1. Introduction
As an important power machinery in production and life, high-pressure common rail diesel engine plays an indispensable role in various industries, and the normal operation of diesel engine is a solid guarantee for our rich life [
1,
2]. The diesel injector is a key component of the diesel engine, and its working state will directly affect the operating power of the diesel engine [
3]. Because the injector works in the high-temperature and high-pressure environment inside the cylinder, faults occur frequently such as nozzle clogging, solenoid valve failure, needle valve stuck, etc. [
4]. These failures of the injector may result in abnormal fuel injection and uneven operation of each cylinder and may even result in further reduction in combustion efficiency and more exhaust emissions. Therefore, it is very necessary to diagnose the injector of high-pressure common rail diesel engine. Because the vibration signal combines a variety of information, including gas depletion, combustion shock, structural vibration, noise, etc., resulting in complex data processing and low diagnostic accuracy of fault feature extraction, the fuel pressure in common rail pipe and high-pressure oil pipe can acquire directly injection process information in cylinder [
5].
The injector operating status information can be reflected by the common rail fuel pressure wave, but the fuel pressure wave is a non-linear and non-stationary signal. In order to extract time domain and frequency domain information at the same time, it is necessary to apply joint time–frequency analysis method for fault detection [
6,
7,
8]. Wigner–Ville distribution (WVD), wavelet transform (WT), empirical mode decomposition (EMD), and local average decomposition (LMD) are commonly used representative methods. However, each representative method has some inherent defects in the extraction of non-stationary signal information. For example, WVD has unavoidable cross-interference terms, which has become an obstacle to its widespread application in signal processing [
9]. The wavelet transform can divide the frequency band into multiple layers, but it cannot further decompose the high frequency part, and the selection of the wavelet base has a significant influence on the decomposition effect, and the adaptive ability is insufficient [
10]. The EMD method has the advantages of intuitiveness and a-posteriori adaptability, but the shortcomings of mode aliasing, end effect, envelope overshoot and undershoot cannot be solved well [
11]. Compared with EMD, the Iterative Filtering method (IF) [
12,
13,
14] and its fast implementation (Fast Iterative Filtering, FIF) based on the Fast Fourier Transform [
15] has a solid mathematical background and all the advantages of the original EMD method and without the well-known shortcomings of mode aliasing, end effects, envelope overshoot and undershoot. The LMD method is an improvement of the EMD method, but when the original signal is disturbed by noise and the harmonic components with similar frequency components are decomposed, there are certain modal aliasing, end effect, and false component, which reduces the resolution and detection effect of the algorithm [
16].
Dragomiretskiy and Zosso proposed a variational mode decomposition (VMD) processing algorithm in 2014 [
17]. Unlike the traditional recursive algorithms of EMD, EEMD, and LMD (local mean decomposition), the VMD algorithm is free of the recursive screening and stripping mode of traditional signal decomposition. It can alleviate modal aliasing and boundary effects with a solid mathematical foundation, as well as the advantages of high-computational efficiency and robustness. However, VMD decomposition number K and α the penalty parameter α need to be preset, and inappropriate parameters may cause information loss or excessive decomposition problems, which may affect subsequent feature extraction [
18,
19,
20]. In 2011, Wen-Tsao Pan was inspired by the foraging behavior of Drosophila, and thus proposed the Fruit Fly Optimization Algorithm (FOA) [
21]. Compared with existing heuristic algorithms (such as genetic algorithm (GA), particle swarm optimization (PSO), and ant colony algorithm), FOA has few parameters to adjust, and it is easy to understand and implement by virtue of the simple computational process. Moreover, it has many advantages such as less control variables, strong control ability, and good global optimization. The optimization solves these problems and has been used by many scholars to solve complex parameter optimization problems [
22,
23,
24]. However, when a fruit fly in the fruit fly population finds the optimal solution for this iteration, all the fruit flies gather at the location of the fruit fly, but if the fruit found by the fruit fly is not the global optimal solution, then the algorithm falls into local optimum. Based on this, this paper proposes a variational mode decomposition based on the improved fruit fly algorithm (IFOA), which can not only adaptively optimize the internal parameters K and α of VMD but also improve the convergence speed and convergence precision of the algorithm.
