Shock and Vibration

Volume 2017 (2017), Article ID 4896056, 13 pages

https://doi.org/10.1155/2017/4896056

## Enhanced Orthogonal Matching Pursuit Algorithm and Its Application in Mechanical Equipment Fault Diagnosis

The Key Laboratory of Metallurgical Equipment and Control of Education Ministry, Wuhan University of Science and Technology, Wuhan 430081, China

Correspondence should be addressed to Jie Luo

Received 29 May 2017; Accepted 30 July 2017; Published 7 September 2017

Academic Editor: Giosuè Boscato

Copyright © 2017 Yong Lv et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The vibration signal measured from the mechanical equipment is associated with the operation of key structure, such as the rolling bearing and gear. The effective signal processing method for early weak fault has attracted much attention and it is of vital importance in mechanical fault monitoring and diagnosis. The recently proposed atomic sparse decomposition algorithm is performed around overcomplete dictionary instead of the traditional signal analysis method using orthogonal basis operator. This algorithm has been proved to be effective in extracting useful components from complex signal by reducing influence of background noises. In this paper, an improved linear frequency-modulated (Ilfm) function as an atom is employed in the proposed enhanced orthogonal matching pursuit (EOMP) algorithm. Then, quantum genetic algorithm (QGA) with the OMP algorithm is integrated since the QGA can quickly obtain the global optimal solution of multiple parameters for rapidly and accurately extracting fault characteristic information from the vibration signal. The proposed method in this paper is superior to the traditional OMP algorithm in terms of accuracy and reducing the computation time through analyzing the simulation data and real world data. The experimental results based on the application of gear and bearing fault diagnosis indicate that it is more effective than traditional method in extracting fault characteristic information.

#### 1. Introduction

According to the statistics, the fault derived from the mechanical parts, such as the shaft and the gear, mainly causes mechanical accidents. Commonly, these faults are often accompanied by nonlinear vibration phenomenon and its vibration signals are often complicated as the strong background noise [1]. Effective analysis method for vibration signal is the focus of much research in mechanical fault monitoring and diagnosis. In order to realize the mechanical fault monitoring and intelligent diagnosis, the method of feature extraction is widely adopted.

In view of nonstationary and nonlinear signal processing, many studies have been implemented. Gabor [2] proposed the STFT (short-time Fourier transform), which overcomes the inadequacy of the Fourier transform. However, it is unsuitable for dynamic signals due to the fixed time window width. Mallat [3] proposed the WT (wavelet transform), which overcomes the shortcoming that the window does not change with frequency. However, it lacks self-adaptability, because the results are closely related to the choice of wavelet basis function. Then, in order to completely reduce the limitations of time-frequency analysis method and better analyze the local time-frequency characteristics of nonstationary or nonlinear signals, Huang et al. [4] presented EMD (Empirical Mode Decomposition). It owns the advantages of orthogonality and completeness and has been widely applied in the fields of biomedical engineering, mechanical fault diagnosis, and analysis of seismic signal [5–7]. However, it creates problems such as envelope, owe envelope [8], mode mixing [9], and end effects [10].

The most traditional signal analysis methods are based on the inner product operation using orthogonal basis. Thus, it tries to use a fixed base function or the same base function of the properties that represents arbitrary signals and ignores the characteristics of the signal itself. Moreover, the energy of the decomposed signal will be distributed on different bases because of the orthogonality. This characteristic is bad for the signal recognition and compression. In order to achieve the fact that the representation of signal is self-adaptive, more flexible, and concise, Mallat and Zhang [11] proposed the sparse decomposition and created a novel method of signal sparse representation in view of wavelet analysis. It breaks the traditional ways of orthogonal basis selection and decomposes the signal into the linear combination of a series of the basic signal (atoms), in which overcomplete dictionary of atoms replaces orthogonal basis. It can not only efficiently realize the representation of signal, but also reflect the intrinsic characteristics of signal. Since this approach has advantages of self-adaptability and flexibility, it is widely used in the fields of image processing [12, 13], biomedical engineering [14, 15], communication engineering [16], and seismic signal processing [17, 18]. To date, through many years of development in signal sparse representation, many decomposition algorithms have been proposed, such as the MP (Matching Pursuit) algorithm, the OMP (Orthogonal MP) algorithm, the BP (Basis Pursuit) algorithm [19], the GBP (Greedy BP) algorithm [20], the ROMP (Regularized Orthogonal MP) algorithm [21, 22], the Co-SaMP (Compressive Sampling MP) algorithm [23], and the StOMP (Stage wise Orthogonal MP) algorithm [24].

