**1. Introduction**

Rolling bearing is one of the widely used parts in rotating machinery and equipment in the manufacturing system. It widely exists in the power end, transmission end, and execution end. Due to its complex working environment, it is very prone to failure. According to relevant statistics, about 30% of failures in rotating machinery are related to rolling bearings. Therefore, the research on fault diagnosis of rolling bearing is of grea<sup>t</sup> significance to ensure the production safety of relevant enterprises. However, in the actual operation process, due to the influence of external noise, receiving distance, and sensor working environment, the fault characteristics of this component are submerged in the interference of intensebackground noise, which has a significantimpact on fault diagnosis.

**Citation:** Yang, J.; Zhou, C.; Li, X. Research on Fault Feature Extraction Method Based on Parameter Optimized Variational Mode Decomposition and Robust Independent Component Analysis. *Coatings* **2022**, *12*, 419. https:// doi.org/10.3390/coatings12030419

Academic Editor: Alessandro Latini

Received: 30 January 2022 Accepted: 15 March 2022 Published: 21 March 2022

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At present, the bearing fault diagnosis method based on signal processing has experienced three processes: time–domain analysis, frequency domain analysis, and time– frequency domain analysis. Because the fault signal of rolling bearing is non-stationary and nonlinear, the time–frequency domain analysis method is relatively suitable for processing this signal. Since the development of the time–frequency analysis method, the algorithms often used by relevant experts and scholars include Short-time Fourier transform (STFT) [1,2], S-transform [3], Wigner Ville distribution (WVD) [4], wavelet transform (WT) [5,6], etc., singular spectrum decomposition (SSD) [7], spectral kurtosis (SK) [8], morphological filtering (MF) [9], and singular spectrum analysis (SSA) [10], etc. However, these methods havetheir ownlimitations. For example, the window function of STFT is fixed, which is not conducive to the analysis of non-stationary bearing fault signals. The standard deviation of the S-transform in the Gaussian window function is fixed as the reciprocal of the frequency, which makes the time–frequency aggregation of the high-frequency part of the signal not ideal.WVD has good time–frequency aggregation, but there is cross-term interference. Although WT has the time–frequency local analysis ability of adjustable window, the scale of wavelet transform does not have a good corresponding relationship with the frequency of signal. In SK algorithm, how to set the parameters of passband center frequency and resonance bandwidth will affect its application effect. In MF algorithm, it is difficult to effectively select the type and size of the structuring element. In SSA algorithm, the embedding dimension and time delay of phase space reconstruction cannot be set automatically. After continuous development, relevant scholars have proposed many adaptive signal processing methods based on previous research results and applied them to the fault feature extraction of rolling bearing. For example, empirical mode decomposition (EMD) [11], ensemble empirical mode decomposition (EEMD) [12], local mean decomposition (LMD) [13], etc., Jiang et al. [14] proposed a fault feature extraction method combining EMD and 1.5-dimensional spectrum. In this method, the signal components decomposed by EMD are screened and reconstructed Then the reconstructed Hilbert envelope signal is analyzed by a 1.5-dimensional spectrum to obtain the characteristic fault frequency of bearing. The effectiveness of the proposed method is proved by experimental analysis. Fan et al. [15] used EMD and pesudo-Wigner-Ville distribution (PWVD) to convert rolling bearing vibration signals with different fault degrees into contour time–frequency images, then extracted energy distribution values as features, and constructed a fault diagnosis model based onfuzzy c-means (FCM). Experimental results show that this method has high recognition accuracy. Hou et al. [16] proposed a fault diagnosis method composed of EEMD, permutation entropy (PE), and Gath Geva (GG) clustering algorithm to solve the problem that it is difficult to identify the fault type of rolling bearing. Experimental results show that the proposed fault diagnosis method can achieve better clustering results. Liang et al. [17] proposed a fault diagnosis method based on long-term and short-term memory network (LSTM) and LMD, to improve the defects of the LMD method. The results show that the method successfully extracts the characteristic frequencies of rolling bearing. Although the above time–frequency analysis methods have achieved certain results, they are based on the principle of recursive decomposition, so they have not been well solved in the aspects of modal aliasing and endpoint effect. To solve this problem, Dragomiretskiy et al. [18] proposed a new adaptive decomposition method variational modal decomposition. The algorithm makes the decomposition result stable through the construction of the variational problem and is applied to the fault diagnosis of rotating machinery. Ye et al. [19] decomposed the bearing vibration signal by the VMD method and introduced the characteristic capability ratio criterion to screen the qualified signal components for reconstruction. Then, the multi-dimensional features of the signal are extracted and input into the particle swarm optimization (PSO) andsupport vector machine (SVM) classification model for fault diagnosis. The results show that the proposed method has higher recognition accuracy than the existing methods. Li et al. [20] proposed a fault diagnosis method combining VMD and fractional Fourier transform (FRFT) to solve the problem that it is difficult to extract fault features and over decomposition when applying

the VMD method to rolling bearing fault diagnosis. By analyzing the results of simulation experiments, the method has a good effect. Xing et al. [21] combined VMD, Tsallis entropy, and fuzzy c-means clustering (FCM) and applied them to fault diagnosis. Through the analysis of the measured vibration signal of rolling bearing, the results show that this method can obtain better results than EMD and LMD methods.

The signal processed by time–frequency analysis contains a lot of noise, which has a specificimpact on the fault feature extraction. Therefore, the signal needs further processing. In recent years, the technology based on blind source separation has become one of the research hotspots. This technology optimizes multiple observation signals according to the principle of statistical independence and decomposes them into several independent components, so as to achieve the purpose of signal enhancement. Robust independent component analysis (RobustICA), as an algorithm with outstanding advantages in blind source separation methods, has been widely used in signal analysis, fault diagnosis, and other fields because of its good effect in signal-to-noise separation effect and calculation efficiency [22–24]. Yang et al. [25] proposed a signal noise-reduction method based on the combination of complementary ensemble empirical mode decomposition (CEEMD) and RobustICA to reduce the noise of pipeline blockage signals. Through the processing and analysis of simulation signals and pipeline blockage detection signals, the results verify the effectiveness of the proposed method. Yao et al. [26] studied the noise reduction of internal combustion engine signals by using the combination of Gammatone filter bank and RobustICA. Experiments show that the classification effect of signal and noise obtained by this method is good. Zhao et al. [27] combined EEMD, RobustICA, and Prony algorithms and applied them to the identification of low-frequency oscillation signals in the power system. Experiments show that the proposed method has a strong anti-interference ability and can effectively suppress noise.

In this paper, a fault feature extraction method based on VMD optimized with information entropy and RobustICA is proposed. Firstly, the fault signal is decomposed by VMD, and the number of modal components *k* and penalty parameters α are selected according to the optimization principle of minimum information entropy. Then, the optimal parameters are substituted into VMD and the signal decomposition operation is carried out. Secondly, the signal components are filtered through the constructed signal component screening criteria, and the observation signal channel is constructed, so as to realize the signal-to-noise separation based on RobustICA. Finally, the denoised signal is demodulated by Hilbert envelope, and the fault characteristic frequency is extracted.

The main contributions of this paper are summarized as follows:


The structure of this paper is as follows. Section 2 introduces the basic principles of VMD, information entropy, and the RobustICA algorithm. Section 3 introduces the specific implementation process of the fault feature extraction method based on information entropy optimization VMD and RobustICA. In Section 4, the stimulation signal is experimentally studied by using the proposed method. In Section 5, the effect of the method is further verified by the actual bearing fault signal experiment. Section 6 is the discussion and conclusions.

## **2. Basic Methods**