How to extract fault characteristic information from nonlinear time series is the key to fault diagnosis of high-pressure common rail injector. In recent years, with the development of nonlinear scientific theories, various information entropy methods have emerged. Pincus [
25] proposed the concept of approximate entropy in 1991. Then, for the self-matching defect of approximate entropy, Richman et al. [
26] proposed the concept of sample entropy. Sample entropy is a commonly used feature extraction method, which has the advantages of strong anti-noise ability and short time series, but the method fault feature states can only be described from a single scale. Costa et al. [
27,
28] proposed a multi-scale entropy (MSE) based on sample entropy to measure the complexity of time series at different scales. Aiming at the sample entropy similarity measure in MSE, the mutation occurs easily. Zheng et al. [
29], combined with the concept of fuzzy entropy, proposed multiscale fuzzy entropy (MFE) and applied it to the fault diagnosis of rolling bearings. In order to extract the fault information of high-frequency components in the signal, Jiang Ying [
30] introduced the concept of hierarchical fuzzy entropy. Compared with multi-scale fuzzy entropy, hierarchical fuzzy entropy considers both the low-frequency component and the high-frequency component of the signal, thus providing more comprehensive and accurate time mode information. In order to alleviate the shortcomings of sample entropy, fuzzy entropy, and permutation entropy, Azami [
31] proposed dispersion entropy (DE) and proposed to measure the complexity of time series from different scale factors. Multiscale Dispersion Entropy (MDE) [
32] does not need to rank the amplitude values of one embedded vector, nor does it need to calculate the distance between any two composite delay vectors in different embedding dimensions. Compared with sample entropy and fuzzy entropy, dispersion entropy has the advantages of simple and fast calculation [
33,
34]. At the same time, dispersion entropy overcomes the main defects of permutation entropy and effectively solves the influence of medium amplitude of embedding vector [
35]. The study of analog signals and biosignals shows that dispersion entropy is relatively insensitive to noise and excellent in anti-interference compared to sample entropy and fuzzy entropy because small changes in amplitude values do not change their class labels; dispersion entropy is more sensitive to changes in the synchronization frequency, amplitude value, and signal bandwidth. Therefore, based on the superiority of analytic hierarchy process and dispersion entropy, hierarchical dispersion entropy (HDE) based on hierarchical entropy and dispersion entropy is proposed. The method describes the complexity and uncertainty of the sequence from different levels and reduces the deviation of the single scale, which realizes the feature richness of the signal sequence from many aspects. Compared with multi-scale sample entropy, hierarchical sample entropy [
36], multi-scale fuzzy entropy, hierarchical fuzzy entropy, and multi-scale dispersion entropy, it can not only consider the high-frequency and low-frequency components of the original sequence but also improve the anti-interference and signal bandwidth variation sensitivity. Finally, combining the IFOA-VMD algorithm with hierarchical dispersion entropy, a new fault diagnosis method for high-pressure common rail injector is proposed and applied to the analysis of engineering test data.
The rest of this article is organized as follows. In
Section 2, the principle of the VMD algorithm is briefly introduced and the flow of the IFOA-VMD algorithm is described. Then, in
Section 3, the hierarchical dispersion entropy algorithm flow is described, and the effectiveness of the proposed method is verified by numerical simulation signals. In
Section 4, the fuel pressure wave signal of the high-pressure common rail injector is analyzed by the fault diagnosis method proposed in this paper, and the effectiveness and superiority of the method are verified by SVM. Finally, the conclusion is given in
Section 5.
3. Hierarchical Dispersion Entropy
The IFOA-VMD method can obtain the IMF component sensitive to the injector fault characteristics, but how to accurately extract the fault state characteristic information from the nonlinear fuel pressure wave signal is an urgent problem to be solved. Comprehensive and accurate reflection of fault characteristics information is the premise of high-precision fault diagnosis. Therefore, this paper proposes a method to extract the hierarchical dispersion entropy (HDE) as the fault feature, which can reflect the fault characteristics of the injector as a whole, providing more comprehensive and accurate time mode information.
3.1. Hierarchical Dispersion Entropy Algorithm Flow
Referring to the advantages of hierarchical segmentation in hierarchical entropy, combined with the definition of dispersion entropy, the concept of hierarchical dispersion entropy is proposed. The calculation process of hierarchical dispersion entropy is as follows:
(i) Given a time series
with the length
N(
,
n is a positive integer), define the averaging operators
and
for the time series as follows:
where
and
carry the low-frequency and high-frequency features of u at scale 2, respectively.
When j = 0 or j = 1, the matrix operator
is defined as follows
(ii) In order to perform the hierarchical analysis on the signal
u(
i), the above operators have to be employed iteratively. Let
construct a vector
, then the integer e can be expressed as
In the formula, the vector corresponding to the positive integer e is ; k and e are the layer number and node number, respectively.
