Research on SVM-Based Bearing Fault Diagnosis Modeling and Multiple Swarm Genetic Algorithm Parameter Identification Method

Due to the complexity of its working environment, rolling bearings are inevitably affected by a variety of factors such as friction, impact and damping during operation, which leads to a large amount of noise in the obtained vibration signal and a strong non-linearity and non-smoothness. For this reason, it is necessary to denoise the signal to improve the extraction of its features. He et al. [1] proposed an improved wavelet total variation (IWATV) denoising method, which uses the wavelet coefficients obtained from wavelet decomposition of a noisy signal for local variance analysis to estimate the noise variance and provide scientific evaluation metrics. Luo et al. [2] proposed a probabilistic principal component analysis (PPCA) method based on GOA method and a comprehensive rolling bearing fault diagnosis method: Adaptive Probabilistic Principal Component Analysis-Enhanced morphological difference filter (APPCA-EMDF). APPCA can make up for the lack of EMDF in denoising ability and solve the interference problem caused by strong background noise. Yuan et al. [3] proposed a method based on multiple fractal fractional trend fluctuation analysis (MF-DFA) and smooth intrinsic time scale decomposition (SITD). This SITD method decomposes the vibration signal into several appropriate rotational components, overcomes the noise effect, maintains the effective signal and improves the signal-to-noise ratio. Huang et al. [4] proposed a new wavelet shrinkage operator to eliminate noise in nonlinear time series, which introduces a generic threshold to classify the details of signal and noise, and the proposed operator was used to solve the problems such as discontinuity of hard operators and large deviation of soft operators, which successfully reduced the noise in the nonlinear time series and also improved the classification accuracy of bearing faults in a noisy environment. REN et al. [5] proposed an in-band noise-reduction method based on iterative SVD (ISVD). The optimal frequency band of the vibration signal was determined using the enhanced wavelet packet transform kurtosis map, the kurtosis of each node was calculated using the envelope spectrum of the wavelet packet coefficient reconstruction signal, the envelope of the reconstruction signal was calculated using the Hilbert transform and the ISVD method was applied to reduce the in-band noise. Nguyen et al. [6] proposed a robust method for monitoring rolling bearing conditions, which uses a new empirical modal decomposition (EMD)-based method to eliminate high levels of noise from acoustic emission (AE) signals and a discrete wavelet packet transform (DWPT)-based envelope analysis technique to effectively identify bearing fault states. Ma et al. [7] proposed a new rolling bearing fault diagnosis method based on the improved VMD adaptive wavelet threshold, decomposing the VMD to obtain the modal components and constructing a dual determination criterion of sample entropy and correlation coefficient to filter the components. Then the adaptive wavelet thresholding function is proposed and quadratic noise reduction is applied to the hybrid imf, which in turn reconstructs each component to achieve joint noise reduction. Hu et al. [8] proposed a new method of rolling bearing fault diagnosis based on variational modal decomposition (VMD) and wavelet threshold denoising optimized by a genetic algorithm. Through theoretical calculation, numerical simulation and application research, the effectiveness and superiority of the method were verified and the method has some application prospects in rotating machinery fault diagnosis. Lee et al. [9] proposed an effective, noise-resistant fault diagnosis model for rolling elements. Signal noise processing uses product function (PF) selection and wavelet packet decomposition (WPD) for noise reduction, and the signal noise-reduction step can effectively remove high-frequency noise and extract fault information hidden under the noise. Since wavelets have the properties of low entropy, de-correlation and multi-resolution, it makes them very superior in analyzing and de-noising non-stationary signals. The wavelet thresholding method is simple in principle and effective in denoising, which is a common method for data denoising nowadays and is widely used for practical fault diagnosis.

