A Condition-Monitoring Approach for Diesel Engines Based on an Adaptive VMD and Sparse Representation Theory

Abstract

This paper presents a novel method for condition monitoring using the RMS residual of vibration signal reconstruction based on trained dictionaries through sparse representation theory. Measured signals were firstly decomposed into intrinsic mode functions (IMFs) for training the initial dictionary. In this step, an adaptive variational mode decomposition (VMD) was proposed for providing information with higher accuracy, and the decompositions were used as discriminative atoms for sparse representation. Then, the overcomplete dictionary for sparse coding was learned from IMFs to reserve the highlight feature of the signals. As the dictionaries were trained, newly measured signals could be directly reconstructed without any signal decompositions or dictionary learning. This meant errors likely introduced by signal process techniques, such as VMD, EMD, etc., could be excluded from the condition monitoring. Moreover, the efficiency of the fault diagnosis was greatly improved, as the reconstruction was fast, which showed a great potential in online diagnosis. The RMS of the residuals between the reconstructed and measured signals was extracted as a feature of condition. A case study on operating condition identification of a diesel engine was carried out experimentally based on vibration accelerations, which validated the availability of the proposed feature extraction and condition-monitoring approach. The presented results showed that the proposed method resulted in a great improvement in the fault feature extraction and condition monitoring, and is a promising approach for future research.

1. Introduction

For power machinery such as diesel engines, effective condition monitoring can be confirmed by both intrusive and indirect measurement methods [1,2], including in-cylinder pressure [3], acoustic emission [4], engine speed fluctuation [5], vibration signal sensors [6], and so on. Among these approaches, methods using vibration signals have attracted extensive attention due to their simplicity, high signal-to-noise ratio, and abundant information provided.

However, vibration signals excited by faults or improper parameter changes are often mixed with those of normal operation. Thus, accurate signal-processing techniques are required in general condition-monitoring technology. At present, various effective algorithms have been proposed to process the vibration signal and extract features from the time–frequency domain, such as wavelet transform (WT) [7], empirical mode decomposition (EMD) [8], short-time Fourier transform (STFT) [9], and so on. Bi et al. [10] proposed a combination of wavelet denoising and EMD to detect knocking in gasoline engines from measured acceleration. Moosavian et al. [11] used STFT and continuous wavelet transform (CWT) to investigate the vibration behavior of piston scratching faults. However, traditional time–frequency analysis methods, such as STFT, cannot consider time-domain resolution and frequency-domain resolution simultaneously. WT is not a self-adaptation decomposition method either [12]. The selection of a WT basis has a significant influence on the analysis. Although EMD is adaptive and uses a recursive algorithm in principle, errors produced by envelope calculations will accumulate during iteration, and results in the mode-mixing phenomenon. The ensemble empirical mode decomposition method (EEMD) [13,14] was proposed to solve this problem, but it resulted in new problems, such as difficult white-noise removal and a decrease in computing speed. To overcome these problems, Dragomiretskiy and Zosso [15] proposed the variational mode decomposition (VMD) algorithm in 2014. As an alternative to EMD, VMD was mainly established using Wiener filtering, which can avoid the shortcomings derived from the recursive model. VMD has been widely applied in the fault diagnosis of mechanical systems [16,17]. Zhang et al. [17] compared the performances of VMD and EMD in the extraction of rolling bearing defect characteristics. The results verified the advantages of VMD in the extraction of bearing defect characteristics. M. Civera and C. Surace [18] reviewed the signal-decomposition techniques for structural health monitoring. The experimental results showed that the VMD algorithm was found to outperform both CEEMDAN and Hilbert vibration decomposition (HVD) for applications in the vibration recordings of mechanical systems. Similar comparisons were made in [19,20], and the same conclusions were drawn. Although VMD has shown great potential in fault diagnosis, the key parameters—decomposition level K and quadratic penalty α—still need to be determined thoroughly. Ni et al. [21] proposed a fault-information-guided VMD method to extract the weak bearing repetitive transient. The mode number was determined using two nested statistical models, and the optimal bandwidth control parameter was determined using the ratio of the fault characteristic amplitude. Dibaj et al. [22] determined the optimal values of decomposition parameters K and alpha by judging the adaptive indices, including the mean correlation coefficients between the adjacent modes and the energy loss coefficient between the original signal and the reconstructed signal, which should not exceed the defined thresholds for optimal values. With the rapid development of bionic optimization technology, various metaheuristic algorithms were used to simultaneously optimize the K and α of VMD, such as a genetic algorithm (GA) [23], particle swarm optimization (PSO) [24], a grasshopper optimization algorithm (GOA) [25], and a whale optimization algorithm (WOA) [26]. Although these parameter-adaptive VMD methods have achieved good results in their respective research fields, the inefficiency of the optimization process is an urgent problem.

