# Research on Feature Extraction of Performance Degradation for Flexible Material R2R Processing Roller Based on PCA

Authors: Yaohua Deng; Huiqiao Zhou; Kexing Yao; Zhiqi Huang; Chengwang Guo
Journal: International Journal of Rotating Machinery
 (2020)

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## Abstract

Performance feature extraction is the primary problem in equipment performance degradation assessment. To handle the problem of high-dimensional performance characterization and complexity of calculating the performance indicators in flexible material roll-to-roll processing, this paper proposes a PCA method for extracting the degradation characteristic of roll shaft. Based on the analysis of the performance influencing factors of flexible material roll-to-roll processing roller, a principal component analysis extraction model was constructed. The original feature parameter matrix composed of 10-dimensional feature parameters such as time domain, frequency domain, and time-frequency domain vibration signal of the roll shaft was established; then, we obtained a new feature parameter matrixZorg∗ by normalizing the original feature parameter matrix. The correlation measure between every two parameters in the matrix Zorg∗ was used as the eigenvalue to establish the covariance matrix of the performance degradation feature parameters. The Jacobi iteration method was introduced to derive the algorithm for solving eigenvalue and eigenvector of the covariance matrix. Finally, using the eigenvalue cumulative contribution rate as the screening rule, we linearly weighted and fused the eigenvectors and derived the feature principal component matrix F of the processing roller vibration signal. Experiments showed that the initially obtained, 10-dimensional features of the processing rollers’ vibration signals, such as average, root mean square, kurtosis index, centroid frequency, root mean square of frequency, standard deviation of frequency, and energy of the intrinsic mode function component, can be expressed by 3-dimensional principal components F1,F2, and F3. The vibration signal features reduction dimension was realized, and F1,F2, and F3 contain 98.9% of the original vibration signal data, further illustrating that the method has high precision in feature parameters’ extraction and the advantage of eliminating the correlation between feature parameters and reducing the workload selecting feature parameters.


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## Body

## 1. Introduction

The object of flexible material roll-to-roll processing is flexible electronic film material; once the core components of processing equipment begin to degrade, flexible electronic film products will be deformed to varying degrees, resulting in processing quality problems [1, 2]. Therefore, it is necessary to predict the performance and health status of the core components of flexible material R2R processing equipment. However, the number of rollers of the flexible material R2R processing equipment is large, and there are correlations between the movement state variables of the collected rollers, which lead to complications in the process of extracting performance degradation feature. If the performance degradation feature parameters of each roller are analyzed separately, the results are often isolated; if the parameters are blindly reduced, it may lead to incorrect performance state prediction conclusions due to the loss of too much raw information. Thus, it is extremely necessary to fuse and reduce the dimensions of the feature data for roll shaft performance degradation prediction and extract representative feature [3].Previous studies have shown that principal component analysis (PCA), based on the idea of spatial transformation, achieves the purpose of optimal variance without reducing the information content contained in the original data and describes the high-dimensional data information with less principal component information, which has incomparable advantages over other algorithms [4–6]. In reference [4] (2018), spearman grade correlation coefficient and PCA were used for feature fusion to obtain the health index representing the declining state of rolling bearing performance. The analysis results of an example show that the proposed method can accurately identify the declining state of the rolling bearing performance. In reference [5] (2018), a frequency analysis method based on the residuals generated by SOM-PCA algorithm is proposed. This method is effective for fault detection and diagnosis of most of the unsupervised classification experimental data sets. In reference [6] (2018) proposed a sensor fault detection method of water chiller sensor based on empirical mode decomposition (EMD) threshold denoising and principal component analysis (EMD-TD-PCA). PCA model is established by EMD threshold denoising data. Q statistic is used to detect sensor fault. Compared with the traditional PCA method, EMD-TD-PCA method can effectively improve the efficiency of fault detection. Therefore, on the basis of previous research, this paper proposes a PCA method for extracting the degradation feature of R2R roll shaft of flexible material, constructs the principal component analysis extraction model of the degradation feature of processing roll shaft, establishes the feature matrix of the original vibration signal of roll shaft and the covariance matrix of the degradation feature, and introduces Jacobi method. The eigenvalues and eigenvectors of the covariance matrix, as well as the principal component matrix algorithm of the vibration signal of the processing roll, are studied. Finally, the validity and effectiveness of the proposed method for extracting the performance degradation eigenvalues are verified by R2R processing experiments.


## 2. Flexible Material R2R Processing Roller Performance Degradation Feature Extraction Based on PCA

The principle framework of flexible material R2R performance degradation feature extraction is shown in Figure1. As shown in Figure 1, the vibration acceleration data of roller axle during R2R processing were collected by sensors. After filtering and denoising the vibration acceleration signal data, the eigenvalues of the vibration acceleration data in the time domain, frequency domain, and time-frequency domain were calculated by integrating mathematical statistics, power spectrum, and empirical mode decomposition (EMD) [7–9]. On this basis, the eigenvalue normalized feature matrix and the covariance matrix of eigenvalue were established. Finally, using the eigenvalue cumulative contribution rate as the screening rule, we linearly weighted and fused the eigenvectors, then obtained the degradation feature parameters principal component of processing roll shaft performance, i.e., the R2R processing roller shaft performance degradation indicator characterizing amount.Figure 1
R2R processing roller shaft performance degradation feature extraction model framework.


## 3. Algorithm Derivation of PCA Extraction Model of Processing Roller’s Performance Degradation Feature

Considering the correlations between the vibration signal features of the unwinding roller, the guide roller, the driving roller, and the rewinding roller of R2R processing equipment, the principal component analysis (PCA) method is introduced to study the weighted fusion and dimensionality reduction of processing roller vibration feature data, i.e., the extraction of the feature parameters of roller performance degradation [10].


### 3.1. Establishing the Matrix of Original Vibration Signal of Feature Parameters

Time-domain analysis and frequency-domain analysis are easy to calculate, and physical meaning of their feature parameters is clear. The time-frequency eigenvalues of the original vibration signal of R2R roller are listed as follows. The original vibration signal of roller in the flexible material processing system is represented byx. The sample length is L, the sliding window width is M, and the sample mean value of smooth filter output is yL. According to the definition of sliding average filtering [11], yL can be obtained as:(1)yL=xL+xL−1+xL−2+⋯+xL−M+1M.Based on the data after smoothing filtering, matrices of primitive feature parameters of processing roll axis vibration signal that were analyzed in the time domain, frequency domain, and time-frequency domain can be established.We postulated the number of samples isn, and feature parameter value of roller vibration signal is Zij (Zij represents the No j feature variable value of the No i sample data), so the feature parameter matrix of roller vibration signal was expressed as follows:(2)Z=z11z12⋯z1mz21zij⋯z2m⋮⋮⋯⋮zn1zn2⋯znm=Z1,Z2,⋯,Zm.In equation (2), Z=Z1,Z2,⋯,Zm is a simplified expression of the feature parameter matrix, and its No j column vectors of different feature parameters are Zj=z1j,z2j,⋯,znjT.Furthermore, the mean value, root mean square, and kurtosis indices of roller vibration signals are expressed byY¯, YRMS, and KV, respectively. The frequency of barycenter, root mean square, and standard deviation of frequency related to the power spectrum of roller vibration signals are expressed by FC, RMSF, and RVF, respectively [12]. The energy of intrinsic mode function components of vibration signals after EMD is represented by EIMF1, EIMF2, EIMF3, and EIMF4 [13]. Take n roller vibration signals and with Y¯, YRMS, KV, FC, RMSF, RVF, EIMF1, EIMF2, EIMF3, and EIMF4 as parameters in equation (2), the original feature parameter matrix of the roller shafts’ vibration signal Zorg can be obtained as follows:(3)Zorg=Y¯1YRMS1KV1FC1RMSF1RVF1EIMF11EIMF21EIMF31EIMF41Y¯2YRMS2KV2FC2RMSF2RVF2EIMF12EIMF22EIMF32EIMF42⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯nYRMSnKVnFCnRMSFnRVFnEIMF1nEIMF2nEIMF3nEIMF4n.In equation (3), the center of gravity frequency Fc=∫0+∞fsfdf/∫0+∞sfdf, root mean square frequency RMSF=∫0+∞f2sfdf/∫0+∞sfdf, and frequency standard deviation RVF=∫0+∞f−Fc2sfdf/∫0+∞sfdf, where f is the vibration frequency and sf is the power spectrum of the vibration signal.