Feature extraction is the basis of rolling bearing fault diagnosis. YU et al. [10] proposed a new feature extraction method combining the K-means method and standard deviation to select the most sensitive features, and also proposed an improved feature dimensionality reduction method to achieve a low-dimensional representation of the high-dimensional feature space and improve the accuracy of rotating machinery bearing fault diagnosis. Ju et al. [11] proposed a feature extraction method for improving multiscale entropy (IMSE) diagnosis, which improves the recognition accuracy of IMSE-extracted features and overcomes the information leakage problem when calculating the similarity of mechanical systems. Xue et al. [12] proposed a feature extraction method based on hierarchical analysis: Hierarchical Dispersive Entropy (HDE), which extracts features with better feature clustering effect and higher recognition rate than other methods, and can more accurately characterize the operating condition of bearings. Kuncan et al. [13] used a new feature extraction method for bearing faults, i.e., the one-dimensional ternary pattern (1D-TP) feature extraction method, which is an efficient and robust feature extraction method with high robustness for fast and real-time feature extraction of bearing faults. Li et al. [14] proposed a feature extraction and evaluation method for rotating machinery fault diagnosis, based on the central limit theory, and gave an extraction method to extract statistical features using existing signal processing tools, which can significantly improve the performance of fault classification and ensure the accuracy of fault classification. He et al. [15] proposed a new repetitive transient extraction method with group sparse structure for detecting rolling bearing faults. The proposed repetitive transient extraction algorithm (RTEA) can successfully extract the repetitive transients caused by bearing faults, and the method can effectively extract the features of outer ring defects and inner ring defects. Zhu et al. [16] constructed a multi-weight fusion-based singular value feature extraction method based on the contribution of singular value and entropy weights to solve the problem of it being difficult to effectively extract feature vectors in rolling bearing fault diagnosis, and combined it with SVM classifier to propose a feature fusion-based bearing fault diagnosis method, which can effectively extract sensitive features of bearings and reduce the interference of non-sensitive features to the diagnosis model. Xia et al. [17] proposed a spectral regression (SR) based method to extract fault features from the original features such as time, frequency, and time–frequency domain features of bearing accelerometer sensor signals, which can reduce the computational cost and preserve more structural information about different bearing faults and severity. There are many methods of feature extraction and various evaluation indexes based on different methods. Kumar et al. [18] proposed a migration learning based method for bearing fault detection, using a pre-trained ResNetV2 model as the base model to establish an effective bearing fault detection strategy, which reduces the need for manual feature extraction and selection of bearing faults. Combined with the actual working conditions of petrochemical production, exploring the fault feature extraction method of rotating mechanical equipment bearings is the focus of current research.

