1. Introduction
Rolling bearing plays an important role in modern equipment [
1], relating to the health condition and the remaining service life. Amplitude-modulated and/or frequency-modulated (AM-FM) multi-component signals would be introduced when local damage such as cracks and exfoliation corrosion occurred during the operation [
2]. It becomes intractable to extract the fault feature as the fact of that the measured signals are usually immersed with intensive background noise, especially in the incipient failure. Thus, robust fault feature extraction method should perfectly eliminate the noisy components and acquire the useful features generated from the non-stationary time series, which provides the proof for the maintenance plan before the whole machine gets disabled.
Recent decades have witnessed significant advances in the feature extraction of fault data collected from either experiments or numerical simulations. Short-time Fourier Transform (STFT) [
3] adopts the method of piecewise averaging of the unstable signal. Several segments are sliced by a constant sliding window in accordance with the order of the original timeline. Spectrum characteristics are then received by Fourier Transform (FT) on each segment. However, the time-frequency resolution is constant and cannot transform over the change of time and frequency, which is the shortcoming of STFT. Wavelet Transform (WT) [
4], a method of local time and frequency conversion, was proposed to represent the original signal with a mother wavelet by scaling and translation operation. While it can effectively extract information with multiscale refinement analysis, WT has an inherent problem of low frequency resolution of high frequency band. Wavelet Packet Transform (WPT) [
5] was an extension of WT. Its basic idea was to make information energy concentration and decompose the signal into low-frequency components and high-frequency components. With further decomposition of the high frequency part, the time-frequency resolution of WPT is higher than that of WT. However, the choice of mother wavelets for WPT is hard to determine. Empirical Mode Decomposition (EMD) [
6] was proposed to decompose the original time series to different intrinsic mode functions (IMFs) by means of adaptive low frequency and high frequency components. EMD is a method of adaptive signal processing that can process nonlinear and non-stationary signals. However, the EMD algorithm has many imperfect problems that need to be solved and improved, such as the effect of the endpoint and mode aliasing problems [
7]. At the same time, the algorithm lacks physical significance. Singular value decomposition (SVD) [
8] can also be used in processing the noise-corrupted signal, but the decomposition effect differs from the various choice of singular values. SVD may not be helpful when the noise energy reaches a certain level [
9].
Dynamic mode decomposition (DMD), stemming from the fluid mechanics community [
10], has been characterized as a powerful tool for Koopman spectrum analysis technology based on equation-free and data driven methods. It can extract the spatiotemporal coherent characteristics by decomposing the complex flow field into a series of simple expressions based on its inherent space-time. Recently, DMD is successfully used for prediction, state estimation and complex system control in the field of fluid mechanics [
11]. Schmid [
12] firstly defined the DMD algorithm by demonstrating that the theory was capable of describing the internal mechanism based on the numerical simulation and measured experimental data in a high-dimensional flow field. The core of the DMD algorithm can be regarded as a space dimension reduction technique, similar to the algorithm of proper orthogonal decomposition (POD). While POD decomposes data into a series of multiple orthogonal modal frequency components, and ranks them in terms of energy content, the DMD algorithm can describe the dynamic characteristics in a series of single non-orthogonal frequency modes and superpose them with coefficients according to the time scale [
13]. Rowley [
14] pointed out that DMD was closely related to the Koopman operator theory, a linear but infinite-dimensional operator that represented the action of a nonlinear dynamic system. Thus, DMD is applicable to deal with nonlinear systems such as roller bearing signal (which we attempt in this paper) based on the consideration that DMD is a numerical approximation to Koopman spectral analysis. The standard DMD algorithm for mechanical time-series signal processing is outlined in
Section 2 with the theory provided by literature [
12]. Since the appearance of the DMD algorithm, widespread use and great achievements have been made in the field of fluid application [
11,
13,
14,
15,
16,
17,
18,
19,
20]. Nowadays, research regarding DMD mainly focuses on two aspects. Firstly, noise suppression algorithms were presented to correct the decomposition results due to the influence of noise. DMD cannot accurately extract dynamic characteristic information when the snapshots, periods of small data, are immersed by noise. These strategies range from a noise-corrected dynamic mode decomposition with control (ncDMD) [
15], forward-backward DMD (fbDMD) [
16], total least squares DMD (tlsDMD) [
17], compressed DMD (cDMD) [
18], and sparsity-promoting DMD (spDMD) [
19]. Though most of the varietal DMD algorithms can extract fluid dynamic information more accurately, they are not self-adaptive as they need to give a rank truncation order in their processes. Secondly, it aims at computational complexity reduction in pre-processing and/or post-processing when big data are processed. This problem does not apply to the mechanical signals, since the amount of mechanical signals is usually smaller relative to the fluid domain.