In the practice of condition monitoring and fault diagnosis, the methods mentioned above are mostly focused on impulsive feature extraction. However, when incipient faults occur, a pseudo-cyclostationary impulsive signal is produced and submerged in noise or other harmonics [27]. Thus, the relationships between the obtained signal component and the impulsive features need to be further explored. Sparse representation theory is an effective approach to determine the information between the basis (atoms) of a dictionary and the target vibration signals, of which the basic principle is to represent a signal by a linear combination of a few atoms [28]. There are two steps in performing sparse representation: dictionary design and sparse coefficient solving. The dictionary can be predefined or learned by using algorithms such as k-singular value decomposition (K-SVD) [29] or the method of optimal directions (MOD) [30]. After the dictionary is obtained, the sparse coefficients are calculated. The commonly used solution includes orthogonal matching pursuit (OMP) [31], basis pursuit algorithm (BP) [32], etc. To extract an accurate impulsive feature excited by a fault or improper parameter change, the atoms should be adopted properly to reconstruct the characteristic patterns hidden in the vibration signals. One of the most commonly reported methods of dictionary learning is construction from wavelets [33,34]. Qin [33] constructed an impulsive wavelet dictionary to extract repetitive impulses. The performance of the dictionary atoms was validated by simulated signal and real vibration signals with bearing fault information. Perhaps the most serious disadvantage of this method was that the dictionary was learned on a wavelet basis. The representation results were mainly determined by the dictionary construction. Since wavelet transform is not an adaptive method, the predefined basis should be selected or designed artificially, which may cause inefficiencies and inaccuracies in condition monitoring. The intrinsic mode functions (IMF) of VMD were also well-matched with the components of the original signal, with specific sparsity and good resolution in both the time and frequency domains [35,36]. Thus, it was feasible to construct a dictionary from IMFs. For a certain condition, the corresponding dictionary could properly represent the measured signals. However, for a totally different condition, the same representation would show dissimilarities. Under this consideration, these dissimilarities were considered as the index for condition identification.

The core of the proposed mode identification approach was to use the dictionary developed to directly reconstruct the measured signal without further decomposition in order to improve the efficiency of the diagnosis and avoid errors in signal processing. In the stage of dictionary development, proper frequency components from sample signals were still essential to provide the available mode information to train the dictionary. Once the validity of dictionary was verified, the signal decomposition step could be excluded, and the residual from the signal reconstruction by the dictionary could be used to identify the condition of system. The scheme diagram of the proposed approach is shown in Figure 1.

The rest of the paper is organized as follows. Section 2 introduces the proposed condition-monitoring method. Section 3 demonstrates the proposed approach with a case study. Some comparisons with the existing methods and discussions of the performance of our method are given in Section 4. Finally, the conclusions and outlook are presented in Section 5.

2. Methods

2.1. Variational Mode Decomposition

VMD can decompose any signal f into K discrete number modes $u_k(t)$, otherwise known as the intrinsic mode function [15]. Each mode compacts around a center frequency ${\omega }_k$ while satisfying that the sum of all modes equals f.

Firstly, the estimated unilateral spectrum of $u_k(t)$ and its center frequency ${\omega }_k$ was obtained by Hilbert transform. To obtain the bandwidth of $u_k(t)$, VMD was required to solve the constrained variational problem as follows:

$$\{\begin{array}{l}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{{\displaystyle \sum _k\Vert {\partial }_t[(\delta (t)+\frac{j}{\pi t})\ast u_k(t)]e^{-j{\omega }_{k}t}\Vert }}_2^2\right\}\\ s.t.{\displaystyle \sum _k{u}_k}=f\end{array}$$

(1)

where t is the time, $\delta (t)$ is the Dirac distribution, ∗ represents a convolution symbol, and $\left\{{\omega }_k\right\}=\{{\omega }_1,{\omega }_2,\dots ,{\omega }_k\}$ indicates center frequencies.