The traditional bearing condition identification mainly relies on human experience to judge, which makes it difficult to diagnose the operation condition of the bearing quickly and accurately, and now the emergence of automated and intelligent algorithms has brought a new way of thinking for fault diagnosis and condition identification. Toumi et al. [19] designed an embedded intelligent system to identify the type and severity of bearing faults using envelope analysis to process the raw vibration signals obtained from the CWRU dataset to obtain the peak amplitudes corresponding to the bearing fault frequencies used as input to the MLP network classifier. Yan et al. [20] proposed a fault classification algorithm based on multi-domain feature optimization SVM, which consists of three main stages: multi-domain feature extraction, feature selection and fault identification. The method solves the problem that the current fault classification framework generally focuses on the pattern of single-domain feature classifier, which easily leads to insufficient feature extraction and low recognition accuracy. Liu et al. [21] introduced the higher-order cumulants (HOC) that can quantitatively describe the nonlinear feature signals with close relationships between mechanical faults, achieve noise reduction of the original bearing vibration signals to obtain a bispectral estimation image, and proposed a rolling bearing fault identification method that combines the basic higher-order spectrum (HOS) theory and the fuzzy clustering method in the field of data mining. Wan et al. [22] proposed a method based on group decomposition (SWD), morphological envelope dispersion entropy (MEDE) and random forest (RF) based on the ideas of signal denoising, feature extraction and pattern classification to achieve effective detection and intelligent identification of weak faults in rolling bearings. Fatima et al. [23] proposed a general method for fault classification of bearing imbalance in rotor bearing systems, which is based on a compensated distance-evaluation technique for feature selection, support vector machine and grid search methods for optimization of parameters in the fault classification process, which is a robust SVM technique that uses only time domain data and does not require any additional preprocessing for the balanced faults for classification. JIANG et al. [24] proposed a new convolutional neural networks (CNNs)-based bearing fault classification method in order to eliminate noise interference and take into account the possible connection between signal frames. Using the sensitivity to spectral kurtosis (SK) pulses, a SK-based filtering method was proposed to suppress the noise, which has strong classification ability under the interference of plant noise and Gaussian noise. Tajeddini et al. [25] used vibration and current sensors as a multi-sensor framework for high-precision bearing fault detection, and used wavelet packet transform for each sensor as an effective pre-processing method for signal denoising, and established a generic threshold based on the Steyn unbiased risk estimation method in the denoising algorithm, and trained a support vector machine for fault classification after extracting appropriate time-domain features from the denoised signal. Gong et al. [26] proposed a bubble entropy-based rolling bearing fault diagnosis method without parameter discussion by extracting fault features based on the refined composite multiscale bubble entropy (VMD-RCMBE) of variational pattern decomposition, which improves the fault identification accuracy of bearings while reducing or even eliminating the influence of parameter discussion. Attoui et al. [27] proposed a new time-frequency method for feature extraction based on classical wavelet packet decomposition features by using WPD and STFT as a time-frequency domain fault feature extraction scheme, LDA, LSDA or feedback envelope method as a feature reduction technique, and ANFIS as an automatic classification system to develop the proposed procedure, which achieves accurate detection and classification of bearing faults with different bearing conditions, different onset times, fault locations, fault types and fault severity. Tan et al. [28] proposed an intelligent bearing fault identification method based on SMCMDE and equilibrium optimizer-based support vector machine (EO-SVM), which not only extracts sensitive features of bearing faults, but also has higher accuracy and stability than other methods. Lu et al. [29] investigated an effective and reliable deep learning method based on convolutional neural networks (CNN) using cognitive theory to introduce the advantages of image recognition and visual perception into bearing fault diagnosis by simulating the cognitive processes in the cerebral cortex, and the method improved the accuracy of failure mode classification of rolling bearings with respect to environmental noise and fluctuating operating conditions. Tong et al. [30] proposed a multi-view learning (DAML)-based bearing fault diagnosis method which obtains a multi-view data set involving vibration and acoustic signals by performing a fast Fourier transform (FFT), and then implements a multi-view feature (MVF) representation including view-invariant information and class discrimination information in a common subspace based on typical correlation analysis (CCA) and uncorrelated linear discriminant analysis (ULDA). The multi-view feature (MVF) representation including view invariant information and class discriminant information is then implemented in a common subspace based on typical correlation analysis (CCA) and uncorrelated linear discriminant analysis (ULDA), and finally a bearing fault is identified using a K-nearest neighbor (KNN) classifier based on the multi-view feature. The petrochemical production process has complex working conditions and various types of faults, and intelligent algorithms are uniquely effective in fault diagnosis, which has always been the focus of scholars’ attention. Yang et al. [31] proposed a sample entropy based on mutual information to select the effective intrinsic mode function (IMF), and with the help of minimum entropy deconvolution, decomposed the denoised signal into several IMF by ICEEMDAN method as a pre-filter to increase the kurtosis value by a factor of about 3.2. The results show that compared with ICEEDAN and MED, the proposed hybrid scheme has better performance in weak fault feature extraction under strong noise background. Zou et al. [32] based on improved adaptive noise fully integrated Empirical Mode decomposition (ICEEMDAN) and inverse residual convolutional neural networks integrated with Ghost modules. An intelligent bearing fault diagnosis method is proposed. In this method, ICEEMDAN is used to decompose the signal, and the kurtosis, correlation coefficient and margin factor index are proposed to filter and reconstruct the important modal components, and then multi-scale sample entropy is used to construct the feature vector, and finally Ghost module, convolution block attention module and inverse residual bottleneck are combined to construct Ghost-IRCNN and realize bearing fault classification and diagnosis. Zhao et al. [33] proposed multi-point optimal minimum entropy deconvolution adjustment (MOMEDA). Bearing simulators were used to collect vibration signals of bearing inner and outer rings, enhanced by MOMEDA, decomposed by ICIEDAN into several intrinsic mode functions (IMFs), and analyzed by envelope demodulation. Finally, the shaft speed frequency, BPFI (ball-pass frequency, inner ring) and harmonic frequency, sideband interval, BPFO (ball-pass frequency, outer ring) and harmonic are obtained. This method can be used to accurately extract different frequency components of bearing fault vibration signals and diagnose different bearing fault locations. Li et al. [34] proposed a combined noise-reduction method for cut acoustic signals based on improved adaptive noise fully integrated Empirical Mode decomposition (ICEEMDAN) and singular value decomposition (SVD). The proposed method can effectively eliminate the influence of background noise in the signal and retain the characteristic frequency of the original cutting sound signal. Compared with traditional noise-reduction methods, ICEEMDAN-SVD combined noise-reduction method performs better in the noise-reduction evaluation criteria of signal-to-noise ratio and RMS error. Qin et al. [35] proposed a rolling bearing troubleshooting method based on improved adaptive white noise fully integrated Empirical Mode decomposition (ICEEMDAN) and Sparrow Search algorithm (SSA) to optimize correlation vector machine (RVM). The ICEEMDAN method is used to decompose the original vibration signal, and the ratio of power entropy and intrinsic mode component (IMF) power to the total power of the signal is taken as the fault characteristic. The width coefficient and hyperparameters of RVM were optimized by the SSA algorithm, and the RVM model was established to realize fault identification of rolling bearings.