Whether the DMD reconstruction matrix best approximates the original system dynamics’ characteristics largely depends on two aspects: the optimal rank truncation order of the companion/similarity matrix and the dominant modes selection strategy. In a standard DMD algorithm, a certain rank order
of the similarity matrix needs to be determined for POD decomposition. It will increase the calculation time if over-estimating the
value because of a large amount of data. Simulation results showed that the cross-correlation coefficient between the reconstruction data and the original synthetic signal even get a bit smaller along with the increase of the selected
. Conversely, the calculation result may not be accurate if the
value is too small, which would deduct some useful signal characteristics. In the past, most scholars presented the rank truncation criteria based on SVD, including selecting an “elbow” of the singular value with a logarithmic curve and a threshold such as a certain proportion of selected data variance. They mainly use the idea of discarding noises by substituting a smaller (compressed) matrix that captures the essential characteristics of the signal. A recent breakthrough by Gavish [
21] provided a hard optimal threshold foundation in theory whether the noise magnitude is known or unknown. The methods of dominant mode selection strategy with a reduced order model are not identical. For a review of criterion to select dominant modes, we refer the reader to [
22], which summarized the recent research situation and analyzed their respective advantages and disadvantages. At the same time, Kou [
22] developed a method that multiplied each normalized DMD mode by its time coefficient. This method considered the evolution of each mode, but neglected the faint modes that should be treated as noise modes. Dang [
23] introduced a tool, multiscale permutation entropy, to calculate the complexity of each DMD mode. Modes whose entropies were smaller than a threshold were chosen to recover the signal. However, for filtering the noise components, an entropy threshold needs to be set by multiple attempts.
In this paper, a DMD framework for mechanical fault signal processing is proposed, solving the intractable problems of optimal rank truncation threshold and dominant modes selection strategy based on the conventional DMD algorithm. The optimal rank truncation of the similarity matrix is calculated with parametric non-convex penalty function for low-rank approximation. Inspired by WPT, which splits the signal up into a collection of different signals according to its level of decomposition, DMD modes are divided into slow modes and fast modes by splitting the continuous-time eigenvalues of the similarity matrix. Hierarchical application of the basic DMD algorithm is applied with a given level . DMD modes are reconstructed by a number of bottomed modes. Fast Fourier Transform (FFT) is performed on the reconstructed signal for the purpose of extracting the fault features. The improved DMD algorithm is applied to process the rolling bearing’s simulation signal for noise reduction and feature extraction. Amazingly, by comparing the signal-to-noise ratio (SNR), our method has tremendous advantages in noise removal and fault feature extraction, in comparison with traditional mature noise reduction methods such as EMD, SVD and WT. Then, the proposed method is applied to the bearing signal in practice; as a result, it can successfully extract the fault characteristics that are highly consistent with the actual field.
This paper is organized as follows.
Section 2 introduces the basic DMD algorithm for mechanical fault signal processing.
