Adaptive feature selection method with FF-FC-MIC for the detection of mutual faults in rotating machinery

2.1. Sparse representation

Assuming that the signal is $y$ and its length is $P$, it can be regarded as a vector of $R^p$, the redundant dictionary $D=\left[d_1,d_2,d_3,\cdots ,d_n\right]$, where $n>p$, the signal y can be expressed as the computational superposition of the basis function as shown in Eq. (1):

where $x_i$ is the coefficient of the basis function, and $x=\left\{x_1,x_2,\cdots x_n\right\}$ is the sparse representation of the signal.

The sparse representation method adopted in this paper is the adaptive Gabor sub-dictionary Orthogonal Matching Pursuit (OMP) algorithm. Gabor atom is a kind of time – frequency atom and has good time-frequency approximation performance. It has a good tracking effect on the stationary components in the signal and can match the cyclic characteristics in the signal. Eq. (2) is the basic definition:

where, $g\left(t\right)=e^{-\pi t^2}$ is the Gaussian window function; $\gamma =(s,u,\xi )$ is the atomic parameter; $s$ is the scale factor; $u$ is the displacement factor; $\xi$ is the frequency factor. Let $\widehat{g}\left(\omega \right)$ be the Fourier transform of $g\left(t\right)$, then the Fourier transform of Gabor atom can be obtained according to Eq. (1), as shown in Eq. (3):

The calculation process of OMP is to build the over-complete dictionary of Gabor atom through the original signal, and use OMP algorithm to match the original signal, find the best matching atom in the over-complete dictionary and generate the residual signal. A new over-complete dictionary is generated according to the residual signal and the next matching calculation is performed until the residual signal meets the shutdown condition [19].

2.2. MIC

The calculation of MIC is based on the concept of mutual information and meshing method of information theory [20]. Mutual information is used to measure the correlation between two variables. Let $A=\left\{a_i,i=\mathrm{1,2},...,n\right\}$, $B=\left\{b_i,i=\mathrm{1,2},...,n\right\}$, $n$ represent the number of samples, and the mutual information between $A$ and $B$ is defined as Eq. (4):

where, $P\left(A,B\right)$ is the joint probability density of $A$ and $B$. $P\left(A\right)$ and $P\left(B\right)$ are the edge probability density of $A$ and $B$ respectively.

When MIC based on the concept of mutual information is calculated, the set of two data points is distributed in two-dimensional space. $G$ is defined as $A$ grid of size $x\times y$, where the range of $A$ is divided into segment $x$ and the range of $B$ is divided into segment $y$, and the limit of the grid is $x\times y<B$, $B=n^{0.6}$. The maximum value of mutual information between variable $A$ and variable $B$ in different meshes is calculated and called maximum mutual information. Eq. (5-6) are the definition formula and calculation formula of maximum mutual information respectively:

where $x$ and $y$ are the number of cells divided on the range $A$ and $B$, respectively. $MI\left(x,y\right)$ is the mutual information of variable $x$ and variable $y$.

2.3. Adaptive feature selection method based on MIC

Traditional feature selection only relies on calculating the correlation between features to remove redundant features without considering the signal-independent features. MIC can not only measure the correlation between features and features, but also between features and signal categories [21]. In order to reduce the complexity of the classification calculation caused by the irrelevant feature information in the multi-dimensional feature quantity, this paper proposes the FF-FC-MIC method to realize the adaptive feature selection of the coupled fault multi-dimensional information. Firstly, calculate the MIC between the feature and the feature, select the irrelevant feature of the feature and the feature in order to eliminate the redundant features; secondly, calculate the MIC between the feature and the signal category, and get the feature based on the strong correlation between the feature and the signal category. Finally, the features obtained by the two methods are intersected, and the feature intersection is the final feature set for classification.

According to the FF-MIC feature selection method, the correlation between features is calculated and the uncorrelated features are selected. $M$ kinds of signals whose signal category is $C=\left\{c_1,c_2,...c_M\right\}$ are set. $N$-dimensional features are selected for each type of signals, and the complete set of features is $F=\left\{f_1,f_2,...f_N\right\}$. The maximum information coefficient between a feature of a single signal and other features is calculated as shown in Eq. (7-8), where $j$ is the serial number of each feature:

where $j$ is the serial number of each feature.

According to Eq. (1) and Eq. (2), the maximum information coefficients between different eigenvalues of signals are calculated and listed as FF-MIC matrix:

Select the minimum value $f_{\mathrm{m}\mathrm{i}\mathrm{n}}$ of each column in the FF-MIC matrix, and all the minimum values form the set $F{F}_{\mathrm{m}\mathrm{i}\mathrm{n}}$, and then set the maximum value in this set as the threshold $F{F}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}$:

The elements in each column of matrix FF-MIC are compared with the threshold value $F{F}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}$, and the number of elements in each column less than the threshold value $F{F}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}$ is set to $f{f}_j$ to form the set $F{F}_j$. The more elements in each column are less than the threshold value, the weaker the correlation between the column features and other features is. The average of the number of elements in each column that are below the threshold is calculated, and features that are greater than the average are selected:

where, $F{F}_j$ is the set of elements in each column less than the threshold $F{F}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}$, and $AV{G}_{FF}$ is the average value of elements in each column less than the threshold.