Based on the petrochemical site environment and the above-mentioned literature, this paper proposes a rotating machinery bearing fault diagnosis method based on the combination of ICEEMDAN-wavelet threshold joint noise reduction, mutual dimensionless index and MPGA-SVM and designs the relevant experiments to obtain the raw data with the faulty bearings of multi-stage centrifugal fans of petrochemical rotating machinery. The raw data are processed by ICEEMDAN-wavelet threshold joint noise reduction and mutual dimensionless index feature extraction to obtain the standard input data set (including training set and test set), which is the key of the subsequent model and model parameter identification algorithm. Finally, based on the standard data set, different models and algorithms are applied to carry out the fault classification of bearings, so as to verify the superior performance of the MPGA-SVM model proposed in this paper.

The rest of this paper is organized as follows. The second part introduces the research framework of this paper, which focuses on three aspects of wavelet-based threshold denoising, mutual dimensionless index extraction features and the MPGA-SVM-based bearing fault diagnosis method, and discusses the research content, technical routes and innovation points. The third part conducts experiments on a petrochemical multistage centrifugal fan fault diagnosis platform to obtain bearing fault data, verify the effectiveness of the designed denoising, feature extraction and fault diagnosis methods and compare them with traditional algorithms to prove the superiority of the research method. The fourth part discusses the main work and contributions of this paper.

Bearing vibration signals are characterized by nonlinearity, multi-source heterogeneity and randomness, and there is serious noise interference. The common methods for noise reduction of the original bearing signal are time-domain filtering noise reduction, wavelet threshold noise reduction, empirical modal decomposition noise reduction, independent component analysis noise reduction and neural network method noise reduction. Each method has certain advantages and disadvantages, and there are many related studies, among which wavelet analysis is more suitable for the processing of local non-stationary signals. The advantages are, firstly, the wavelet transform achieves the improvement of local resolution in the frequency domain, and it is relatively easy to analyze the transients in the asymptotic signal and noise. Secondly, the wavelet transform can compress the signal into a local region in time and frequency, so that each wavelet term obtained after wavelet decomposition of a non-stationary signal can correspond to a local part of the analyzed signal, and the temporal changes of the signal can be well captured, which is very useful for analyzing the local characteristics of the signal. Thirdly, the wavelet decomposition algorithm outperforms other classical analysis methods in terms of efficiency and speed of time and frequency analysis.

Wavelet threshold denoising is a very widely used signal noise-reduction processing method. First, the original signal is decomposed into a series of wavelet coefficients. Second, a threshold is applied to each wavelet obtained after the decomposition; when the wavelet coefficients are below the threshold, they are considered noise to be eliminated. Finally, the wavelet coefficients after wavelet threshold processing is completed are reconstructed to obtain the noise-reduced signal. The principle of the method is shown in Figure 2.

Research shows that, although the wavelet threshold denoising method is effective in some aspects of noise processing, the method ignores that the decomposed wavelet coefficients with smaller thresholds also have valid information. Although the signal is weak, it cannot be ignored. If these weak signals are ignored, the denoising pre-processing will cause the loss of useful information in the weak energy.

Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN): Adaptive white noise is added several times in the decomposition stage. The decomposition process is averaged by the ensemble to have better completeness. Then, the residual components are used to reconstruct a more accurate intrinsic modal component (intrinsic mode function, IMF). The previous literature proposes an Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN), which solves the intrinsic modal component in a different manner from CEEMDAN. ICEEMDAN is based on the residual signal obtained from the previous iteration minus the residual signal of the current iteration and subsequently averaged together. The final decomposition of the eigenmode component shows a high frequency to low frequency ordering. In addition, the appearance of spurious modal components is effectively suppressed.

Based on the above theoretical basis, this paper proposes an ICEEMDAN decomposition algorithm combined with wavelet thresholding as a signal noise-reduction method to establish a bearing vibration signal noise-reduction model. After ICEEMDAN has decomposed the bearing signal collected in the field, the IMF components arranged from high frequency to low frequency are obtained, and the method only denoises the high-frequency IMF components, which contain more noise by wavelet thresholding, while retaining the low-frequency IMF components, which contain less noise. Finally, the high- and low-frequency IMF components are reconstructed. Figure 3 shows the process of the ICEEMDAN-wavelet threshold joint noise reduction.

The original signal with nonlinearity, which contains more noise, is decomposed by ICEEMDAN to obtain a series of IMF components from high frequency to low frequency.

The correlation coefficient method is used to calculate the correlation coefficients of the IMF components obtained from the ICEEMDAN decomposition. The high-frequency IMF components, which contain more noise (correlation coefficient value greater than 0.1), are selected.

The correlation coefficient enables one to measure the correlation between the IMF components after the ICEEMDAN decomposition and the original signal of the bearing after adding noise. A larger correlation coefficient value proves that the two are more similar and the IMF component contains more noise. On the contrary, a smaller correlation coefficient value proves that the two are less similar and the IMF component contains less noise. The correlation coefficient value is used as the basis to select the high-frequency components with more noise in the IMF components obtained by the ICEEMDAN decomposition. The Pearson correlation coefficient is used, and the correlation is defined as follows:

In Equation (1), where $cov\left(\right)$ represents the covariance, f is the signal after white noise has been added to the bearing signal, $\overline{f}$ is the mean value of the signal, and ${\sigma }_{IM{F}_i}$ and ${\sigma }_f$ are the standard deviations of $IM{F}_i$ and f respectively.

The IMF components with high frequency and more noise are wavelet-decomposed, and the appropriate decomposition scale and wavelet function should be selected before the decomposition.

A suitable threshold function is selected; then, a threshold is applied for the wavelet coefficients obtained after the wavelet decomposition for quantization.

The soft threshold value is shown in Equation (3):

The hard threshold is shown in Equation (4): where $sgn\left(\right)$ is the symbolic function; $|IM{F}_i|$ is the unprocessed wavelet coefficients; $\lambda$ is the threshold; $\widetilde{IM{F}_i}$ is the wavelet coefficients after the threshold quantization process. Threshold $\lambda$ is defined as:

In Equation (5), ${\sigma }_i$ is the standard deviation of the noise component layer i; 0.6745 is the empirical value; ${\tau }_i$ is the absolute median of the wavelet coefficients in layer i. The noise-reduction effect is more satisfactory when the hard-threshold signal is used.

The wavelet coefficients after wavelet thresholding are analyzed. The coefficients in the wavelet coefficients that are larger than the wavelet threshold are selected. The wavelet reconstruction function waverec is used to reconstruct the signal. The high-frequency IMF component after noise reduction is obtained.

The noise-reduced high- and low-frequency IMF components are summed and reconstructed. Then, the noise-reduced signal $\dot{x\left(t\right)}$ is obtained.

Let the collected rotating machinery bearing fault observable vibration signal be $z\left(t\right)$, which is composed of the fault-free vibration signal $s\left(t\right)$, bearing fault characteristic signal $x\left(t\right)$ and Gaussian white noise $v\left(t\right)$. The fault-free vibration signal $s\left(t\right)$ is the fault-free vibration signal collected by the sensor at the early stage of operation of the new rotating machinery after break-in. Figure 4 shows the combined model of the observable vibration signal $z\left(t\right)$.