### 3.2. Normalization of Feature Parameter Matrix

Since the parameters of feature parameter matrixZorg have different dimension and dimension units, in order to eliminate the influence of dimension between indexes on the accuracy of data analysis results, it is necessary to normalize feature parameter matrix so that each index is in the same order of magnitude [14].Forn samples collected, zkj is an element of the feature parameter matrix Z; then, the mean value of feature parameters in column No j of feature parameter matrix Z is obtained by Z¯j=EZj=∑k=1nzkj/n. For normalization of feature matrix Z, the variable zkj of No j column and No k row in feature matrix Z is divided by the mean value z¯j of the feature parameter in No j column in turn. That is, the normalized value zkj∗ of zkj is calculated by the following equation (4):(4)zkj∗=zkjEZj=zkj∑k=1nzkj/n,j=1,2,⋯,m.According to the above normalization method, the feature parameter matrixZ represented by equation (2) is normalized to Z1 as follows:(5)Z1=z11∗z12∗⋯z1m∗z21∗z22∗⋯z2m∗⋮⋮⋯⋮zn1∗zn2∗⋯znm∗=Z1∗,Z2∗,⋯,Zm∗.Equations (4) and (5) are used to normalize the original feature parameter matrix Zorg expressed by equation (3). Equation (4) is used to obtain the average value, root mean square, kurtosis index, center of gravity frequency, root mean square of frequency, standard deviation of frequency, and the energy of intrinsic mode function components of roller vibration signal processed by flexible material R2R system: Y¯j∗, YRMSj∗, KVj∗, FCj∗, RMSFj∗, and RVFj∗ and EIMF1j∗, EIMF2j∗, EIMF3j∗, and EIMF4j∗. Equation (5) is used to obtain the normalized matrix Zorg∗ of the original feature parameter matrix Zorg of the vibration signal of R2R processing roll for flexible material is expressed as equation (6).(6)Zorg∗=Y¯1∗YRMS1∗KV1∗FC1∗RMSF1∗RVF1∗EIMF11∗EIMF21∗EIMF31∗EIMF41∗Y¯2∗YRMS2∗KV2∗FC2∗RMSF2∗RVF2∗EIMF12∗EIMF22∗EIMF32∗EIMF42∗⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯n∗YRMSn∗KVn∗FCn∗RMSFn∗RVFn∗EIMF1n∗EIMF2n∗EIMF3n∗EIMF4n∗.Based on equation (6), it can be seen in Table 1 that each column vector in the feature parameter matrix Zorg∗ represents one of the eigenvectors of the vibration signal of R2R processing roll for flexible material.Table 1
Feature vector table of roll vibration signal for flexible material R2R system.
NumEigenvector typeEigenvector expression1Average value of vibration signalY¯∗=Y¯1∗,Y¯2∗,⋯,Y¯n∗T2RMS of vibration signalYRMS∗=YRMS1∗,YRMS2∗,⋯,YRMS1n∗T3Kurtosis of vibration signalKV∗=KV1∗,KV2∗,⋯,KVn∗T4FC (frequency domain parameters)FC∗=FC1∗,FC2∗,⋯,FCn∗T5RMSF (frequency domain parameters)RMSF∗=RMSFC1∗,RMSFC2∗,⋯,RMSFCn∗T6RVF (frequency domain parameters)RVF∗=RVF1∗,RVF2∗,⋯,RVFn∗T7The energy of IMF1EIMF1∗=EIMF11∗,EIMF12∗,⋯,EIMF1n∗T8The energy of IMF2EIMF2∗=EIMF21∗,EIMF22∗,⋯,EIMF2n∗T9The energy of IMF3EIMF3∗=EIMF31∗,EIMF32∗,⋯,EIMF3n∗T10The energy of IMF4EIMF4∗=EIMF41∗,EIMF42∗,⋯,EIMF4n∗T


### 3.3. Construction of Covariance Matrix of Feature Parameters and Screening of Feature Values

The feature parameter matrix of vibration signal of R2R processing roll for flexible material has been deduced above. However, in order to obtain the principal component variable of the feature parameters of the vibration signal of the processing roller, it is necessary to construct the feature quantity covariance matrix ofZorg∗ and screen the feature values.According to the definition of the covariance matrix, the correlation covariance matrix between two parameters in feature parameter matrixZorg∗ can be expressed as CZ of equation (7):(7)CZ=covY¯∗,Y¯∗⋯covY¯∗,EIMF4∗⋮⋱⋮covEIMF4∗,Y¯∗⋯covEIMF4∗,EIMF4∗.Because the covariance matrix is a symmetric matrix and the diagonal line of the matrix is the variance of each dimension, it not only reflects the correlation measure between all feature dimensions but also reflects the degree of data redundancy. The larger the value on the diagonal line of the matrix, the variables are more important. Conversely, the smaller the value, the smaller the corresponding variable is the secondary variable of the noise signal [15–17].Next, the covariance matrixCZ is transformed into a diagonal matrix by Jacobi iteration method [18], and then, all eigenvalues are solved. Assuming that the eigenvalue of the covariance matrix is λii=1,2,⋯,m, it is regarded as a diagonal matrix composed of principal diagonal elements, and Λ is used to represent the orthogonal matrix composed of the eigenvectors corresponding to eigenvalues. According to the property of symmetric matrix RTCZR=Λ, thus the problem of finding eigenvalues of covariance matrix CZ is transformed into solving orthogonal matrix R [19]. Furthermore, according to the iteration principle of Jacobi’s method, the orthogonal transformation of covariance matrix can be realized by constructing a series of planar rotating orthogonal matrices so that the proportion of diagonal elements of the covariance matrix CZ increases step by step, while the nondiagonal elements become smaller. When the nondiagonal elements approach zero, the principal diagonal elements can be approximated as matrices. Eigenvalues of CZ and expressions of matrix Λ obtained from diagonalization of covariance matrix CZ are as follows:(8)Λ=λ10⋯00λ2⋯0⋮⋮⋱000⋯λm.In order to solve the diagonal matrixΛ of equation (8), it is necessary to construct a rotating orthogonal matrix to transform covariance matrix CZ orthogonally. The rotation angle of the covariance matrix [20] is θ, and the subscripts of the absolute maximum element of covariance matrix CZ on nonprincipal diagonal line are expressed by p and q, respectively. The rotating orthogonal matrix Rpqθ is constructed:(9)Rpqθ=1⋱pth→cosθsinθ⋱qth→−sinθcosθ⋱↑pth↑qth1.Further, if the initial value of the covariance matrix isCZ0 and the first rotating orthogonal matrix Rpq1θ is constructed, the covariance matrix CZ is transformed into CZ1 by the first orthogonal transformation.(10)CZ1=Rpq1TθCZ0Rpq1θ.The element values of No j column and No i row of matrixCZ1 and CZ0 are represented by c1ij and c0ij, respectively. Equation (9) is substituted in equation (10) and expanded to obtain as equation (11)(11)c1ip=c1pi=c0picosθ−c0qisinθ,i≠p,q,c1iq=c1qi=c0pisinθ+c0qicosθ,i≠p,q,c1pp=c0ppcos2θ+c0qqsin2θ−c0pqsin2θ,c1qq=c0ppsin2θ+c0qqcos2θ+c0pqsin2θ,c1pq=c1qp=c0pqcos2θ+c0pp−c0qq2sin2θ.Furthermore, to reduce nondiagonal element component, setc1pq=c1qp=0, introduce the variables s and t, and assume s=c0qq−c0pp/2c0pq,t=tanθ. According to trigonometric function t=t2+2ts−1=0,s≠01,s=0, g is used to denote cosθ=1/1+t2=g, and h is used to represent sinθ=t/1+t2=h; then, equation (11) is simplified as [21] equation (12). Equation (12) is the matrix after the first orthogonal transformation.(12)c1ip=c1pi=gc0pi−hc0qi,i≠p,q,c1iq=c1qi=hc0pi+gc0qi,i≠p,q,c1pp=c0pp−tc0pq,c1qq=c0qq+tc0pq,c1pq=c1qp=0.Further, iteratively calculatesCZ1, according to ck−1pq=maxi≠jck−1ij, the subscript of the largest element of absolute value on nonmain diagonal of the rotation matrix Rpqk−1θ is selected as the p and q values of rotation matrix Rpqkθ. The covariance matrix CZ is obtained after k times of orthogonal transformations equation (13).(13)CZk=RpqkTθCZk−1Rpqkθ=RpqkTRpqk−1T⋯Rpq1TCZ0Rpq1θ⋯Rpqk−1θRpqkθ.If the form of the covariance matrixCZk is consistent with the form of the symmetric matrix shown in equation (8), terminate iterating. CZk is obtained as equation (14).(14)CZk=λ10⋯00λ2⋯0⋮⋮⋱000⋯λk.According to the properties of symmetric matrices, we further derive from equation (14) and have equation (15):(15)CZk=Λ=RTCZR.In equation (15), the relationship between the orthogonal matrix R and rotating orthogonal matrix Rpqkθ of equation (13) is R=Rpq1θ⋯Rpqkθ⋯Rpqnθ, and the principal diagonal element of CZk is all the eigenvalues of the covariance matrix CZ.In order to better screen eigenvalues of covariance matrixCZ, it is necessary to calculate the cumulative contribution rate of eigenvalues. Let λi denote eigenvalue of covariance matrix CZ, the cumulative sum of the first eigenvalues is ∑i=1lλi, and cumulative sum of all eigenvalues is ∑i=1mλi,l<m. Then, the principal component value (or variance ratio) of single eigenvalue is calculated by λi/∑i=1mλi. The cumulative contribution rate η of eigenvalues is calculated by the following equation (16). The constant ε=0.85 is introduced as the screening condition in this paper, and the principal components are retained only when η≥ε is satisfied, which is used for subsequent weighted fusion calculation of eigenvectors [4].(16)η=∑i=1lλi∑i=1mλi,l<m.