The penalty parameter $\alpha$ and the Lagrange multiplication operation $\lambda (t)$ were introduced to convert the constrained variational problem to a nonconstrained problem. $\alpha$ guaranteed the accuracy of the reconstructed signal in the presence of Gaussian noise, and $\lambda (t)$ maintained the strict enforcement of the constraint. Then, the augmented Lagrangian could be described as below:

$$\begin{array}{cc}L(\left\{u_k\right\},\left\{w_k\right\},\lambda ):& =a{\displaystyle \sum _k{\Vert {\partial }_t[(d(t)+\frac{j}{\pi t})\ast u_k(t)]e^{-j{w}_{k}t}\Vert }_2^2}\\ & +{\Vert f(t)-{\displaystyle \sum _k{u}_k(t)}\Vert }_2^2+\langle \lambda (t),f(t)-{\displaystyle \sum _k{u}_k(t)}\rangle \end{array}$$

(2)

where $\langle ·\rangle$ denotes the inner product operation.

To solve this equation, the alternating direction method of multipliers (ADMM) was used here to obtain the saddle point of Equation (2). The mode number k was determined in advance. Then, the frequency-domain expression of mode $\left\{{\stackrel{⌢}{u}}_k^1\right\}$, the corresponding center frequency $\left\{{\stackrel{⌢}{\omega }}_k^1\right\}$, and the Lagrangian multiplier $\left\{{\stackrel{⌢}{\lambda }}^1\right\}$ were initialized. The mode $u_k$ and the center frequency ${\omega }_k$ were updated as follows:

$${\stackrel{⌢}{u}}_k^{n+1}(\omega )=\frac{\stackrel{⌢}{f}(\omega )-{\displaystyle \sum {}_{i\ne k}{\stackrel{⌢}{u}}_i(\omega )+\frac{\stackrel{⌢}{\lambda }(\omega )}{2}}}{1+2\alpha {(\omega -{\omega }_k)}^2}$$

(3)

$${\omega }_k^{n+1}=\frac{{\displaystyle {\int }_0^{\infty }\omega {\left|{\stackrel{⌢}{u}}_k(\omega )\right|}^{2}d\omega }}{{\displaystyle {\int }_0^{\infty }{\left|{\stackrel{⌢}{u}}_k(\omega )\right|}^{2}d\omega }}$$

(4)

Then, the Lagrangian multiplier was also updated by:

$${\stackrel{⌢}{\lambda }}^{n+1}(\omega )\leftarrow {\stackrel{⌢}{\lambda }}^n(\omega )+\tau \left[\stackrel{⌢}{f}(\omega )-{\displaystyle \sum _k{\stackrel{⌢}{u}}_k^{n+1}(\omega )}\right]$$

(5)

The implementation of VMD algorithm can be given as follows:

• Initialize $\left\{u_k^1\right\}$, $\left\{{\omega }_k^1\right\}$, ${\stackrel{⌢}{\lambda }}^1$, and n.
• Update $u_k$ by Equation (3).
• Update ${\omega }_k$ by Equation (4).
• Update $\lambda$ by Equation (5).
• Repeat steps (2)–(4) until the termination condition is met:

$${\displaystyle \sum _k\frac{{\Vert {\stackrel{⌢}{u}}_k^{n+1}-{\stackrel{⌢}{u}}_k^n\Vert }_2^2}{{\Vert {\stackrel{⌢}{u}}_k^n\Vert }_2^2}}<e$$

(6)

2.2. Adaptive VMD

2.2.1. Correlated Kurtosis

The correlated kurtosis criterion was used to indicate the impulsive feature of the decomposition result. Improper parameters of VMD will influence the correlated kurtosis criterion of the decomposition results. Compared to the kurtosis index, which tends to a maximum with a single impulse, the correlated kurtosis was only sensitive to the periodic impulses of a specific period T. Thus, it could effectively avoid the interference terms, and was suitable for the decomposition of multisource vibration signals. For a periodic shock signal y with length N, the correlated kurtosis is given as follows [37]:

$$C{K}_M\left(T\right)=\frac{{\displaystyle {\sum }_{n=1}^N{\left({\displaystyle {\prod }_{m=0}^M{y}_{n-mT}}\right)}^2}}{{\left({\displaystyle {\sum }_{n=1}^N{y}_n^2}\right)}^{M+1}}$$

(7)

where M is the order of shift (in this paper M = 1) and T is the period of interest.

