在强烈外界噪声下或轴承故障早期发展阶段，从轴承非平稳故障信号中提取微弱冲击成分是一个难点，针对这一问题，提出了一种新的基于非凸罚正则化稀疏低秩矩阵（Non-convex penalty regularization sparse low-rank matrix，NPRSLM）的轴承微弱故障特征提取方法。该方法不依赖振动信号结构的先验知识，也无需采集大量的样本信号来训练字典，避免了传统稀疏表示设计冗余字典带来的缺乏物理意义，通用性差等缺陷。该方法的核心思想是把采集的振动信号与待提取的故障脉冲看作一维矩阵（向量），通过求解稀疏正则化的反问题得到故障脉冲信号。在建模上，通过引入非凸罚函数代替了传统最小化L1-norm融合套索算法，建立非凸罚正则化稀疏低秩矩阵模型，理论推导了所建立模型的严格凸性，并利用交替方向乘子法（Alternating direction method of multipliers，ADMM）对模型进行求解，同时讨论了模型参数对模型算法的收敛性问题、凸性与非凸性边界取值问题等。仿真算例与大型减速机圆锥滚子轴承诊断实例表明：该方法不仅能提取隐藏在强烈外界噪声中的微弱冲击特征，而且改善了传统最小化L1-norm融合套索算法在提取微弱故障冲击时产生的脉冲能量大幅衰减与脉冲数目丢失问题。
Generally, it is a difficulty to extract weak periodical impulses from the bearing non-stationary vibration signals that are often corrupted by heavy background noise, especially at the early stage of bearing faulty development. In this paper, a novel fault feature extraction methodology is proposed based on non-convex penalty regularization sparse low-rank matrix (NPRSLM) for taper roller bearing. The proposed NPRSLM model neither relies on the priori knowledge of the analyzed signal, nor requires a large number of sample signals to form a training dictionary before diagnosis, thus it avoids a series of redundant dictionary problems such as physical model matching problem and poor universality. The core idea is that the proposed method considers the observed signal and fault impulses as the one-dimensional matrix (or vector), and the fault impulses can be obtained by solving the anti-problem of the sparse regularization. Specifically, on the modeling system, the non-convex function is introduced to replace the conventional minimization L1-norm fused lasso low-rank matrix (ML1-FLLM) so as to reconstruct a novel non-convex regularization model. Meanwhile, the solution of the proposed NPRSLM model is derived by ADMM technique, and strictly convex property, boundary condition problem and convergence of the proposed model are also provided. The simulation and practical rolling bearing experiments indicate that the proposed method not only successfully extract weak periodical impulses from observed signal corrupted by heavy background noise, but also the energy attenuation problem and impulse loss problem are improved when the conventional ML1-FLLM process the weak fault vibration signals.
[1] YUAN Jing,WANG Yu,PENG Yizhen,et al. Weak fault detection and health degradation monitoring using customized standard multiwavelets[J]. Mechanical Systems and Signal Processing, 2017,94:384-399.
[2] LI Qing,JI Xia,LIANG S Y. Incipient fault feature extraction for rotating machinery based on improved AR-minimum entropy deconvolution combined with variational mode decomposition approach[J]. Entropy, 2017,19,317.
[3] 李宏坤,杨蕊,任远杰,等. 利用粒子滤波与谱峭度的滚动轴承故障诊断[J]. 机械工程学报, 2017, 53(3):63-72. LI Hongkun, YANG Rui, REN Yuanjie, et al. Rolling element bearing diagnosis using particle filter and Kurtogram[J]. Journal of Mechanical Engineering, 2017, 53(3):63-72.
[4] 何国林,丁康,林慧斌. 基于匹配追踪的齿轮箱耦合调制振动信号分离方法研究[J]. 机械工程学报, 2016, 52(1):102-108. HE Guolin,DING Kang,LIN Huibin. Matching pursuit method for coupling modulation signal separation of gearbox vibration[J]. Journal of Mechanical Engineering, 2016,52(1):102-108.
[5] WANG Shibin,HUANG Weiguo,ZHU Zhongkui. Transient modeling and parameter identification based on wavelet and correlation filtering for rotating machine fault diagnosis[J]. Mechanical Systems and Signal Processing,2011,25:1299-1320.