### 3.4. Derivation of Principal Component Selection Algorithms for Feature Matrix

The normalization of the original feature parameter matrix and screening method of covariance matrix eigenvalue of vibration signal are discussed before. The algorithm of selecting principal components of the feature matrix of the vibration signal of processing roll is deduced below.The eigenvector matrix consisting of eigenvectors corresponding to selected eigenvalues is represented byU. The eigenvector matrix is expressed as follows when Ui=u1i,u2i,⋯,uriT is used to represent the eigenvectors of No i column (r is used to represent the number of eigenvalues contained in each column of eigenvectors) shown as equation (17).(17)U=U1,U2,⋯,Ul=u11u12⋯u1lu21u22⋯u2l⋮⋮⋯⋮ur1ur2⋯url.Then, the normalized matrixZorg∗ of original feature parameters shown in equation (6) and the eigenvector matrix U are calculated by linear weighting fusion, and feature principal component matrix F of the vibration signal is obtained as equation (18).Thus, the first principal component, the second principal component,…, the Nol principal component of the vibration signal features F1,F2,⋯,Fl can be obtained, which is calculated as equation (19).To this point, the principal component extraction method of vibration signal features of R2R processing roll for flexible material has been deduced. The validity of the above aspects is verified by experiments.(18)F=Zorg∗•U⇒Y¯1∗YRMS1∗KV1∗FC1∗RMSF1∗RVF1∗EIMF11∗EIMF21∗EIMF31∗EIMF41∗Y¯∗2YRMS2∗KV2∗FC2∗RMSF2∗RVF2∗EIMF12∗EIMF22∗EIMF32∗EIMF42∗⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯n∗YRMSn∗KVn∗FCn∗RMSFn∗RVFn∗EIMF1n∗EIMF2n∗EIMF3n∗EIMF4n∗•u11u12⋯u1lu21u22⋯u2l⋮⋮⋯⋮ur1ur2⋯url,(19)F1=u11Y¯∗1+u21YRMS1∗+⋯+ur1EIMF41∗,F2=u12Y¯∗2+u22YRMS2∗+⋯+ur2EIMF42∗,⋮Fl=u1lY¯∗n+u2lYRMS∗n+⋯+urlEIMF4n∗.


## 3.1. Establishing the Matrix of Original Vibration Signal of Feature Parameters

Time-domain analysis and frequency-domain analysis are easy to calculate, and physical meaning of their feature parameters is clear. The time-frequency eigenvalues of the original vibration signal of R2R roller are listed as follows. The original vibration signal of roller in the flexible material processing system is represented byx. The sample length is L, the sliding window width is M, and the sample mean value of smooth filter output is yL. According to the definition of sliding average filtering [11], yL can be obtained as:(1)yL=xL+xL−1+xL−2+⋯+xL−M+1M.Based on the data after smoothing filtering, matrices of primitive feature parameters of processing roll axis vibration signal that were analyzed in the time domain, frequency domain, and time-frequency domain can be established.We postulated the number of samples isn, and feature parameter value of roller vibration signal is Zij (Zij represents the No j feature variable value of the No i sample data), so the feature parameter matrix of roller vibration signal was expressed as follows:(2)Z=z11z12⋯z1mz21zij⋯z2m⋮⋮⋯⋮zn1zn2⋯znm=Z1,Z2,⋯,Zm.In equation (2), Z=Z1,Z2,⋯,Zm is a simplified expression of the feature parameter matrix, and its No j column vectors of different feature parameters are Zj=z1j,z2j,⋯,znjT.Furthermore, the mean value, root mean square, and kurtosis indices of roller vibration signals are expressed byY¯, YRMS, and KV, respectively. The frequency of barycenter, root mean square, and standard deviation of frequency related to the power spectrum of roller vibration signals are expressed by FC, RMSF, and RVF, respectively [12]. The energy of intrinsic mode function components of vibration signals after EMD is represented by EIMF1, EIMF2, EIMF3, and EIMF4 [13]. Take n roller vibration signals and with Y¯, YRMS, KV, FC, RMSF, RVF, EIMF1, EIMF2, EIMF3, and EIMF4 as parameters in equation (2), the original feature parameter matrix of the roller shafts’ vibration signal Zorg can be obtained as follows:(3)Zorg=Y¯1YRMS1KV1FC1RMSF1RVF1EIMF11EIMF21EIMF31EIMF41Y¯2YRMS2KV2FC2RMSF2RVF2EIMF12EIMF22EIMF32EIMF42⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯nYRMSnKVnFCnRMSFnRVFnEIMF1nEIMF2nEIMF3nEIMF4n.In equation (3), the center of gravity frequency Fc=∫0+∞fsfdf/∫0+∞sfdf, root mean square frequency RMSF=∫0+∞f2sfdf/∫0+∞sfdf, and frequency standard deviation RVF=∫0+∞f−Fc2sfdf/∫0+∞sfdf, where f is the vibration frequency and sf is the power spectrum of the vibration signal.