In view of the sparse theory, scholars have done some researches of fault diagnosis. Candes et al. [25] presented the thought of stage wise matching pursuit and then proposed the line frequency modulation wavelet path tracking method. On the support of linear frequency modulation based wavelet path tracing method, Peng et al. [26] proposed a signal sparse decomposition method based on multiscale line frequency modulation. However, there exists problems like loss of local signal amplitude information, huge computation and decomposition of low efficiency, and so on. Moreover, the researchers have combined sparse theory with the other theories and presented many new methods for fault diagnosis [27, 28]. Wang et al. [29] combined minimum entropy solution of convolution with sparse decomposition for feature extraction of the weak fault. Similarly, Yan and Zhou [30] combined relevant cumulant with StOMP algorithm for feature extraction of the weak impulse signal. The computation efficiency is higher than traditional StOMP algorithm, but the performance of the StOMP algorithm is always dependent on the threshold selection. Taking all the above factors into consideration, the OMP algorithm is used in this paper.

In this paper, the enhanced orthogonal matching pursuit (EOMP) algorithm firstly uses an improved linear frequency-modulated (Ilfm) function as atoms, which can improve the precision and computing efficiency of reconstructing signal. Then it combines quantum genetic algorithm (QGA) with the OMP algorithm, as the QGA can quickly solve multiple parameters of the global optimal solution problem. It can further increase the calculation efficiency of the OMP algorithm. Thus, fault features extraction from vibration signal can be recognized. The EOMP algorithm is superior to traditional OMP algorithms on the accuracy and computation efficiency in the analysis of simulation data and experimental signal. It has been proved that the EOMP algorithms are more feasible and effective in real mechanical equipment fault diagnosis.

#### 2. Algorithm Description

##### 2.1. The OMP Algorithm

The MP algorithm belongs to the greedy iterative algorithm. In each iteration, it selects atoms from overcomplete dictionary that best matches with the structure of source signal. Then it requires only a small part of the atoms to accurately represent the source signal until the convergence by multiple iterations. The OMP algorithm is proposed by Pati et al. [31]. Naturally, it is consistent with the MP algorithm on the general idea. However, it adds the atoms orthogonalization to the algorithm, which makes the convergence of the OMP algorithm faster and better. The basic process of the OMP algorithm is briefly introduced as follows.

Given a source signal , where is the length of the signal space, . Define a dictionary of atoms , where is a set of *γ*, . The source signal is decomposed as follows:where is the projection of to and is the residual error after the first iteration. Obviously, is obviously orthogonal to . Hence, there exists the following equation for the decomposition of conservation of energy:

If the energy of the residual is minimized, becomes the maximum. This indicates that the best atom is related to the largest inner product of the signal and the selected atom.

For the MP algorithm, after iterations, the signal can be represented as

However, in the OMP algorithm, the selected atom needs to be transformed to Schmidt orthonormalization as follows:

Thus, the signal can be represented aswhere indicates that the signal can be represented by a very small amount of atoms.