All the selected features are composed into a set $S_{ff}$, where $f$ is the selected feature, $j$ is the serial number of each selected feature, $n$ is the number of the selected feature, $n\in N$:

Multiple types of signals are classified, the selected feature sets of each type of signals are calculated respectively, and the feature sets obtained by all signals are combined to form the feature matrix $S_1$, $x\in N$:

Fig. 1Correlation between feature and signal category

The feature selection method based on FC-MIC selects features with strong correlation between features and signal categories by calculating the correlation between features and signal categories. Establish the feature selection process as shown in Fig. 1, where the $X$-axis represents the fault category (category is represented by a number), and the $Y$-axis represents the size of the eigenvalue of all samples if the sample type is the value of the $X$-axis. The maximum system information coefficients between features and signal categories are calculated, and the FC-MIC matrix with size of $1\times m$ is obtained:

The average value of all MIC is calculated as the threshold value $F{C}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}$, and those greater than the average value are strongly correlated features, which are selected. All strongly correlated features are formed into an eigenmatrix $S_2$, and $y$ is the number of selected features, $y\in N$:

where $y$ is the number of selected features, $y\in N$.

According to two parallel feature selection methods, eigenmatrices $S_1$ and $S_2$ are obtained. Then find the intersection of $S_1$ and $S_2$ to obtain the feature subset as shown in Eq. (20):

4.1. Experimental simulation analysis

In this section, the effect of the proposed method is preliminarily verified by constructing simulation signals:

As shown in Eqs. (21-24), bearing normal, single bearing fault, single unbalance fault and unbalanced-bearing coupling fault are simulated respectively, where $x_0$ represents Gaussian white noise signal. Simulate these four kinds of signals each with 60 groups, a total of 240 groups, of which 30 groups of data are selected as training samples for each signal, and 30 groups of data are used as test samples. Fig. 3 shows the imbalance-bearing coupling fault simulation signal and its signal composition. The improved sparse denoising algorithm is used to preprocess the simulated coupling fault signal and obtain the reconstructed signal with high SNR, as shown in Fig. 4.

Fig. 3Simulation signal of coupling fault

Next, Multi-class features of reconstructed signals are required, including 14 dimensional features such as time domain feature, frequency domain feature and entropy value feature. Table 1 describes the detailed information of these features. Finally, use SVM for classification, use radial basis kernel function (RBF) as the kernel function, and optimize the parameters of SVM using grid optimization algorithm. Fig. 5 shows the fault classification under the one-dimensional feature when the feature is the root mean square. Fig. 6 shows the fault classification situation under the two-dimensional feature when the features are the root mean square value and kurtosis.

Fig. 4Coupling fault based on sparse denoising reconstruction signal

Fig. 5Fault classification under one-dimensional characteristics of the simulated signal

Fig. 6Fault classification under two-dimensional characteristics of the simulated signal

By comparing Fig. 5 and Fig. 6, it can be seen that the effect of fault classification using two-dimensional feature is higher than that using one-dimensional single feature. It shows that the multi-dimensional features of coupling faults express more comprehensive information, so the classification effect is higher than that of single-dimensional features. But which feature vectors to select for fault classification is a problem.

The FF-FC-MIC method proposed in this paper is used to select suitable features. First, feature ordinals with high correlation between feature-feature and feature-category are selected by FF-MIC and FC-MIC methods respectively, and then the intersection of these ordinals is calculated. The result of feature selection is shown in Table 2. Five features are selected using FF-FC-MIC, and the signal features are represented by numbers 1-14.

The five types of features selected by FF-FC-MIC are input into SVM for fault diagnosis. At the same time, the effects of FF-MIC method, principal component analysis (PCA) and this method are compared. The results of the full feature classification without feature selection and the classification effects of the three feature selection methods are shown in Fig. 7. It can be seen from Fig. 7 that the classification accuracy of the FF-FC-MIC feature selection method proposed in this paper is up to 100 %, and the shortest time to complete the classification is only 14.83 s. The simulation experiment analysis proves that this method not only improves the accuracy of fault classification, but also reduces the time of fault classification.