## 3.2. Normalization of Feature Parameter Matrix

Since the parameters of feature parameter matrixZorg have different dimension and dimension units, in order to eliminate the influence of dimension between indexes on the accuracy of data analysis results, it is necessary to normalize feature parameter matrix so that each index is in the same order of magnitude [14].Forn samples collected, zkj is an element of the feature parameter matrix Z; then, the mean value of feature parameters in column No j of feature parameter matrix Z is obtained by Z¯j=EZj=∑k=1nzkj/n. For normalization of feature matrix Z, the variable zkj of No j column and No k row in feature matrix Z is divided by the mean value z¯j of the feature parameter in No j column in turn. That is, the normalized value zkj∗ of zkj is calculated by the following equation (4):(4)zkj∗=zkjEZj=zkj∑k=1nzkj/n,j=1,2,⋯,m.According to the above normalization method, the feature parameter matrixZ represented by equation (2) is normalized to Z1 as follows:(5)Z1=z11∗z12∗⋯z1m∗z21∗z22∗⋯z2m∗⋮⋮⋯⋮zn1∗zn2∗⋯znm∗=Z1∗,Z2∗,⋯,Zm∗.Equations (4) and (5) are used to normalize the original feature parameter matrix Zorg expressed by equation (3). Equation (4) is used to obtain the average value, root mean square, kurtosis index, center of gravity frequency, root mean square of frequency, standard deviation of frequency, and the energy of intrinsic mode function components of roller vibration signal processed by flexible material R2R system: Y¯j∗, YRMSj∗, KVj∗, FCj∗, RMSFj∗, and RVFj∗ and EIMF1j∗, EIMF2j∗, EIMF3j∗, and EIMF4j∗. Equation (5) is used to obtain the normalized matrix Zorg∗ of the original feature parameter matrix Zorg of the vibration signal of R2R processing roll for flexible material is expressed as equation (6).(6)Zorg∗=Y¯1∗YRMS1∗KV1∗FC1∗RMSF1∗RVF1∗EIMF11∗EIMF21∗EIMF31∗EIMF41∗Y¯2∗YRMS2∗KV2∗FC2∗RMSF2∗RVF2∗EIMF12∗EIMF22∗EIMF32∗EIMF42∗⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯n∗YRMSn∗KVn∗FCn∗RMSFn∗RVFn∗EIMF1n∗EIMF2n∗EIMF3n∗EIMF4n∗.Based on equation (6), it can be seen in Table 1 that each column vector in the feature parameter matrix Zorg∗ represents one of the eigenvectors of the vibration signal of R2R processing roll for flexible material.Table 1
Feature vector table of roll vibration signal for flexible material R2R system.
NumEigenvector typeEigenvector expression1Average value of vibration signalY¯∗=Y¯1∗,Y¯2∗,⋯,Y¯n∗T2RMS of vibration signalYRMS∗=YRMS1∗,YRMS2∗,⋯,YRMS1n∗T3Kurtosis of vibration signalKV∗=KV1∗,KV2∗,⋯,KVn∗T4FC (frequency domain parameters)FC∗=FC1∗,FC2∗,⋯,FCn∗T5RMSF (frequency domain parameters)RMSF∗=RMSFC1∗,RMSFC2∗,⋯,RMSFCn∗T6RVF (frequency domain parameters)RVF∗=RVF1∗,RVF2∗,⋯,RVFn∗T7The energy of IMF1EIMF1∗=EIMF11∗,EIMF12∗,⋯,EIMF1n∗T8The energy of IMF2EIMF2∗=EIMF21∗,EIMF22∗,⋯,EIMF2n∗T9The energy of IMF3EIMF3∗=EIMF31∗,EIMF32∗,⋯,EIMF3n∗T10The energy of IMF4EIMF4∗=EIMF41∗,EIMF42∗,⋯,EIMF4n∗T


## 3.3. Construction of Covariance Matrix of Feature Parameters and Screening of Feature Values

The feature parameter matrix of vibration signal of R2R processing roll for flexible material has been deduced above. However, in order to obtain the principal component variable of the feature parameters of the vibration signal of the processing roller, it is necessary to construct the feature quantity covariance matrix ofZorg∗ and screen the feature values.According to the definition of the covariance matrix, the correlation covariance matrix between two parameters in feature parameter matrixZorg∗ can be expressed as CZ of equation (7):(7)CZ=covY¯∗,Y¯∗⋯covY¯∗,EIMF4∗⋮⋱⋮covEIMF4∗,Y¯∗⋯covEIMF4∗,EIMF4∗.Because the covariance matrix is a symmetric matrix and the diagonal line of the matrix is the variance of each dimension, it not only reflects the correlation measure between all feature dimensions but also reflects the degree of data redundancy. The larger the value on the diagonal line of the matrix, the variables are more important. Conversely, the smaller the value, the smaller the corresponding variable is the secondary variable of the noise signal [15–17].Next, the covariance matrixCZ is transformed into a diagonal matrix by Jacobi iteration method [18], and then, all eigenvalues are solved. Assuming that the eigenvalue of the covariance matrix is λii=1,2,⋯,m, it is regarded as a diagonal matrix composed of principal diagonal elements, and Λ is used to represent the orthogonal matrix composed of the eigenvectors corresponding to eigenvalues. According to the property of symmetric matrix RTCZR=Λ, thus the problem of finding eigenvalues of covariance matrix CZ is transformed into solving orthogonal matrix R [19]. Furthermore, according to the iteration principle of Jacobi’s method, the orthogonal transformation of covariance matrix can be realized by constructing a series of planar rotating orthogonal matrices so that the proportion of diagonal elements of the covariance matrix CZ increases step by step, while the nondiagonal elements become smaller. When the nondiagonal elements approach zero, the principal diagonal elements can be approximated as matrices. Eigenvalues of CZ and expressions of matrix Λ obtained from diagonalization of covariance matrix CZ are as follows:(8)Λ=λ10⋯00λ2⋯0⋮⋮⋱000⋯λm.In order to solve the diagonal matrixΛ of equation (8), it is necessary to construct a rotating orthogonal matrix to transform covariance matrix CZ orthogonally. The rotation angle of the covariance matrix [20] is θ, and the subscripts of the absolute maximum element of covariance matrix CZ on nonprincipal diagonal line are expressed by p and q, respectively. The rotating orthogonal matrix Rpqθ is constructed:(9)Rpqθ=1⋱pth→cosθsinθ⋱qth→−sinθcosθ⋱↑pth↑qth1.Further, if the initial value of the covariance matrix isCZ0 and the first rotating orthogonal matrix Rpq1θ is constructed, the covariance matrix CZ is transformed into CZ1 by the first orthogonal transformation.(10)CZ1=Rpq1TθCZ0Rpq1θ.The element values of No j column and No i row of matrixCZ1 and CZ0 are represented by c1ij and c0ij, respectively. Equation (9) is substituted in equation (10) and expanded to obtain as equation (11)(11)c1ip=c1pi=c0picosθ−c0qisinθ,i≠p,q,c1iq=c1qi=c0pisinθ+c0qicosθ,i≠p,q,c1pp=c0ppcos2θ+c0qqsin2θ−c0pqsin2θ,c1qq=c0ppsin2θ+c0qqcos2θ+c0pqsin2θ,c1pq=c1qp=c0pqcos2θ+c0pp−c0qq2sin2θ.Furthermore, to reduce nondiagonal element component, setc1pq=c1qp=0, introduce the variables s and t, and assume s=c0qq−c0pp/2c0pq,t=tanθ. According to trigonometric function t=t2+2ts−1=0,s≠01,s=0, g is used to denote cosθ=1/1+t2=g, and h is used to represent sinθ=t/1+t2=h; then, equation (11) is simplified as [21] equation (12). Equation (12) is the matrix after the first orthogonal transformation.(12)c1ip=c1pi=gc0pi−hc0qi,i≠p,q,c1iq=c1qi=hc0pi+gc0qi,i≠p,q,c1pp=c0pp−tc0pq,c1qq=c0qq+tc0pq,c1pq=c1qp=0.Further, iteratively calculatesCZ1, according to ck−1pq=maxi≠jck−1ij, the subscript of the largest element of absolute value on nonmain diagonal of the rotation matrix Rpqk−1θ is selected as the p and q values of rotation matrix Rpqkθ. The covariance matrix CZ is obtained after k times of orthogonal transformations equation (13).(13)CZk=RpqkTθCZk−1Rpqkθ=RpqkTRpqk−1T⋯Rpq1TCZ0Rpq1θ⋯Rpqk−1θRpqkθ.If the form of the covariance matrixCZk is consistent with the form of the symmetric matrix shown in equation (8), terminate iterating. CZk is obtained as equation (14).(14)CZk=λ10⋯00λ2⋯0⋮⋮⋱000⋯λk.According to the properties of symmetric matrices, we further derive from equation (14) and have equation (15):(15)CZk=Λ=RTCZR.In equation (15), the relationship between the orthogonal matrix R and rotating orthogonal matrix Rpqkθ of equation (13) is R=Rpq1θ⋯Rpqkθ⋯Rpqnθ, and the principal diagonal element of CZk is all the eigenvalues of the covariance matrix CZ.In order to better screen eigenvalues of covariance matrixCZ, it is necessary to calculate the cumulative contribution rate of eigenvalues. Let λi denote eigenvalue of covariance matrix CZ, the cumulative sum of the first eigenvalues is ∑i=1lλi, and cumulative sum of all eigenvalues is ∑i=1mλi,l<m. Then, the principal component value (or variance ratio) of single eigenvalue is calculated by λi/∑i=1mλi. The cumulative contribution rate η of eigenvalues is calculated by the following equation (16). The constant ε=0.85 is introduced as the screening condition in this paper, and the principal components are retained only when η≥ε is satisfied, which is used for subsequent weighted fusion calculation of eigenvectors [4].(16)η=∑i=1lλi∑i=1mλi,l<m.