(iii) Based on the vector
, the hierarchical component
is expressed as
where
k represents the
k-layer in the hierarchical segmentation, and the original time series
u(
i) is represented by
and
in the low-frequency portion and the high-frequency portion of the k + 1 layer. For different
k and
e, the signals
consist of the hierarchical decomposition of signal
u(
i) in different scales. In
Figure 4, the hierarchical decomposition of
u(
i) in 4 scales is illustrated in the form of a hierarchical tree.
(iv) Find the dispersion entropy of each hierarchical component obtained and obtain the dispersion entropy of
hierarchical component [
31]. The hierarchical component sequence
is mapped to
by introducing a normal cumulative distribution (NCDF). The calculation formula is
, and the value ranges from 0 to 1. Next, we assign each y to an integer class with labels from 1 to
c. For each member of the mapped signal, we use
, where
shows the jth member of the classification time series.
(v) Introducing the embedded dimension m and the delay parameter
d and reconstructing the sequence,
is
Each time series is mapped to a dispersion pattern , where . The number of dispersion patterns that can be assigned to each time series is equal to since the signal has m members and each member can be one of the integers from 1 to c.
(vi) For each of
dispersion pattern, relative frequency is obtained as follows:
In fact, shows that the number of dispersion patterns that are assigned to divided by the total number of embedding signals with embedding dimension m.
(vii) Based on the definition of information entropy, the single dispersion entropy is
Hierarchical dispersion entropy can be expressed as
3.2. Parameter Selection
According to the definition of hierarchical dispersion entropy, four parameters need to be set before the calculation of hierarchical fuzzy entropy: signal length N, embedding dimension m, class number c, and decomposition layer number k. Since the k value is too large, it affects the computational efficiency and causes the points involved in each hierarchical component calculation to decrease. At the same time, if the k value is too small, the original sequence band division is not detailed enough to obtain sufficient gradation components from low frequency to high frequency. This paper sets the number of decomposition layers k = 3. In order to evaluate the sensitivity of hierarchical dispersion entropy to signal length N, embedding dimension m and class number, 40 sets of hierarchical dispersion entropy of white noise and 1/f noise of different lengths are calculated, and 40 different levels are calculated. The mean and standard deviation of the nodes are determined by the coefficient of variation (CV), where the coefficient of variation is CV = standard deviation/mean.
As shown in
Figure 5, it can be concluded from a and b that the larger the signal length
N is, the higher the stability is, and the smaller the error bar is, the difference between
N = 1024 and
N = 4096 is not obvious; from
Table 1 it can be seen that as the signal length is larger, the smaller the CV value is, the more stable the calculation of HDE is. In this paper,
N = 1024 is selected as the optimal signal length. It can be seen from
Table 2 and
Figure 6 that the CV value of the embedding dimension
m = 2 is small, indicating that the HDE value of
m = 2 is high and the error is small. In this paper,
m = 2 is selected as the optimal embedding dimension. It can be seen from the CV values of different classes in
Table 3 and
Figure 7. As
C increases, the CV value increases, the coefficient of variation of
c = 3 is the smallest, and the error rate is the lowest.
C = 3 is the best class.
3.3. Comparison with Other Methods
In order to verify that the proposed hierarchical dispersion entropy method is better than the current information entropy method, this paper compares the hierarchical dispersion entropy with multi-scale sample entropy, hierarchical sample entropy, multi-scale fuzzy entropy, hierarchical fuzzy entropy, and multi-scale dispersion entropy. A total of 40 sets of white noise and 1/
f noise were used as information entropy to calculate the samples, and the coefficient of variation CV of the same decomposition node of each information entropy was compared. The information entropy parameter selection is as follows: the decomposition layer number
k = 3, the signal length
N = 1024, the class number
c = 3, the embedding dimension
m = 2, the scale factor
. The comparison results are shown in
Figure 8 and
Table 4. The MSE and HSE are compared as examples. As the decomposition scale increases, the stability of HSE is significantly higher than that of MSE, and the CV value of HSE of decomposition node 4 is also smaller than MSE, indicating that the hierarchical entropy performance is significantly better than multi-scale entropy; the hierarchical entropy can consider the high and low frequency components, and extract the time mode information more comprehensively and accurately; taking MDE, MSE and MFE as examples to compare, with the increase of the decomposition scale, MDE is more stable than MSE and MFE. The CV value of the decomposition node 4 is also the smallest MDE and the error bar is the smallest, indicating that the performance of the dispersion entropy is significantly better than the sample entropy and the fuzzy entropy. The anti-noise of the dispersion entropy is better, and the bandwidth variation is more sensitive, and it is able to map status information more accurately. It shows that the CV value of HDE is the smallest and the calculation is the most stable. The HDE method is better than the existing public information entropy method, which not only improves the stability of entropy calculation but also reduces the bit error rate of entropy calculation.