Fig. 7Simulation signal classification accuracy

4.2. Real experimental signal analysis

In order to analyze and verify the proposed method, a rotor system was simulated by SQ company's mechanical fault comprehensive test bench. The rotor unbalance fault, load end bearing fault, motor end bearing fault and their coupling faults are simulated respectively by using the test bed. Fig. 8 shows the structure composition and fault setting of the experimental platform. The bearings used in the experiment were load bearing of Type ER-12K and motor bearing of type 6203, and the unbalanced mass was composed of a screw and two gaskets mounted on the wheel. In the experiment, the motor speed was set at 1800 r/min and the sampling frequency was set at 12800 Hz. The vibration signal in this paper was collected by acceleration sensor B.

Fig. 8Test bench structure diagram and faults setting

Three experimental cases were used to verify the method proposed in this paper, and each case was composed of signals of four different fault types. The detailed information of signal fault types in the three experimental cases is shown in Table 3. The fault signal types in case 1 are coupling faults between rotor unbalance and bearings; Case 2 is the coupling fault signal between motor bearing and load bearing; Case 3 includes single rotor unbalance fault, single bearing fault, unbalance – bearing coupling fault, and bearing – bearing coupling fault. 240 groups of signals were collected for each experimental case, including 60 groups of each type of fault signal (30 groups of training samples, 30 groups of test samples).

The coupling fault signal in case 1 is analyzed experimentally using the method proposed in this paper. Firstly, denoising preprocessing was performed on the collected multi-type coupling fault signals, using the improved sparse representation algorithm mentioned in this paper. Fig. 9-12 show the reconstructed signals after denoising. It can be seen from Fig. 9-12 that the reconstructed signal ensures the useful impact features in the original signal while removing the useless features in the signal.

Fig. 9The reconstructed signal of coupling fault U-IL

Fig. 10The reconstructed signal of coupling fault U-RL

Fig. 11The reconstructed signal of coupling fault U-IM

Fig. 12The reconstructed signal of coupling fault IM-IL

Similarly, 14 feature vectors of 14 dimensions were calculated by using 14 feature indexes mentioned in the simulation experiment, and fault classification was analyzed by combining SVM. Fig. 13 shows the fault classification results under one-dimensional single feature vector, and Fig. 14 shows the fault classification results under two-dimensional feature. As can be seen from Fig. 13 and 14, for complex coupled faults, the fault classification effect of single feature vector and randomly selected low-dimensional feature vector is very poor.

Fig. 13Fault classification under the one-dimensional feature of the actual signals

Fig. 14Fault classification under the two-dimensional feature of the actual signals

Furthermore, the FF-FC-MIC method proposed in this paper is used for adaptive feature selection of coupling faults. The results of FF-FC-MIC feature selection for two different experimental cases are shown in Table 4. For the coupling fault signals between unbalance and bearings in case 1, FF-FC-MIC adaptively selects 4 types of features. For coupling fault signals between motor bearings and load bearings in Case 2, FF-FC-MIC selects 8 types of features. And for coupling fault signals and single fault signal in Case 3, FF-FC-MIC selects 6 types of features. Then the selected feature vectors are input into the SVM model for fault classification.

Fig. 15 shows the classification results of unbalanced-bearing coupling fault signals in case1, which are selected by different feature selection methods and then classified by SVM model. It can be seen from Fig. 15 that the classification accuracy of the 4-dimensional features selected by FF-FC-MIC in case 1 is 94.17 %, and the classification time is only 14.95 s. In addition, PCA and FF-MIC feature selection methods as well as full features not selected by feature selection were used for comparative analysis, as shown in Fig. 15. Compared with the full-feature method without feature selection, although the classification accuracy is the same, the classification time of ff-FC-MIC method is significantly reduced due to the removal of useless features after feature selection. And compared with other two feature selection methods, FF-FC-MIC has great advantages in ensuring accuracy and reducing classification time for unbalanced-bearing coupling faults.

Fig. 15Classification effects of different feature selection methods on case1 coupling faults

Case 2 is aimed at the coupling fault of motor bearing-load bearing, and Fig. 16 shows the fault classification results of different feature selection methods. According to the comparison in Fig. 16, while FF-FC-MIC is equal to the other three methods in classification time, it greatly improves the accuracy of SVM classification. Case 3 integrated single fault and coupling fault in rotor system. Fig. 17 is the comparison of classification accuracy and classification time of case 3 using this method and other three methods. It can be seen from Fig. 17 that this method still has obvious advantages in classification accuracy and time in this case.

Fig. 16Classification effects of different feature selection methods on case 2 coupling faults

Fig. 17Classification effects of different feature selection methods on case 3

In the above analysis of three real cases of rotor system faults, the method proposed in this paper has excellent classification results compared with other methods. Experimental results show that this method can not only improve the accuracy of fault classification but also reduce the time of fault classification. Case 1 and Case 2 mainly deal with the applicability of this method when complex coupling faults exist in the rotor system. Case 3 includes single faults and coupled faults, but the classification effect still reaches 98.33 %. The three experiments show that the proposed method has excellent performance in fault diagnosis of the rotor system.