## 3.4. Derivation of Principal Component Selection Algorithms for Feature Matrix

The normalization of the original feature parameter matrix and screening method of covariance matrix eigenvalue of vibration signal are discussed before. The algorithm of selecting principal components of the feature matrix of the vibration signal of processing roll is deduced below.The eigenvector matrix consisting of eigenvectors corresponding to selected eigenvalues is represented byU. The eigenvector matrix is expressed as follows when Ui=u1i,u2i,⋯,uriT is used to represent the eigenvectors of No i column (r is used to represent the number of eigenvalues contained in each column of eigenvectors) shown as equation (17).(17)U=U1,U2,⋯,Ul=u11u12⋯u1lu21u22⋯u2l⋮⋮⋯⋮ur1ur2⋯url.Then, the normalized matrixZorg∗ of original feature parameters shown in equation (6) and the eigenvector matrix U are calculated by linear weighting fusion, and feature principal component matrix F of the vibration signal is obtained as equation (18).Thus, the first principal component, the second principal component,…, the Nol principal component of the vibration signal features F1,F2,⋯,Fl can be obtained, which is calculated as equation (19).To this point, the principal component extraction method of vibration signal features of R2R processing roll for flexible material has been deduced. The validity of the above aspects is verified by experiments.(18)F=Zorg∗•U⇒Y¯1∗YRMS1∗KV1∗FC1∗RMSF1∗RVF1∗EIMF11∗EIMF21∗EIMF31∗EIMF41∗Y¯∗2YRMS2∗KV2∗FC2∗RMSF2∗RVF2∗EIMF12∗EIMF22∗EIMF32∗EIMF42∗⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯n∗YRMSn∗KVn∗FCn∗RMSFn∗RVFn∗EIMF1n∗EIMF2n∗EIMF3n∗EIMF4n∗•u11u12⋯u1lu21u22⋯u2l⋮⋮⋯⋮ur1ur2⋯url,(19)F1=u11Y¯∗1+u21YRMS1∗+⋯+ur1EIMF41∗,F2=u12Y¯∗2+u22YRMS2∗+⋯+ur2EIMF42∗,⋮Fl=u1lY¯∗n+u2lYRMS∗n+⋯+urlEIMF4n∗.


