comparison between the hybrid SVM and back propagation network (BPN) has been done. Wavelets ‘‘Db2”, ‘‘Sym5” and ‘‘Dmeyer” are the mother wavelets to build and perform the proposed WPA technique. Vibration signals are used to classify by the BPN and hybrid SVM after feature selection of ‘‘Shannon entropy”. The features of ‘‘Shannon entropy” in each fault condition are summarized for the input of the BPN and hybrid SVM. The number of experimental data is 240. The 40 data sets are used to training and the 200 data sets are used to test the identification accuracy of the two methods.

7

6x10

7

7x10

Table 2 Fault recognition accuracy for different fault types using different mother wavelets to construct WPA. Sym5

Dmeyer

BPN

Hybrid SVM

BPN

Hybrid SVM

BPN

Hybrid SVM

85.00 83.33 80.00 81.67

88.33 86.67 83.33 86.67

90.00 88.33 85.00 86.76

91.67 90.00 86.67 88.33

93.33 91.67 86.67 86.67

95.00 93.33 88.33 93.33

Average (%)

82.50

86.25

87.50

89.17

89.58

92.50

Recognition time (s)

12.17

4.93

14.90

5.11

13.56

4.28

4.2. Parameters of the SVM

4.3. Comparison between hybrid SVM and BPN of three different mother wavelet conditions

7

5x10

Fig. 6. Comparison of accuracy acquired with different C and g values for RBF kernels.

Db2

Currently in the literature, there is no method available for deciding the value of C, for choosing the best kernel function and for setting the kernel functions. As a result, the most appropriate kernel function and the values of kernel function parameters C as well as of the parameter are decided by trial and error procedure (Parikh & Das, 2007). The selection of RBF kernel width ð1=2gÞ is one of the major problems in SVMs for good classification performance. Small kernel width may cause over-fitting, and large one under-fitting (Chang et al., 2005). The kernel width determines the radius of the hypersphere enclosing part of the data as a classifier boundary in a multi-dimensional feature spaces (Yang, Hwang, Kim, & Tan, 2005). For choosing the optimum values of the parameters C and g of the RBF kernel, a large number of studies had been carried out by varying the values of these two parameters. The range of variation of these two parameters, which had been considered, is as follows: (i) C, from 1 to 7  107 ; and ii) g, from 102 to 102 . The classification accuracies acquired for different combinations of C and g are shown in Fig. 6. From Fig. 6., it can be observed that, the maximum classification accuracy (92.85%) is obtained for C ¼ 2:34  107 and g ¼ 0:5. Therefore, as already mentioned earlier, these values had been finally chosen in this paper.

7

4x10

C

Fault type Fig. 5. Fault identification accuracy for different fault types.

7

3x10

Fault Fault Fault Fault

type type type type

I (%) II (%) III (%) IV (%)

Table 2 summarized the identification accuracy of three different mother wavelet conditions using both the BPN and hybrid SVM algorithms. From this table, it is observed that among different mother wavelets, Dmeyer gives the best accuracy, and, therefore, it has been used as the mother wavelet function for feature extraction in this research. Furthermore, the results show that the accuracy of hybrid SVM is better than those of BPN and hybrid SVM spent less time than BPN in classification. So we can use this procedure for fault classification of rotating machinery. 5. Conclusion The aim of this paper is to intelligent diagnosis the fault type on rotating machinery accurately and quickly. WPA is a well-known signal processing technique for fault detection and identification. In this paper, a 1-v-r multi-class support vector machine is presented. This paper describes a new approach using WPA for extraction of features from vibration signals of the rotating machinery in time-frequency domain and hybrid SVM to classify the patterns inherent in the features extracted through the WPA of different fault types. The results show the applicability and effectiveness of this method to detect the fault in the rotating machinery. Acknowledgement The work was supported by the South China Normal University, Republic of China.

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