In this paper, y is the extracted mode of VMD. M = 1 because an M that was too large not only would increase the computational complexity, but also would easily lead to the drift problem. The value of impact sensitive period T was determined by $T=\lceil \raisebox{1ex}{$f_s$}\!\left/ \!\raisebox{-1ex}{${\omega }_k$}\right.\rceil$, where f_s is the sampling frequency, $\lceil ·\rceil$ represents rounding up to the nearest integer, and ${\omega }_k$ is the center frequency of the IMFs defined in Section 2.1.

2.2.2. Grey Wolf Optimization

The grey wolf optimization algorithm has the advantages of a simple principle, fast convergence, and a high accuracy [38]. GWO has been proved to be superior to the common genetic algorithm, particle swarm optimization algorithm, and ant colony optimization algorithm in exploitation and exploration, high local optima avoidance, and fast convergence. In recent years, it has been widely used in the field of machinery condition monitoring [39,40].

The grey wolf algorithm is an intelligent algorithm that simulates the collective hunting process of gray wolves. In the grey wolf algorithm, wolves are divided into four levels: α wolf, β wolf, δ wolf, and ω wolf. GWO defines the optimal solution, suboptimal solution, and the third optimal solution as the α wolf, β wolf, and δ wolf, respectively. These three wolves are used to guide hunting. Other alternative solutions are defined as the ω wolf, following these three to search, track, and surround their prey.

In the searching process, the behavior of wolves can be described the following formula:

$$\{\begin{array}{l}D=\left|C\cdot X_p(t)-X(t)\right|\\ X(t+1)=X_p(t)-A\cdot D\end{array}$$

(8)

where D is the distance between the wolf and the target prey, X(t) and X_p(t) are the current position of the wolf and the target prey, t represents the current iteration, and A and C are the coefficient vectors determined by:

$$\{\begin{array}{l}a=2(1-t/M)\\ A=2a\cdot rand1-a\\ C=2\cdot rand2\end{array}$$

(9)

where rand1 and rand2 are random vectors with values between [0, 1], and a is the convergence factor, which decreases from 2 to 0 in the iterative process.

In the surrounding process, the position of the α wolf, β wolf, and δ wolf are updated using Equations (10) and (11):

$$\{\begin{array}{l}{X}_1=X_{\alpha }-A_1{D}_{\alpha }\\ X_2=X_{\beta }-A_2{D}_{\beta }\\ X_3=X_{\delta }-A_3{D}_{\delta }\end{array}$$

(10)

$$X(t+1)=(X_1+X_2+X_3)/3$$

(11)

where D_α, D_β, and D_δ are the distance between the corresponding wolf and prey; and X_α, X_β, and X_δ represent the positions of the respective wolves.

After surrounding, the group of grey wolves begin to attack the prey. This process can be simulated by decreasing A from 2 to 0 with a in the iterative process. When $\left|A\right|$ ≤ 1, the grey wolf attacks the prey, achieving a local search. When $\left|A\right|$ > 1, the wolves give up hunting and begin a new round of search, and perform a global search.

2.2.3. Adaptive VMD Based on GWO

The proposed method realized parameter optimization of VMD using the GWO algorithm. The objective function was constructed as follows:

$$\{\begin{array}{l}fitness=\underset{\gamma =\left(K,\alpha \right)}{\min }\left\{-CK\right\}\\ s.t.K\in \left[2,10\right]\\ \alpha \in \left[200,5000\right]\end{array}$$

(12)

where fitness represents the fitness function, CK is the correlation kurtosis of the VMD mode u_i, and $\gamma =\left(K,\alpha \right)$ is the parameter group of VMD to be optimized. The details of this method can be described as follows:

• Load the measured vibration signal.
• Initialize the parameters of GWO, such as the maximum iteration number t = 20 and population size n = 100.
• Extract the mode using VMD, and calculate the fitness of all modes. Use Equation (7) to evaluate the fitness value.
• Determine whether the termination condition was achieved. If not, update the positions of wolves until reaching the termination criterion and continue the iteration.
• Obtain and save the optimal parameters and the minimum fitness.
• Use the optimal parameters for VMD to extract the modes.
• Use the decomposition results further for dictionary learning.