## 4. Verification Experiment

In order to verify the validity of the principal component extraction method for vibration signal feature parameters of R2R processing roll for flexible material, an experimental platform for flexible material roll-to-roll processing was built as shown in Figure2. The experimental platform was mainly composed of mechanical structure and driving control unit. The mechanical structure used standard reinforced aluminum alloy profiles as the support frame, including unwinding module, driving module, and rewinding module. The unwinding, rewinding, and driving module all adopt 120 W speed regulating motor; the driving module adopts counter-roll mode, and the roll rotates under the motor driving. The upper roll was controlled by a hand-operated lifting handle. The upper rubber-covered roll should be lifted to make the material go around the upper rubber-covered roll and then flat the material. After the correct alignment, the upper rubber-covered roll should be put down, and the upper and lower roll rotates simultaneously to realize the material transmission. The adaptive maximum width of experimental platform is 450 mm, which can be used to transport material thickness of 0.1-5 mm. Blue PET polyester film with a width of 50 mm, a thickness of 0.05 mm, a density of 1450 kg/m3, an elastic modulus of 3495 MPa, and a Poisson’s ratio of 0.3 was used in the experiment. The magnet adsorption triaxial accelerometer was selected to realize data acquisition. The sensitivity is 20 mv/g, the frequency response is 5-5000 Hz, and the size is 24∗24∗15mm. The adsorption installation method will not destroy the measured object, so it was convenient to move. In addition to the hardware mentioned above, the experimental platform also includes a professional version of the Lenovo desktop computer in Windows 7 and software data processing software MATLAB R2014a.Figure 2
Flexible material roll-to-roll processing equipment.Considering that the driving module is the core of the whole experimental platform as well as the roll and the bearing are closely connected, the vibration of the roll will be fully reflected in the bearing, so the three-axis accelerometer was adsorbed on the bearing seat on one side of the driving roll to realize data acquisition of vibration signal. The sampling frequency is 10 kHz, data acquisition was conducted every 60 minutes, each time for 1 s, and a total of 984 groups of samples were collected. With the gradual progress of the experiment, performance of driving rolls declines in varying degrees.(1) Sample data denoising processing. According to equation (1), sliding average filtering method yL=xL+xL‐1+xL‐2+⋯+xL‐M‐1/M, the sliding window width M was 5, input 984 sets of sample data for calculation. By using the sliding smoothing algorithm, the original sample data was further smoothed and denoised which means the systematic errors and random errors of the sample data were finally eliminated and reduced. Figure 3 is a comparison of the effects before and after noise reduction of partial data of the first group of samples.Figure 3
Moving average filtering effect diagram.(2) Constructed the original feature parameter matrixZorg. Substitute the noise-reduced sample data into the original feature parameter matrix Zorg of flexible material R2R processing roller vibration signal of Section 1 (see equation (3)), shown in the equation below(20)Zorg=Y¯1YRMS1KV1FC1RMSF1RVF1EIMF11EIMF21EIMF31EIMF41Y¯2YRMS2KV2FC2RMSF2RVF2EIMF12EIMF22EIMF32EIMF42⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯nYRMSnKVnFCnRMSFnRVFnEIMF1nEIMF2nEIMF3nEIMF4n.The average value, root mean square, kurtosis index, center of gravity frequency, root mean square of the frequency, standard deviation of frequency, and the energy of intrinsic mode function components of sample data were calculated. The calculation results of the above indexes are given in Figures4–6, respectively, and some calculation results of each feature index are given in Table 2.Figure 4
The calculation results of mean value, root mean square, and kurtosis index.
(a)
Average value(b)
Root mean square(c)
Kurtosis valueFigure 5
Calculations of center of gravity frequency, frequency root mean square, and frequency standard deviation.
(a)
Centre of gravity frequency(b)
RMS of frequency(c)
Frequency standard deviationFigure 6
Calculation results of intrinsic mode function components.
(a)
Energy of IMF1(b)
Energy of IMF2(c)
Energy of IMF3(d)
Energy of IMF4Table 2
Original feature parameter matrix for flexible material R2R system roller vibration signal.
NumY¯YRMSKVFCRMSFRVFEIMF1EIMF2EIMF3EIMF41-0.01010.04482.78991050.31450.93.66481.54567.6518.4664.31432-0.00250.04382.94851090.81500.43.72971.4646.385918.3015.11653-0.00240.04502.72061086.41475.13.74811.61746.509719.8654.85364-0.00220.04562.81541124.91511.93.77372.04478.972119.3974.69975-0.00200.04502.9041083.814653.79341.68429.384218.7355.59276-0.00160.04542.82321099.81474.43.74141.8498.020519.8995.26317-0.00160.04632.75881092.51457.13.78461.55088.907119.2525.99948-0.00200.04512.71631100.21461.93.70181.48528.162220.4174.4579-0.00190.04642.74591097.11459.33.75451.64557.625322.4064.506110-0.00200.04522.71311064.51412.23.59011.28879.37419.0363.845911-0.00170.04612.72011076.21425.33.67591.69329.057820.6794.704812-0.00150.04562.83121085.71439.53.70561.37837.005521.2925.749913-0.00240.04552.70191084.21443.53.63661.42987.812320.8564.897414-0.00100.04582.88151114.114763.74541.83628.80920.0214.920715-0.00230.04642.6731086.41448.93.71691.56787.034522.4175.774116-0.00150.04562.844510931464.73.71091.44366.275321.6395.834217-0.00200.04602.79721072.21415.83.63751.48329.834519.7334.1918-0.00160.04592.73161065.11415.73.64431.55499.430819.6524.685319-0.00270.04672.64661074.41422.43.67131.59059.216920.2134.99420-0.00220.04622.80961086.81440.83.7551.73859.091819.1745.439821-0.00140.04602.72791076.614323.74791.47536.208522.2145.146922-0.00180.04612.81241060.71421.73.71941.5237.176421.7494.908823-0.00180.046212.76991076.61449.63.76431.57596.351121.8325.176624-0.00220.04542.85731087.61469.23.76471.57097.995819.2435.558525-0.00190.04712.79461057.51434.73.79891.62545.817223.3425.864926-0.00210.04522.71711048.31421.43.62661.63517.57818.7113.789727-0.00150.04552.66521068.91428.33.69841.46817.60121.6444.226828-0.00220.04632.73771042.71392.93.63351.37317.217820.9235.39529-0.00230.04682.65931033.11400.73.73001.36398.979921.3674.47630-0.00240.04582.78411063.21438.53.72851.57066.746521.8674.3497……………………………721-0.00270.05303.14911656.52201.34.98168.102614.20222.8928.0045722-0.00210.05283.090816412192.94.96867.750911.63724.2138.4416723-0.00210.05323.1971643.12182.14.94557.838813.5523.998.3561724-0.00210.05443.14981593.52131.65.02327.918612.45226.3666.6946725-0.00220.05383.35121644.62172.74.93718.009112.50124.4898.3043726-0.00260.05403.21071637.62179.15.02317.850112.89424.8379.0401727-0.00210.05343.123916562201.24.93768.842812.42324.1956.4097728-0.00150.05403.17961650.721804.9797.726813.34422.719.5417729-0.00230.05343.10971621.92164.14.95237.717513.51824.3636.6699730-0.00190.05493.2711595.32125.15.0326.847512.8527.1368.3782……………………………965-0.00090.09913.03641487.92097.59.5421.22532.31582.16337.627966-0.00060.12682.81091515.11937.210.97627.71592.465140.5140.1967-0.00190.11622.88571421.41958.310.57627.59928.219161.6527.7968-0.00080.11392.57981555.52121.710.54634.98840.41134.5536.982969-0.00190.11752.41471624.72152.210.74233.98284.17182.75649.921970-0.00060.13463.36941529.62086.013.28439.086113.03126.6256.488971-0.00300.16043.29121541.52029.415.44148.641148.64170.3784.156972-0.00210.17415.62451529.02118.317.82463.5277.414191.18126.03973-0.00220.15373.93131676.32168.114.94457.251190.61137.9462.819974-0.00230.14789.1981712.6227816.21948.64499.23793.87386.746975-0.00240.19313.84361721.92111.317.81972.926400.09178.8464.7399760.00330.319413.9251586.22060.731.139175.56276.13498.11459.64977-0.00520.225712.0911587.22047.122.02388.389245.69171.25209.03978-0.00320.19316.15321688.32100.017.4764.441349.32160.57114.69979-0.00030.14674.33591568.52057.814.03843.37790.334178.9684.912980-0.00270.373713.9551515.81912.732.214177.87832.19504.77552.48981-0.00440.20268.56651535200819.34264.232200.84197.54175.44982-0.00160.235510.8191473.81907.320.92897.985323.58299.29201.229830.00180.00191.451567.2464.930.10350.00030.00080.00100.0009984-0.00110.00141.7458104.61492.940.075410.00020.00040.00060.0010The root mean square and kurtosis indicators in Figure4 have a slight upward trend in the early stage of the fault, and the mutation phenomenon appeared until the 970th sample point. The sensitivity of immediate domain feature to the early fault and medium fault was general, and the fault characterization of late failure phase was relatively strong. In Figure 5, the frequency-domain feature curves of the center of gravity frequency were slightly increased at about 500th sample point and rose sharply at 700 points. Both were in the upper and lower oscillating unstable state in subsequent time period and were significantly mutated at the 970th sample point. The trend of the frequency standard deviation feature curve was like the trend of the time domain feature change. In Figure 6, the energy of different eigenmode functions has a great difference in energy variation when the fault occurs and can characterize the energy distribution of the roller vibration signal in the frequency domain.(3) Normalization of the original feature parameter matrix. Substituting the original feature data of Table 2 into the equation (6) of Section 2, as shown below(21)Zorg∗=Y¯1∗YRMS1∗KV1∗FC1∗RMSF1∗RVF1∗EIMF11∗EIMF21∗EIMF31∗EIMF41∗Y¯∗2YRMS2∗KV2∗FC2∗RMSF2∗RVF2∗EIMF12∗EIMF22∗EIMF32∗EIMF42∗⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Y¯n∗YRMSn∗KVn∗FCn∗RMSFn∗RVFn∗EIMF1n∗EIMF2n∗EIMF3n∗EIMF4n∗.The original feature matrix was normalized and calculated, and the normalized average value, root mean square, kurtosis index, center of gravity frequency, frequency root mean square, frequency standard deviation, and energy of intrinsic mode functions component were obtained, as shown in Table3.Table 3
Normalized feature parameter matrix of roller vibration signal of flexible material R2R processing.
NumY¯YRMSKVFCRMSFRVFEIMF1EIMF2EIMF3EIMF415.247800.851090.953670.879380.910960.834450.351410.500690.657050.4869121.332100.832351.007900.913290.942040.849230.332860.417960.651180.5774431.275900.855430.929990.90960.926160.853430.367740.426060.706830.5477741.170400.867330.962390.941840.949260.859250.464890.587230.690180.530451.237800.866180.992680.907430.919820.863740.382930.61420.666620.6311960.834980.863180.965060.920820.925720.851890.42040.524940.708040.5939970.827870.879670.943040.914710.914860.861730.35260.582970.685020.6770981.045400.857180.928520.921160.917870.842880.337680.534220.726470.5030190.986810.881720.938630.918560.916240.854870.374130.499080.797240.50855101.076700.858090.927420.891270.886670.817440.2930.613530.677330.43404110.924840.875780.929810.901060.894890.836970.384970.592840.735790.53098120.801990.865650.967790.909020.903810.843750.313380.458510.75760.64893131.262000.864410.923590.907760.906320.828020.325090.511320.742090.55272140.530330.87030.984990.93280.926720.852790.417490.576550.712380.55535151.219200.881880.913710.90960.909710.846310.356460.460410.797630.65166160.785260.86730.972340.915130.919630.844940.328220.410720.769950.65844171.038300.873620.956170.897720.888930.828240.337230.643670.702130.47288180.857570.872020.933750.891770.888860.829780.353530.617250.699250.52878191.419700.886960.904690.899560.893070.835920.361620.603250.719210.56362201.162300.877360.960410.909940.904620.854990.395270.595060.682240.61393210.744340.873580.932480.90140.89910.853380.335430.406350.790410.58087220.930860.876940.961370.888090.892630.846880.346280.46970.773860.5540230.930450.877210.946840.90140.910150.85710.35830.415680.776820.58423241.157000.861850.976710.910610.922450.857190.357170.523330.68470.62733251.016600.894880.955280.885410.900790.864990.369560.380740.830550.66191261.118000.858190.928790.87770.892440.825750.371760.495980.665770.4277270.780110.864370.911050.894950.896770.842110.333790.497490.770130.47703281.154400.879940.935830.873020.874550.827330.312190.472410.744470.60887291.211800.888560.909030.864980.879450.849290.31010.587740.760270.50516301.256600.869750.951690.890180.903180.848950.35710.441560.778060.4909……………………………7211.427101.007401.076501.386901.382101.134301.842200.929520.814530.903387221.115401.002901.056501.374001.376801.131301.762300.761640.861540.952717231.091101.010501.092801.375701.370101.126101.782300.886850.85360.943067241.093701.032901.076701.334