##### 2.2. The Enhanced OMP Algorithm

In essence, the OMP algorithm is also greedy iterative algorithm and requires a lot of inner product operations to obtain the best atom in each decomposition, so the computation burden is too heavy. The EOMP method introduces quantum genetic algorithm into the OMP algorithm. It firstly codes the parameters of dictionary of atoms in the way of quantum bit and normalizes these parameters of atoms. Secondly, it makes the inner product of the atom and the signal as the fitness function. Then, in each iteration, it makes the best solution in the process of population evolution instead of the best solution in the contemporary evolution to update quantum gate. Finally it gets the optimal value of each parameter and the best atoms. In the end, the required signal is obtained by repeating the above iterative algorithm.

Quantum Genetic Algorithm (QGA), proposed by Han and Kim [32], is a probability evolutionary algorithm combining quantum computing with genetic algorithm. Compared with the traditional genetic algorithm, it has advantages of small population size, fast convergence rate and global optimization ability, and so on. However, the population evolution is the key to this kind of evolutionary algorithm. Consequently, quantum gate has a great influence on the performance of the QGA. In this paper; the quantum rotation gate is used in population evolution operationThe process for updating qubits is summarized as follows:where and , respectively, denote the fore-and-aft probability of the th quantum rotation gate updating and is rotation angle of the th quantum rotation gate, the size and direction of which determine the performance of the QGA.

The rotation angle of quantum gate is generally fixed and can be obtained from the table. To address this problem, a novel quantum genetic algorithm is proposed by Zhang et al. [33]. It can adaptively adjust the size of rotation angle. The rotation angle of the quantum rotation gate is described as follows:where has a great influence on the convergence rate of the QGA, so the value must be reasonable. In this paper, is defined as a variable related to the evolution iteration and the optimal solution, as described in (9). The EOMP can adaptively define the grid-search size in accordance with the iterations and then adaptively adjust through the comparison of the global best solution and the local best solution. Therefore, if the global best solution is better than the local best solution, it will appropriately reduce the value to diminish grid-search size and improve search precision. If the local best solution is better than the global best, it will appropriately increase the value to expand grid-search size and improve convergence rate. This guarantees the accuracy of the OMP algorithm at the same time and improves the convergence rate of the OMP algorithmwhere is the remainder of and ; and are the current iteration and the max iteration in the QGA; and denote the global best solution and the local best solution, respectively.

##### 2.3. The Dictionary Design

In addition to improving the process of the OMP algorithm, researchers have been the subjects of intense research for selection of the dictionary. Studies show that the dictionary has a great influence on the performance of the OMP algorithm [34, 35]. To date, Gabor dictionary is widely used and has a better performance in signal processing. It is determined by four factors of . But there exists a problem that the frequency of the time-frequency atom does not change with time. It also means that there is much truncation and mixed distortion among the signal component in the decomposition of the chirp signal. Moreover, the more parameters form, the more atom clusters and computation time are required.

A novel dictionary is recently proposed by Nagaraj et al. in the analysis of EEG signals [14] and is used to simulate the time-frequency response function of duffing oscillator through the improvement of linear frequency modulation function (Ilfm). It has a good effect on biomedical engineering due to duffing oscillator sensitive to weak characteristic signal. So, this dictionary is introduced into the proposed method for realizing the processing of mechanical fault signal.

This function of Ilfm dictionary is defined aswhere , , and , respectively, denote time resolution, central frequency, and the rate of change of frequency. This dictionary is determined by three factors of .

##### 2.4. The Simulation Evaluation

In this paper, the simulation platform is listed as follows: CPU (i7-4770k), Memory (8 G), and SSD (128 G). The simulation software is MATLAB. In order to test the effectiveness and accuracy of the EOMP method, the following evaluation indicators are introduced in this paper.(1)signal-to-error ratio (SER)(2)comparability index ()(3)mean square error (MSE)where is length of the signal, , , and denote the source signal, the residual after the th iteration, and the reconstruction signal after the th iteration, respectively.

The simulated signal model is described in (14) and (15) and shown in Figure 1.where is sine frequency modulation signal, of which the frequency is 25 Hz and is random noise, of which the amplitude is 0.3.