Figure 4, $z\left(t\right)$ [36] is expressed as

In Equation (6), $\tau$ is the delay time constant, c is the correlation coefficient and ${\displaystyle \sum _{P=1}^P}{x}_p\left(t\right)$ represents a series of fault characteristic signals in the bearing, such that $x\left(t\right)={\displaystyle \sum _{P=1}^P}{x}_p\left(t\right)$ is transformed to:

To construct the mutual dimensionless index, the fault features are first extracted from the bearing vibration signal. Figure 5 shows the specific process.

Fault feature extraction takes the observable vibration signal $z\left(t\right)$ and fault-free vibration signal $s\left(t\right)$ as the starting point and performs fast Fourier transform (FFT) on them to obtain $Z\left(k\right)$ and $S\left(k\right)$ in the frequency domain. Then, the method takes the complex conjugate $S\left(k\right)$ of $S{\left(k\right)}^{*}$. The correlation function $M\left(k\right)$ in the frequency domain is the product of $Z\left(k\right)$ and $S{\left(k\right)}^{*}$. Then, the method performs a fast Fourier inverse transform (IFFT) on $M\left(k\right)$, and their correlation function $m\left(t\right)$ in the time domain can be obtained. Afterward, the correlation coefficient c is calculated by correlation analysis of the delay time $\tau$. Finally, $y\left(t\right)$ is obtained by linearly subtracting $z\left(t\right)$ and $s\left(t\right)$ to complete the fault feature extraction. Where $p\left(y\right)$ is the amplitude probability density of fault characteristic signal $y\left(t\right)$ and $p\left(s\right)$ is the amplitude probability density of non-fault vibration signal $s\left(t\right)$. The definition of the mutual dimensionless index is shown in Equation (8).

Mutual waveform indicator: let $l=2$ and $n=1$

Interpulse indicators: let $l=\infty$ and $n=1$

Mutual margin index: let $l=\infty$ and $n=\frac{1}{2}$

Mutual peak metrics: let $l=\infty$ and $n=2$

Mutual peak metrics: let $l=4$ and $n=2$

After the ICEEMDAN-wavelet threshold noise reduction and mutual dimensionless index processing, the mutual dimensionless indices of the bearing signals under different fault states are obtained.

In solving the classification problem of linear data samples, SVMs establish classification surfaces at relatively distant locations from the two types of samples. In solving the classification problem of nonlinear data samples, the SVM transforms the low-dimensional nonlinear classification problem into a high-dimensional spatially divisible one by performing a high-dimensional spatial transformation through the kernel function and effectively classifies linearly indivisible samples. In other words, it aims to find a hyperplane to partition the data samples with a partitioning principle of interval maximization. Figure 6 shows the SVM classification surface.

The classification performance of SVMs is mainly determined by the penalty factor c and parameters r (where $r=1/2{\sigma }^2$) of the kernel function. Currently, there is no specific approach to select c and r. Grid search and cross-test are the traditional approaches, but they are inefficient. If the range and step size of the parameters to be optimized are not properly set, the selection of optimal parameters will be affected. Thus, this paper will introduce the multiple swarm genetic algorithm to solve the optimal values of key parameters c and r, which affect the classification performance of SVMs, by using the parallelism of the multiple swarm genetic algorithm and powerful global search ability to establish the optimal classification model of SVMs.

The multiple population genetic algorithm performs the simultaneous optimization search for multiple populations, and the parameters are set differently for different subpopulations to realize the diversity of optimization results.

Meanwhile, each subpopulation can be linked using a migration operator to realize the co-evolution of multiple subpopulations, which yields the optimal solution. The optimal solution in the evolutionary generation of each subpopulation is preserved by the manual selection operator, which is also the criterion for determining whether the algorithm converges. In this paper, the minimum number of generations of optimal individuals is maintained as the basis for the end of algorithm operation. The principle of the multi-population genetic algorithm is shown in Figure 7.

The steps of the MPGA-optimized SVM are as follows:

Multiple swarm genetic algorithms can effectively search for the optimal values of key parameters c and r, which affect the classification performance of SVMs to obtain the optimization model of SVM. Hence, the methods mentioned in Section 2 are fused. Then, a bearing fault diagnosis method based on the combination of the ICEEMDAN-wavelet threshold, mutual dimensionless index and MPGA-SVM is proposed and experimentally validated using a dataset of petrochemical rotating machinery bearings under different fault states.