Section 3 measures the solving of the optimal rank truncation order of the similarity matrix, and the strategy of dominant modes selection are outlined. Analysis results of dynamic simulation signal and practical bearing fault signal are described in
Section 4. Conclusions are summarized in
Section 5.
2. DMD Algorithm for Mechanical Signal Processing
DMD is an equation-free Koopman frequency analysis technique based on the theory of SVD and mode decomposition. Essentially, it is a kind of order reduction method that can represent potential dynamics’ characteristics of a complex high-dimensional system by extracting a series of single frequency. An order reduction algorithm decomposed from DMD in flow fields is divided into two kinds of mathematical expressions. One introduces a companion matrix approximating to the infinite dimensional linear Koopman operator, and another presents a similarity matrix for a linear mapping operation on the singular values of POD. For detailed description of the two methods, we suggest reading [
22]. Both methods are capable of implementing the core algorithm, but the latter is more commonly used in numerical calculations for its robustness. We generalize the DMD algorithm framework for mechanical time-series signal processing based on the similarity matrix.
Supposing that a size of
sample points is acquired by a sensor with equal intervals between two successive sampling points, we define the one-dimensional time series as
, while
.
can also be made from a digital simulation signal. A
shift-stack Hankel matrix
can be constructed by Equation (1):
In order to achieve the maximum spatial and temporal complexity of the original noisy signal (As in Equation (7), When the matrix is close to the square matrix, the maximum characteristic frequency components can be obtained with SVD), the product of
and
should be as large as possible. According to the principle of inequality, the product achieves maximum when
and
are equal or close to each other. The dimension
is defined as follows:
where
is positive integer sequence that
.
It is possible to arrange the
column vectors into two
data matrices:
DMD is algorithmically a regression by assuming an optimal local linear approximation for mapping the current data to the subsequent data. The best-fit linear operator
is then used in terms of these signal matrices as:
Apparently, the evolution of the sequence and is determined by the eigenvalues of for the linear system. It can also be taken as an operator to approximate the dynamic characteristics when the data are produced in nonlinear systems, such as bearing vibration signal.
This solution minimizes the error:
where
is the Frobenius norm, given by
In what follows, we outline the key steps of the DMD algorithm for mechanical signal processing.
1. Firstly, seek an invertible matrix decomposition by SVD on matrix X:
where U and V are orthonormal, called the left and right singular vector, respectively,
. The symbol
denotes the complex conjugate transpose.
contains a number of non-zero singular values
by descending sequence in its diagonal.
The matrix
of Formula (4) may be obtained by the pseudoinverse of
:
2. It is quite normal that the matrix
contains such a large amount of data that leads to the calculating processing reluctant. Thus, we choose a given number of truncated rank
, and project it onto POD modes with the order of characteristic vectors. Thus, we have a similarity matrix
as follows:
where
. The eigenvalues and eigenvectors of
are then represented by those of
as they process the same dynamical features.
3. Perform eigenvalue decomposition on the similarity matrix
:
where
is the eigenvectors of similarity matrix
, and
is a diagonal matrix containing the corresponding complex eigenvalue
.
4. Compute the reconstruction matrix of DMD. The evolution of the signal characteristics can be characterized by the similarity matrix. In addition, the
i-th eigenvector of original operator is presented by the relevant feature vector of the similarity matrix:
Formula (11), the project DMD approximate solution, is often called standard DMD mode. By firstly rewriting for convenience
, the approximate solution of reconstruction matrix of DMD is then given by:
where
is a matrix consisting of DMD modes
,
is a diagonal matrix whose enters are continuous-time eigenvalues of the similarity matrix
,
is a vector containing the initial amplitude of each mode, and
,
denotes the Moore–Penrose pseudoinverse.
5. Finally, signal reconstruction and feature extraction can proceed based on the reconstruction matrix . We take the first column of as the recuperative signal that the length is just about half of the original signal. Then, the Fourier transform is applied on the recuperative signal for a spectrogram, determining whether there is a failure frequency.