[6] LI Qing,LIANG S Y. Incipient fault diagnosis of rolling bearings based on impulse-step impact dictionary and re-weighted minimizing nonconvex penalty Lq regular technique[J]. Entropy,2017,19:421.
[7] ZHOU Haitao,CHEN Jin,DONG Guangming,et al. Detection and diagnosis of bearing faults using shift-invariant dictionary learning and hidden Markov model[J]. Mechanical Systems and Signal Processing,2016,72-73:65-79.
[8] FENG Zhipeng,LIANG Ming. Complex signal analysis for planetary gearbox fault diagnosis via shift invariant dictionary learning[J]. Measurement,2016,90:382-395.
[9] CUI Lingli,WANG Jing,LEE S. Matching pursuit of an adaptive impulse dictionary for bearing fault diagnosis[J]. Journal of Sound and Vibration,2014,333(10):2840-2862.
[10] TROPP J A. Just relax:convex programming methods for identifying sparse signals in noise[J]. IEEE Transactions on Information Theory,2006,52(3):1030-1051.
[11] BACH F,JENATTON R,MAIRAL J,et al. Optimization with sparsity-inducing penalties[J]. Foundations and Trends in Machine Learning,2012,4(1):1-106.
[12] JOJIC V,SARIA S,KOLLER D. Convex envelopes of complexity controlling penalties:The case against premature envelopment[J]. Journal of Machine Learning Research,2011,15:399-406.
[13] CHARTRAND R. Nonconvex splitting for regularized low rank + sparse decomposition[J]. IEEE Transactions on Signal Processing,2012,60(11):5810-5819.
[14] WOODWORTH J,CHARTRAND R. Compressed sensing recovery via nonconvex shrinkage penalties[J]. IEEE Transactions on Information Theory,2015,11:1-29.
[15] CANDES E J,WAKIN M B,BOYD S P. Enhancing sparsity by reweighted L1 minimization[J]. The Journal of Fourier Analysis and Applications,2008,14(5):877-905.
[16] LORENZ D A. Non-convex variational denoising of images:Interpolation between hard and soft wavelet shrinkage[J]. Current Development in Theory and Application of Wavelets,2007,1(1):31-56.
[17] WIPF D,NAGARAJAN S. Iterative reweighted L1 and L2 methods for finding sparse solutions[J]. IEEE Journal of Selected Topics in Signal Processing,2010,4(2):317-329.
[18] NIKOLOVA M. Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares[J]. Multiscale Modeling and Simulation, 2005,4(3):960-991.
[19] SELESNICK I W,BAYRAM I. Sparse signal estimation by maximally sparse convex optimization[J]. IEEE Transactions on Signal Processing,2014,62:1078-1092.
[20] ECKSTEIN J,Bertsekas D P. On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators[J]. Mathematical Programming,1992,55(1-3):293-318.
[21] LUO Minnan,ZHANG Lingling,LIU Jun,et al. Distributed extreme learning machine with alternating direction method of multiplier[J]. Neurocomputing,2017,261:164-170.
[22] DONOHO D L,JOHNSTONE L M. Ideal spatial adaptation by wavelet shrinkage[J]. Biometrika,1993, 81:425-455.
[23] DING Yin,HE Wangpeng,CHEN Binqiang,et al. Detection of faults in rotating machinery using periodic time-frequency sparsity[J]. Journal of Sound and Vibration,2016,382:357-378.
[24] Bearing Data Center. Case western reserve university.[2017-09-01]. http://csegroups.case.edu/bearingdatacenter/pages/apparatus-procedures.
[25] GUO Wei,TSE P W,DJORDJEVICH A. Faulty bearing signal recovery from large noise using a hybrid method based on spectral kurtosis and ensemble empirical mode decomposition[J]. Measurement,2012,45(5):1308-1322.
[26] SAWALHI N,RANDALL R B,ENDO H. The enhancement of fault detection and diagnosis in rolling element bearings using minimum entropy deconvolution combined with spectral kurtosis[J]. Mechanical Systems and Signal Processing,2007,21(6):2616-2633.