To verify the ability of the proposed method, a simulated signal was constructed and decomposed using the proposed method and some published objective functions [25,41]. The simulated signal S was composed of two sinusoidal signals (S₁ and S₂), a periodic pulse signal S₃, and a random noise S₄. S is given as below:

$$\{\begin{array}{ll}{S}_1& =\sin (2\pi \times 120t)\\ S_2& =1.2\ast \sin (2\pi \times 230t)\\ S_3& =20\times e^{-1.5\ast {10}^4\ast {\left(t-0.2\right)}^2}\sin (2\pi \times 400t)+20\times e^{-1.5\ast {10}^4\ast {\left(t-0.5\right)}^2}\sin (2\pi \times 400t)\\ & +20\times e^{-1.5\ast {10}^4\ast {\left(t-0.8\right)}^2}\sin (2\pi \times 400t)\\ S_4& =\eta \\ S& =S_1+S_2+S_3+S_4\end{array}$$

(13)

In this signal, $\eta$ was a Gauss white noise of 10 dBw. The sampling frequency of the simulation signal was 1024 Hz.

Figure 2 shows the waveform of the simulated signal and its Fourier spectrum. Figure 3, Figure 4 and Figure 5 show the decomposition results and the reconstructed signal S_R found by using different objective functions. Figure 3 represents the results using the CK index, Figure 4 is the KCI, and Figure 5 is the energy difference between the original signal and the reconstructed signal.

As shown in Figure 4, the pulse component was confused in IMF1, as the two sinusoidal signals were mixed up. Figure 5 shows that an overdecomposition problem occurred, as there were two irrelevant components (IMF1 and IMF4) obtained by VMD. As compared to Figure 2, the decomposition results in Figure 3 well reflected the original frequency distribution of the original signal. No modal aliasing, underdecomposition, or overdecomposition problem occurred in this situation. Hence, the correlation kurtosis was suitable for VMD optimization.

2.3. Sparse Representation Theory

Although the vibration signal could be decomposed by the adaptive accurately, the efficiency of the method presented a problem regarding the iteration of both the GWO and VMD. Sparse representation classification was used to solve the problem.

The basic idea of sparse representation theory is to represent a signal as a linear combination of atoms in a dictionary [28]. A signal f with length n can be viewed as a vector in ${ℝ}^n$. An overcomplete dictionary $D=\left\{d^1,d^2,\cdots ,d^m\right\}$ consists of m atoms, and d is atoms of the dictionary. The length of each atom is also n. The atoms can span the entire space when $m>n$. Thus, signal f can be represented as a combination of basic atoms:

$$f=Dx={\displaystyle \sum _{i=1}^m{d}^i{x}^i}$$

(14)

where $x={\left[x^1,x^2,\cdots ,x^m\right]}^T$ are coefficients of atoms.

To obtain a sparse representation, the solution should meet following criteria:

$$\arg \underset{x}{\min }{\Vert x\Vert }_0,s.t.f=Dx$$

(15)

where ${\Vert ·\Vert }_0$ denotes the $l_0$ norm, which is defined as the number of nonzero elements. Using Equation (15), we can see that there are two problems to solve in sparse representation: the designing of dictionary D and the solving of coefficient x.

2.3.1. Coefficient Solving

The process to obtain a minimum ${\Vert x\Vert }_0$ in Equation (8) is NP-hard. The OMP is a commonly used solution for its efficiency. The OMP is a greedy pursuit algorithm. In every step, the OMP selects atoms with the highest correlations with the residual as follows:

$$i=\arg \max \left|d_i^{T}r\right|$$

(16)

where i is the position of the selected atom and r is the residual.

The coefficient can be calculated as below using the least squares method:

$$x=\arg \min {\Vert f-\varphi x\Vert }_2$$

(17)

where $\varphi$ is the index set and ${\Vert ·\Vert }_2$ is the $l_2$ norm.

All selected atoms were normalized to ensure the efficiency of the calculations.

The OMP algorithm can be described as follows [31]:

• Initialize the residual $r_0=f$ and variable $\varphi$;
• Select most relevant atoms (Equation (16));
• Calculate the coefficient using Equation (17);
• Update the residual r by $r=f-\varphi x$;
• Repeat steps (2)–(4) until one of the following termination conditions is met:

$${\Vert r^l\Vert }_2<\epsilon$$

(18)

$${\Vert x\Vert }_0<l$$

(19)

where l is the sparsity level and $\epsilon$ is the excepted error.

In this paper, Equation (19) was used as the termination criteria, because residuals are more intuitive when evaluating the performance of the sparse representation.

2.3.2. Dictionary Learning

The dictionary of sparse representation can be obtained by a predefinition or training from data. Usually, predefined dictionaries cannot represent the signal properly, because the predefined atoms may not be able to reflect the patterns of the signal. Dictionary learning can grasp the internal structure of the data more accurately, and the essential characteristics of the data can be reflected.