201.338301.143801.800400.814990.938150.755557251.163101.021401.145501.377001.364201.124201.821000.818190.871360.937227261.336801.025601.097501.371101.368201.143701.784800.843920.883741.02037271.101501.014201.067801.386501.382001.124302.010500.813090.86090.723397280.807701.025201.086901.382101.368701.133701.756800.873370.808061.07697291.183801.015101.063001.358001.358801.127601.754700.884760.866880.752767301.01001.025301.085401.368901.362301.130101.818900.85050.86280.86240……………………………9650.471031.882201.037901.245801.316902.172204.825802.115002.923504.2465009660.318252.407400.960851.268501.216302.499206.301406.051904.999604.525609671.014402.206900.986421.190101.229502.408106.275001.846905.751803.126209680.425772.163100.881861.302401.332102.401207.955002.644804.787504.173809690.998492.232200.825421.360301.351302.446007.726305.509002.944605.634009700.329122.556401.151801.280701.309703.024708.886807.397804.505306.375209711.562403.045401.125001.290601.274203.5159011.05909.728506.06209.497809721.093103.30601.922601.280201.330004.0583014.44205.066806.8025014.22409731.173002.917601.343801.403501.361303.4027013.017012.47504.908107.089709741.181802.806403.144201.433901.430303.6930011.0606.495103.340209.790109751.256003.667101.313901.441701.325604.0572016.581026.18606.363407.30640976-1.742606.064204.760001.328101.293807.0903039.916018.073017.724051.87509772.700404.285504.133101.328901.285305.0145020.096016.08006.0934023.59109781.645703.667102.103401.413601.318503.9779014.652022.86305.7133012.94409790.189712.785501.482101.313201.292003.196509.862405.912406.360779.583109801.420607.095504.770301.269101.200907.3349040.441054.467017.961062.35209812.310303.847002.928301.285201.260704.4041014.604013.14507.0288019.80009820.864684.471003.698301.234001.197504.7652022.278021.178010.649022.7090983-0.955670.037580.496170.056260.291910.023560.000080.000050.000030.000109840.598070.027240.596770.087580.309500.017170.000050.000030.000020.00010(4) Constructing the covariance matrix of normalized feature parameters. On the basis of normalized processing of Zorg feature parameter matrix, the normalized feature parameter matrix of roller vibration signals processed by R2R flexible material in Table 3 is divided into columns and substituted into Section 3.3 equation (7) as follows(22)CZ=covY¯∗,Y¯∗⋯covY¯∗,EIMF4∗⋮⋱⋮covEIMF4∗,Y¯∗⋯covEIMF4∗,EIMF4∗.The covariance matrix of the feature parameters was calculated.(23)CZ′=0.0664‐0.00220.00030.00090.0006‐0.0031‐0.04450.0278‐0.0283‐0.0557‐0.00220.19510.08790.03150.02490.21501.01560.96260.48951.13910.00030.08790.06100.01350.01190.10210.54130.47740.21880.66960.00090.03150.01350.02100.01880.03840.15150.12090.06100.10940.00060.02490.01190.01880.01760.03230.12980.09380.04660.0913‐0.00310.21500.10210.03840.03230.24321.15291.05340.53431.2796‐0.04451.01560.54130.15150.12981.15295.97545.42082.61126.91050.02780.96260.47740.12090.09381.05345.42086.03862.40586.4036‐0.02830.48950.21880.06100.04660.53432.61122.40581.30603.0370‐0.05571.13910.66960.10940.09131.27966.91056.40363.03708.8900.(5) Solving the eigenvalues and eigenvectors of the covariance matrix. According to ck−1pq=maxi≠jck−1ij in Section 3.3, p, q, and Rpqθ were determined, and the covariance matrix CZ′ was transformed orthogonally according to the equation(24)CZk=RpqkTθCZk−1Rpqkθ=RpqkTRpqk−1T⋯Rpq1TCZ0Rpq1θ⋯Rpqk−1θRpqkθ.In the experiment, the number of orthogonal transformations isk=18. The eigenvalue of the covariance matrix was calculated as follows:(25)CZ18=21.21300000000000.92400000000000.47200000000000.14500000000000.06300000000000.02700000000000.00900000000000.00500000000000.00100000000000.0000936.Furthermore, the eigenvectors corresponding to the 10 eigenvalues on the diagonal line of the above eigenvalue matrixCZ18 were calculated, respectively. Finally, the eigenvectors corresponding to all eigenvalues were obtained as shown in matrix U′.(26)U′=−0.002−0.0690.0530.0710.965−0.2070.1160.001−0.002−0.0050.089−0.040−0.188−0.1770.1290.277−0.3780.2030.7430.3070.0480.0480.0090.1370.0960.191−0.509−0.813−0.0990.0170.011−0.024−0.1150.1020.0700.5750.400−0.1370.265−0.6250.009−0.013−0.1010.1260.0600.5160.406−0.060−0.2590.6840.100−0.019−0.231−0.0540.1530.389−0.4700.457−0.531−0.2150.5210.109−0.6120.510−0.059−0.2690.0570.0050.0470.0030.504−0.8170.267−0.016−0.0680.0180.008−0.024−0.024−0.0030.2310.028−0.421−0.8040.041−0.1030.184−0.246−0.123−0.0210.6330.5560.511−0.0940.0400.1120.0330.0690.011−0.002.Then, the 10 eigenvalues on the diagonal line of the above eigenvalue matrixCZ18 were substituted into λi/∑i=1mλi,m=10 to calculate the variance ratio (principal component value of a single eigenvalue). The cumulative contribution rate of the eigenvalues was calculated by substituting 10 eigenvalues on the diagonal line of the above eigenvalue matrix CZ18 into η=∑i=1lλi/∑i=1mλi,l=1,2,⋯,10,m=10. The calculation results of the variance ratio and the cumulative contribution ratio are shown in Table 4.Table 4
Cumulative variance contribution rate table of roll vibration signals for flexible material R2R system.
Eigenvalue numberEigenvalueVariance ratioAccumulated contribution rateλ121.2130.9300.930λ20.9240.0410.971λ30.4270.0190.989λ40.1450.0060.995λ50.0630.0030.998λ60.0270.0010.999λ70.0090.0001.000λ80.0050.0001.000λ90.0010.0001.000λ100.0000.0001.000From Table4, it can be seen that the variance ratios of eigenvalues λ1,λ2,λ3,λ4,λ5, and λ6 are 0.930, 0.041, 0.019, 0.006, 0.003, and 0.001, respectively, while the variance ratios of eigenvalues λ7,λ8,λ9, and λ10 are close to 0.000, which indicates that λ7,λ8,λ9, and λ10 contain almost no feature information of vibration signals and can be regarded as secondary variables. Similarly, the variance ratio of the eigenvalues λ5 and λ6 is less than 0.005. Compared with the eigenvalues λ1,λ2,λ3, and λ4, it can also be considered as a secondary variable. So λ1,λ2,λ3, and λ4 were selected to calculate the principal component.(6) Constructing PCA principal component performance degradation index and draw three-dimensional contour map. The normalized feature parameter matrix (see Table 3) and the abovementioned eigenvector matrix U′ of vibration signal of the R2R processing roll for flexible material were used to calculate the principal component of vibration signal feature parameters by equation (19), and the results are shown in Table 5 (part)(27)F1=u11Y¯∗1+u21YRMS1∗+⋯+ur1EIMF41∗,F2=u12Y¯∗2+u22YRMS2∗+⋯+ur2EIMF42∗,⋮Fl=u1lY¯∗n+u2lYRMS∗n+⋯+urlEIMF4n∗.Table 5
Principal component calculation of feature parameters of vibration signals (part).
NumF1F2F3F4NumF1F2F3F41-1.038200.24010-0.45960-0.30863701-0.66965-0.0662060.059745-0.170432-1.02110-0.14789-0.28357-0.0355702-0.58086-0.0948480.083001-0.244883-1.0060-0.12953-0.218810.010187030.21294-0.247270.78798-0.753694-0.88494-0.00550-0.18563-0.052537040.32029-0.274050.71352-0.773685-0.85476-0.02788-0.31445-0.021497050.34432-0.289570.70823-0.833576-0.89577-0.11181-0.210730.016967060.33246-0.403240.75217-0.675107-0.85337-0.10144-0.315790.050717070.34225-0.152560.77493-0.750788-0.99127-0.02931-0.224390.055037080.33916-0.340210.70677-0.678159-0.96615-0.07045-0.155790.101557090.2658-0.284430.63715-0.6477810-1.032800.080777-0.272350.036797100.24144-0.32010.66818-0.73402…………………………500-1.03050-0.03557-0.266150.0022597414.6570-1.514602.20830-2.2781501-1.00070-0.13426-0.29863-0.0462497526.613015.40703.11460-2.1513502-1.01900-0.11944-0.246040.0018397666.1990-19.10702.83770-0.52186503-0.97923-0.18848-0.29467-0.0639897733.8730-2.28540-0.24986-3.1073504-0.92120-0.12961-0.28593-0.0132997827.37709.77000-0.38371-1.3508505-0.99977-0.12605-0.25503-0.0076097914.1770-1.832102.932501.03660506-1.09560-0.11377-0.139280.0366498091.61005.00480-11.76400.95157507-0.99919-0.19812-0.202360.0097398127.1900-2.00060-0.695970.13394508-0.98075-0.14233-0.191720.0361698238.04901.900302.14970-0.32731509-1.03910-0.17238-0.198030.03396983-2.10840-0.32350-0.73284-0.23275510-0.94198-0.02021-0.24582-0.02016984-2.10840-0.22067-0.81421-0.36467For the accuracy of performance degradation assessment, the selection of principal component requires that the dimension of principal component is minimum to reduce the computational complexity of the principal component analysis process while the original signal feature information is included. According to Table4, the variance ratio of λ1 is 0.930. In principle, only the principal component F1 is selected to characterize the decline of the performance of the roll shafts for flexible material R2R processing. The performance declining curve of principal component F1 which plotted according to the calculation results in Table 5 is shown in Figure 7. The performance declining curve of principal component F1 is plotted as shown in Figure 7.Figure 7
Performance declining curve of principal componentF1.As can be seen from Figure7, the changing trend of principal component F1 tends to be stable in the early stage (near 500 sample points) and the middle stage (near 700 sample points) of the performance decline, but it does not show a significant upward trend until the late stage (near 900 sample points) of the performance decline, which cannot clearly distinguish the early performance decline from the midperformance decline of the roll shaft. To this end, the performance degradation curve of the principal components F1,F2,F3, and ,F4 corresponding to eigenvalues λ1,λ2,λ3, and λ4 was drawn, as shown in Figure 8.Figure 8
Performance decline curve of principal componentsF1, F2, F3, and F4.
(a)(b)(c)(d)In Figure8, principal components F1 and F2 in Figures 8(a) and 8(b) tend to be stable in the early stage (near the 500 sample points) and middle stage of performance degradation (near the 700 sample points) and does not show a significant upward trend until the late stage of performance degradation (near the 900 sample points). Principal components F3 and F4 in Figures 8(c) and 8(d) tend to be stable in the early stage of performance degradation and increase slightly in the middle stage of performance degradation and does not show a significant upward trend until the late stage of performance degradation. Compared with the principal components F1 and F2, the changing trend of the principal components F3 and F4 is more obvious, which indicates that the feature information contained solely in the principal components F1 and F2 is not enough to characterize the decline of the performance of the R2R processing roll for flexible material.Considering the computational complexity in dimension reduction of principal component analysis data and the cumulative contribution rate of the eigenvaluesλ1,λ2, and ,λ3 was 0.989, which was much larger than the screening requirement of principal component analysis, three principal components F1,F2, and F3 were selected to achieve the purpose of evaluating the performance degradation of R2R processing roll for flexible material. On the basis of this, a three-dimensional contour map was drawn (see Figures 9(a) and 9(b)).Figure 9
Three-dimensional contour map of principal componentsF1, F2, and F3.
(a)
Principal component three-dimensional global map(b)
Principal component three-dimensional spatial local mapAs can be seen from Figure9(a), it is impossible to distinguish the decline process clearly because the amplitude of principal component feature index in the early stage is smaller and more concentrated, and the amplitude in late degradation stage is larger and more dispersed. For this reason, the red rectangular region in Figure 9(a) is enlarged locally as shown in Figure 9(b). The principal component values are clearly divided into three regions: the leftmost data centralized region, the middle narrow band region, and the right upper scatter region, which represents good performance and early degradation, medium-term degradation, and late degradation, respectively. Among them, the different colored contour lines on the left of Figure 9 realize the division between the good performance state (red area) and the early degradation state (purple area).The experimental results showed that the principal componentsF1,F2, and F3 of the vibration feature parameter of the R2R processing roll for flexible material can not only distinguish the medium-term degradation and the late degradation state but also distinguish the normal state from the early degradation state, and the distinction is much more visualized. Therefore, it is further proved that the principal components F1,F2, and F3obtained by the method in the paper not only realizes data dimensionality reduction but also can characterize the performance degradation state of the R2R processing roll for flexible material.


