Mechanical vibration signal mapped into a high-dimensional space tends to exhibit a special distribution and movement characteristics, which can further reveal the dynamic behavior of the original time series. As the most natural representation of high-dimensional data, tensor can preserve the intrinsic structure of the data to the maximum extent. Thus, the tensor decomposition algorithm has broad application prospects in signal processing. High-dimensional tensor can be obtained from a one-dimensional vibration signal by using phase space reconstruction, which is called the tensorization of data. As a new signal decomposition method, tensor-based singular spectrum algorithm (TSSA) fully combines the advantages of phase space reconstruction and tensor decomposition. However, TSSA has some problems, mainly in estimating the rank of tensor and selecting the optimal reconstruction tensor. In this paper, the improved TSSA algorithm based on convex-optimization and permutation entropy (PE) is proposed. Firstly, aiming to accurately estimate the rank of tensor decomposition, this paper presents a convex optimization algorithm using non-convex penalty functions based on singular value decomposition (SVD). Then, PE is employed to evaluate the desired tensor and improve the denoising performance. In order to verify the effectiveness of proposed algorithm, both numerical simulation and experimental bearing failure data are analyzed.

A fault diagnosis model for rolling bearing based on complementary ensemble adaptive local iterative filtering (CEALIF), Laplacian score (LS) feature selection, and genetic algorithm-based backpropagation neural network (GA-BPNN) is proposed in this article. When the rolling bearing fails, the field-measured waveforms usually shows strong nonlinearity and non-stationary characteristics. Adaptive Local Iterative Filtering (ALIF) is an alternative novel approach to empirical mode decomposition to decompose complex signals into multiple intrinsic mode functions (IMFs), but modal aliasing will occur in actual processing. Aiming at the phenomenon of modal aliasing, a noise-assisted analysis methodology, namely complementary ensemble adaptive local iterative filtering, which could overcome the modal aliasing problem of ALIF. This article applies it to the pre-processing of rolling bearing time series. Then, the time domain (TD) statistical features of the IMFs, their Fourier frequency domain (FD) features, and the time frequency domain (TFD) energy features are extracted to capture the fault information. Meanwhile, to avoid feature redundancy and enhance the diagnostic performance, the LS is adopted to rank the features to improve the fault characteristics. Subsequently, the optimized feature vectors are entered into the GA-BPNN to automatically achieve the fault type recognition. The experimental data analysis results of rolling bearings indicate that the model can effectively diagnose the degree and type of failure.

Tensor decomposition technique has been widely used in multichannel signals processing for its distinct superiority. The early fault feature signal of bearing is weak and easily inundated by ambient noise. In addition, the interference signals generated by other mechanical components also have a serious impact on the result of fault diagnosis. Aiming at the above issues, and built on the methods of generalized non-convex tensor robust principal component analysis (GNCTRPCA) and tensor singular value kurtosis (TSVK), this paper introduces a new fault diagnosis technique for multichannel bearing signals. First, the attractor tensor is formed by reconstructing the acquired multichannel signals in phase space. Second, based on tensor singular value decomposition (TSVD), the tensor robust principal component analysis (TRPCA) can provide a favorable noise reduction performance. However, TRPCA usually lowers the amplitude of useful singular value tubes (SVTs). To tackle this problem, the GNCTRPCA method is proposed to avoid the amplitude reduction. Third, a new TSVK method is introduced to determine the reconstructed order of SVTs, so as to extract the multichannel fault feature signals. Finally, the bearing fault type can be identified by comparing the peak frequencies of the extracted signals with the theoretical fault-related frequencies. Simulation analysis and experiment studies verify the principle and effectiveness of the proposed technique.

The acquired bearing fault signal usually reveals nonlinear and non-stationary nature. Moreover, in the actual environment, some other interference components and strong background noise are unavoidable, which lead to the fault feature signal being weak. Considering the above issues, an effective bearing fault diagnosis technique via local robust principal component analysis (LRPCA) and multi-scale permutation entropy (MSPE) was introduced in this paper. Robust principal component analysis (RPCA) has proven to be a powerful de-noising method, which can extract a low-dimensional submanifold structure representing signal feature from the signal trajectory matrix. However, RPCA can only handle single-component signal. Therefore, in order to suppress background noise, an improved RPCA method named LRPCA is proposed to decompose the signal into several single-components. Since MSPE can efficiently evaluate the dynamic complexity and randomness of the signals under different scales, the fault-related single-components can be identified according the MPSE characteristic of the signals. Thereafter, these identified components are combined into a one-dimensional signal to represent the fault feature component for further diagnosis. The numerical simulation experimentation and the analysis of bearing outer race fault data both verified the effectiveness of the proposed technique.

To solve the problem of multi-fault blind source separation (BSS) in the case that the observed signals are under-determined, a novel approach for single channel blind source separation (SCBSS) based on the improved tensor-based singular spectrum analysis (TSSA) is proposed. As the most natural representation of high-dimensional data, tensor can preserve the intrinsic structure of the data to the maximum extent. Thus, TSSA method can be employed to extract the multi-fault features from the measured single-channel vibration signal. However, SCBSS based on TSSA still has some limitations, mainly including unsatisfactory convergence of TSSA in many cases and the number of source signals is hard to accurately estimate. Therefore, the improved TSSA algorithm based on canonical decomposition and parallel factors (CANDECOMP/PARAFAC) weighted optimization, namely CP-WOPT, is proposed in this paper. CP-WOPT algorithm is applied to process the factor matrix using a first-order optimization approach instead of the original least square method in TSSA, so as to improve the convergence of this algorithm. In order to accurately estimate the number of the source signals in BSS, EMD-SVD-BIC (empirical mode decomposition—singular value decomposition—Bayesian information criterion) method, instead of the SVD in the conventional TSSA, is introduced. To validate the proposed method, we applied it to the analysis of the numerical simulation signal and the multi-fault rolling bearing signals.

Gears are key components in rotation machinery and its fault vibration signals usually show strong nonlinear and non-stationary characteristics. It is not easy for classical time–frequency domain analysis methods to recognize different gear working conditions. Therefore, this paper presents a joint fault diagnosis scheme for gear fault classification via tensor nuclear norm canonical polyadic decomposition (TNNCPD) and multi-scale permutation entropy (MSPE). Firstly, the one-dimensional vibration data of different gear fault conditions is converted into a three-dimensional tensor data, and a new tensor canonical polyadic decomposition method based on nuclear norm and convex optimization called TNNCPD is proposed to extract the low rank component of the data, which represents the feature information of the measured signal. Then, the MSPE of the extracted feature information about different gear faults can be calculated as the feature vector in order to recognize fault conditions. Finally, this researched scheme is validated by practical gear vibration data of different fault conditions. The result demonstrates that the proposed scheme can effectively recognize different gear fault conditions.