Considering a set of P samples $F=\left[f_1,f_2,\cdots ,f_P\right]$, the objective is to find a dictionary that provides the best representation for each individual. The problem can be described mathematically as follows:

$$\arg \underset{D,X}{\min }{\Vert F-DX\Vert }_F^{2}s.t.{\Vert x_i\Vert }_0\le l_0,1\le i\le P$$

(20)

where $X=\left[x_1,x_2,\cdots ,x_P\right]$ is the sparse coefficient. ${\Vert ·\Vert }_F^2$ represents the Frobenius norm:

$${\Vert X\Vert }_F^2=\sqrt{{\displaystyle \sum _{ij}{X}_{ij}^2}}$$

(21)

K-SVD updates the dictionary atom-by-atom, thereby achieving more efficient solution [29]. In K-SVD, OMP is used to calculate the sparse coefficients of randomly chosen examples given by Equation (22):

$${\widehat{x}}_i=\arg \underset{x}{\min }{\Vert f_i-Dx\Vert }_2^{2}s.t.{\Vert x_i\Vert }_0\le l_0,1\le i\le P$$

(22)

Then, the coefficients are employed to calculate the residual matrix between the original samples and the reconstructed signals by using selected atoms:

$$E_{j0}=F-{\displaystyle \sum _{j\ne j_0}{a}_j{x}_j^T}$$

(23)

where j is the column index. In this process, to avoid breaking the sparsity of coefficients, only nonzero elements in $x_j^T$ were processed. The OMP algorithm is given in detail as follows:

• Initialize the iteration counter $k=0$ and column index, and generate $D_{\left(0\right)}\in {ℝ}^{n\times m}$ either by using random entries or m randomly chosen examples.
• Increase k by 1 and calculate sparse coefficients using Equation (22).
• Define a group of examples ${\mathsf{\Omega }}_{j0}$ using atom $x_{j0}=u_1$.
• Calculate the residual matrix using Equation (23).
• Obtain $E_{j0}^R$ by restricting $E_{j0}$ corresponding to ${\mathsf{\Omega }}_{j0}$.
• Apply SVD on $E_{j0}^R=U\Delta V^T$
• Repeat (3)–(6) for $j_0=1,2,\cdots ,M$ to update the columns of the dictionary and obtain $D_{\left(k\right)}$.
• Repeat (2)–(7) until the predefined threshold or iteration count is reached.

2.4. Fault Diagnosis Workflow

A workflow conducted using the proposed approaches for fault diagnosis is given in Figure 6. In the training phase, signals measured from object system were decomposed by the adaptive VMD to construct a set of IMFs. An initial dictionary consisting of those IMFs was developed and trained through signals from the training data set using K-SVD. The dictionaries only needed to be trained once offline. With a trained dictionary, signal decomposition methods (such as VMD) could be excluded from the following fault diagnosis process. In application, the measured signal was directly reconstructed by the trained dictionaries using OMP, and dissimilarities between the reconstructed and tested data were calculated for condition identification based on the RMS of residuals.

3. Case Study: Valve Clearance and Injection Timing of a Diesel Engine

3.1. Bench Test

The experiment was performed on a turbocharged 6-cylinder diesel engine in a semianechoic chamber, as shown in Figure 7. The engine specifications are listed in

Table 1. Main specifications of the diesel engine used.

Items	Specifications
Number of cylinders	6
Displacement	7.14 L
$\mathrm{Bore}\times$ Stroke	$108\mathrm{mm}\times$ 130 mm
Max power	220 kW @ 2300 rpm
Max torque	1160 Nm @ 1200–1600 rpm

. The vibration signals were collected using an LMS Scadas acquisition system (Siemens, Germany) through PCB 621B40 accelerometers (PCB Piezotronics, USA). As shown in Figure 7b, accelerometers were attached to the head and block of each cylinder. Data were recorded every 100 rpm from 800 rpm (idle speed) to 2300 rpm (maximum speed) at the rated power. The sampling frequency was 25,600 Hz to ensure the following analysis could cover the band of the main vibration energy; i.e., 10,000 Hz. Three working modes were tested, including a normal condition, a varying valve-clearance condition, and a varying injection-timing condition, as described in

Table 2. Details of working modes.

Number	Condition Description	Specification
1	Normal	Default value set by manufacturer
2	Small valve clearance	0.25 mm (intake), 0.45 mm (exhaust)
3	Large valve clearance	0.35 mm (intake), 0.55 mm (exhaust)
4	Small advance in injection timing	1 °C A
5	Large advance in injection timing	2 °C A
6	Small delay in injection timing	1 °C A
7	Large delay in injection timing	2 °C A

.

