A Novel Mechanical Fault Diagnosis Scheme Based on the Convex 1-D Second-Order Total Variation Denoising Algorithm
Convex 1-D first-order total variation (TV) denoising is an effective method for eliminating signal noise, which can be defined as convex optimization consisting of a quadratic data fidelity term and a non-convex regularization term. It not only ensures strict convex for optimization problems, but also improves the sparseness of the total variation term by introducing the non-convex penalty function. The convex 1-D first-order total variation denoising method has greater superiority in
... riority in recovering signals with flat regions. However, it often produces undesirable staircase artifacts. Moreover, actual denoising efficacy largely depends on the selection of the regularization parameter, which is utilized to adjust the weights between the fidelity term and total variation term. Using this, algorithms based on second-order total variation regularization and regularization parameter optimization selection are proposed in this paper. The parameter selection index is determined by the permutation entropy and cross-correlation coefficient to avoid the interference by human experience. This yields the convex 1-D second-order total variation denoising method based on the non-convex framework. Comparing with traditional wavelet denoising and first-order total variation denoising, the validity of the proposed method is verified by analyzing the numerical simulation signal and the vibration signal of fault bearing in practice. be non-stationary, non-linear, and non-Gaussian  . Therefore, there are some disadvantages for traditional denoising methods in mechanical fault feature extraction. Many scholars have adopted the wavelet transform as a tool for denoising due to its good time-frequency localization ability, but it is restricted by proper choice of wavelet basis function and decomposition level [6, 7] . Empirical mode decomposition (EMD) is a new non-stationary signal adaptive processing method, which has been widely used in one-dimensional signal processing  . However, this method also has some obvious shortcomings, such as lack of rigorous theoretical support, model-aliasing and the endpoint effect [9,10]. Moreover, some techniques have been heavily employed in fault diagnosis and fault reconstruction for a class of nonlinear systems [11, 12] . Traditional total variation (TV) algorithm is another widely used signal processing method, especially in image processing and one-dimensional signal processing [13, 14] , because it can suppress noise effectively and maintain a good image edge. Commonly, TV algorithm is constituted by a quadratic data fidelity term and a convex regularization term  . Subsequently, novel convex 1-D total variation denoising was introduced by the fact that non-convex regularizers can better recover flat signal regions  . Therefore, the non-smooth convex regularizer is replaced by a non-convex one such as logarithmic penalty or arctangent penalty [17, 18] . It is proven that convex 1-D total variation denoising has better capacity than the traditional total variation denoising method in signal reconstruction. Nevertheless, the regularization parameter is employed to adjust the weights between fidelity constraints and total variation term, which lack objective evaluation criteria. Furthermore, the abovementioned convex 1-D total variation (TV) denoising by the first-order total variation operator always produces undesirable staircase artifacts. Since the second-order difference is sparser than the first-order difference, it has been widely applied in image processing [19, 20] . In order to improve the performance of signal reconstruction and avoid the problem of staircase artifacts, the second-order total variation using non-convex penalty function is firstly employed for vibration signal processing in this paper. Then, this paper distinguishes the differences between unwanted signal such as noisy data and useful signals by permutation entropy in dynamics characters. The concept of permutation entropy (PE) was proposed in the application of measuring the complexity of one-dimensional time series  . It is an algorithm to describe irregular and nonlinear systems, which cannot be quantitatively described in a relatively simple way. The PE has the advantages of simple calculation and high sensitivity to signal change, which can be employed to evaluate the denoising effect [22, 23] . Nevertheless, the phenomena of excessive noise reduction should be avoided, as it causes the details to be lost in the signal. Thus, the reconstructed signal must maintain a certain similarity with original signal, which can be described by cross-correlation coefficient. Therefore, regularization parameter optimization method is proposed based on permutation entropy and cross-correlation coefficient in this paper. It not only will be able to ensure the greatest degree of suppression noise, but also can improve the similarity between the signal after noise reduction and the original signal, which can avoid the problem of excessive denoising. In order to verify the rationality and feasibility of the proposed method, it is used to analyze the numerical simulation and the actual fault vibration signal of rolling bearings in the bearing test rig and the fan in metallurgical industry. The rest of the paper is organized as follows. In Section 2, the basic ideas of convex 1-D second-order total variation denoising method are introduced. The simulation signal analysis is described in Section 3. The fault bearing data from bearing test rig is analyzed in Section 4. The industrial measurement signal analysis of faulty bearing in fan is illustrated in Section 5. The final conclusions are given in Section 6.