## 5. Conclusion

In this paper, a new principal component analysis method for data dimensionality reduction was proposed, to handle the problem of high-dimensional performance characterization and the complexity of calculating the performance indicators in flexible material roll-to-roll processing.Based on the analysis of the performance influencing factors of flexible material roll-to-roll processing roller, a principal component analysis extraction model was constructed. And the research on the derivation of the extraction model algorithm was carried out.The original feature parameter matrix composed of 10-dimensional feature parameters such as time domain, frequency domain, and time-frequency domain vibration signal of the roll shaft was established; then, we used the means of averaging to normalize the original feature parameters matrix. The correlation measure between every two parameters in the matrix was used as the eigenvalue to establish the covariance matrix of the performance degradation feature parameters. The Jacobi iteration method was introduced to derive the algorithm for solving eigenvalue and eigenvector of the covariance matrix. Finally, using the eigenvalue cumulative contribution rate as the screening rule, we linearly weighted and fused the eigenvectors and derived the feature principal component matrixF of the processing roller vibration signal.Experiments results showed that after the extraction by principal component analysis model, the initially obtained 10-dimensional features of the processing rollers’ vibration signals, such as average, root mean square, kurtosis index, centroid frequency, root mean square of frequency, standard deviation of frequency, and energy of the intrinsic mode function component, can be expressed by 3-dimensional principal componentsF1,F2, and F3. These three components contain 98.9% data information of the original vibration signal, which means that the feature parameter dimension reduction of processing roller vibration signal has been realized. The three-dimensional contour map drawn by F1,F2, and F3 value not only can clearly distinguish the degraded state in the middle and late stages but also can monitor the state process of the roller from normal to early degradation and can also provide the roller predictive maintenance subsequent research with strong theoretical support.The above experimental results show that the method in this paper has high accuracy in feature extraction of the performance degradation of flexible material R2R processing; it also has the advantages of eliminating the correlation between the feature variables and reducing the workload of feature selection. The research results of this paper can provide strong theoretical support for the follow-up predictive maintenance research of roll shaft.



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*Source: 8812660-2020-12-28.xml*