3.2. Signal Processing Based on Adaptive VMD

As mentioned above, the decomposition level K and the quadratic penalty α are the key parameters in VMD. To obtain an optimal combination of these two parameters, a signal under Condition 1 was randomly intercepted for 0.06 s and 1538 data points, as shown in Figure 8, which was the length of about one working cycle at 2000 rpm. It was found that the vibration signal was composed of a series of shock responses in the time domain, and the energy was distributed in a wide range of frequencies.

The signal was decomposed by the adaptive VMD. The optimized parameters were K = 6 and α = 6159. Using the optimal K and α, the decomposition and reconstruction of the original signal were carried out, and are shown in Figure 9 and Figure 10. As shown in Figure 9, each IMF was separated clearly without an overdecomposition phenomenon. Each sub-band was restricted to a narrow frequency range. Compared with those in Figure 8, the noise components in Figure 10 were reduced to an acceptable degree, and the impulsive features were reserved well.

To explain the advantages of VMD, the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) was employed to decompose the measured signal; the results are shown in Figure 11. There were 11 IMFs in the results of the CEEMDAN; only the first 7 IMFs are shown, as the rest were obvious illusive components.

As shown in Figure 11, the CEEMDAN could not decompose the frequency components completely, especially for the high-frequency range. The bandwidths of the first four IMFs were too large, as too much noise existed in them. The decomposition results affected the recognition of faults.

To further verify the accuracy of the decomposition, a set of 50 signals under Condition 1 was decomposed using the original VMD. The decomposition process used a different decomposition level K to show the central frequency variation. K was set as 5–7. The central frequencies of the IMF were obtained as shown in Figure 12. As shown in Figure 12, when K = 5, the central frequencies of IMF1 and IMF2 fluctuated greatly because multiple components were mixed into one mode; that is, underdecomposition occurred in this situation. When K = 7, the central frequencies of IMF6 and IMF7 were quite close to each other. Moreover, they had the same trend in variation, which indicated that false components appeared in these cases. When K = 6, the central frequencies of the six decomposed IMFs were relatively smooth for all signals used, and the distances in the central frequencies were proper. Thus, K = 6 was considered as the optimal value.

3.3. Dictionary Learning

The results of VMD; i.e., the IMFs, were narrow band signals with sparsity. Thus, these IMFs could be used for dictionary learning. As discussed in Section 2, to construct an overcomplete dictionary requires m > n, where n is the length of the atoms and m is the number of atoms. We found for IMF1 that the lowest central frequency was around 800 Hz. Thus, the IMFs could be further compressed to a lower dimension. The sampling frequency was 25,600 Hz, which meant at least 25,600/800 = 32 sampling points could contain all the impulses. On the good side, an exact number of n was not essential for building a underdetermined system. Here, the length of the atoms was set as 200. In this section, 100 samples with 1538 points in length of each condition were used, and 600 IMFs were obtained per condition. These IMFs were intercepted with a length of 200, and seven 200 $\times$ 600 dictionaries were obtained. Thereafter, a training set conducted using 1200 IMFs per condition (that is, 200 samples in each condition) was used for dictionary learning. These dictionaries were learned using K-SVD. The sparsity level was 6, and the iteration number was 100. The reconstruction residual for each dictionary learning process was recorded as shown in Figure 13. The reconstruction residuals converged to a small value smoothly in all the cases. The atoms could successfully capture the characteristics of the training set. The learned dictionary could be used for sparse coding.

3.4. Feature Extraction

Another 7000 samples (1000 per condition) were collected as the testing set to verify the ability in feature extraction. Each sample in the training set and the testing set contained 200 points, and 1000 samples of each condition were sparse-coded by all the respective seven dictionaries using OMP. Thus, seven sets of reconstructed signals were obtained, and 7000 residual signals were derived for each dictionary. The RMS value of the residuals were calculated as the index for indicating the discrimination. The results are shown in Figure 14.

As can be seen in Figure 14, the proposed residual feature showed a great clustering performance for each condition. Taking Condition 1 as an example, reconstructions by the seven dictionaries showed the smallest residuals between the test data and the reconstructed signals coded by the dictionary of Condition 1. Moreover, the residuals were distributed in a relatively narrow range, which proved that the feature was a stable index for condition monitoring. For residuals from other dictionaries, as shown in Figure 14a, obvious deviations could be found from the black points. This was mainly caused by the discrimination of the data structure between the dictionary and the test signals of other conditions. This conclusion was appropriate for other scenarios as well.

Furthermore, a threshold set was confirmed for condition monitoring. Given a random obtained signal, the signal was sparse-coded by the respective seven dictionaries, and the residuals were calculated. The threshold should have been able to distinguish the correct condition from a false condition by assuming that the residual of the corresponding dictionary was smaller than a certain value. The residuals from other dictionaries were defined as residual_err. Then, the threshold C was determined by $C=0.95\ast \min \left(residua{l}_{err}\right)$.

In this research, the threshold set was $C=\left[2.71,2.70,2.80,2.12,1.85,2.12,2.07\right]$. The sequence of the threshold was in accordance with the dictionary number. When applying this threshold set, only 74 of 7000 samples were misclassified; i.e., the identification accuracy of the proposed method was 98.9%.

4. Comparison and Discussion

4.1. Comparison

To demonstrate the advantages of the proposed method, the dictionaries were constructed using discrete wavelet transform (DWT) decomposition for comparison. The wavelet base function used was the Sym5 wavelet, due to its advantages in reducing the phase distortion [11]. A soft threshold denoising method was used, and the threshold selection method was ‘sqtwolog’. The final residual features are shown in Figure 15.

Compared with those in Figure 14, the residual features in Figure 15 had the same trend. However, there were no obvious boundaries between the features. Most of the residuals overlapped each other. In addition, the residual could not be distributed in a small range, which meant this index had a poor performance regarding the within-class distribution. In this case, the conditions could not be identified.

4.2. Discussions

This work can be also considered as further improvements of VMD, as it used IMFs to construct overcomplete dictionaries for sparse coding. VMD indeed was more efficient than EMD or other algorithms using a recursive model. However, compared to the proposed method, feature extraction using VMD still took more time. The research was performed on a desktop computer with an Intel i7-10700 CPU @ 2.90 GHz. Once the dictionary was learned, the reconstruction process of 1000 signals only took about 0.03 s. The RMS value of the residual was also calculated immediately. To decompose a signal with 1538 points using K = 6 and α = 6000 took about 1 s, not to mention the GWO process and the feature-calculation process. In sum, the proposed method ensured both the integrity of the target mode and a high operational efficiency, and was suitable to realize the online diesel-engine diagnosis.

There is still more work to be done. As illustrated in Section 3, the proposed method constructed a dictionary for each condition to confirm the classification of the test data. However, in the ideal situation, the condition monitoring would be represented by only one dictionary. This would require a more comprehensive understanding of the test signals, and the design of a more suitable dictionary.

5. Conclusions and Outlook

This paper presented a novel approach to condition monitoring based on an adaptive VMD and sparse representation classification. The given workflow required the learning of dictionaries trained by samples with signal characteristics. As the dictionaries were developed, measured signals were reconstructed directly by the learned dictionaries, and signal-decomposition techniques were no longer needed in the condition-monitoring process. It improves the efficiency of condition monitoring and reduces the risk from an inappropriate signal processing.

In this paper, a correlated kurtosis was used to search for the best combination of VMD parameters simultaneously. The adaptive VMD was integrated with sparse representation to develop dictionaries for the condition monitoring of a diesel engine using measured vibration signals. The results showed that the IMFs from the adaptive VMD could reserve the condition feature with sparsity, and the dictionary for sparse-coding learned from the IMFs could reflect the structure of the signal well.

A residual feature, calculated as the condition index, showed a great clustering capability that could distinguish different conditions based on the discrimination of the data structure between the dictionary and test signals of different conditions.

Seven engine modes were used in this paper, including a normal condition, two valve clearances, and four injection timings. A mode-identification accuracy of 98.9% (6926 corrected from 7000 samples) was obtained.

Much work still needs to be done in the future. In this paper, the method was implemented in a fault-diagnosis case for a diesel engine. Two types of faults—valve wear and improper injection timing—were considered, including seven different conditions in total. We chose two types of abnormal conditions because the valve-clearance fault is a common wear-out failure [1,20], and the injection timing is a key combustion parameter for diesel engines [8]. They both have a significant influence on engine performance. The robustness of the proposed method should be further investigated with other faulty modes. In addition, a hard threshold set was applied in the classification to achieve the condition monitoring. In practical diagnoses, some intelligent classifiers could be used for a better classification performance.