Sparsity-Enhanced Model-Based Method for Intelligent Fault Detection of Mechanical Transmission Chain in Electrical Vehicle

1  Introduction

In recent years, the rapid development of artificial intelligence and advanced signal processing technologies have attracted substantial attention in smart cities, which facilitate related fields from traditional ways to intelligent applications. Electrical vehicles (EVs) are of great importance to global environmental protection because of their zero exhaust emissions [1–3]. Compared with fuel vehicles, the mechanical transmission structure of electric vehicles has changed significantly [4,5]. Because of the use of a motor drive, EVs no longer use complex gearbox. However, gears, bearings and other mechanical components are retained [6]. These parts are still working under harsh conditions, and their condition monitoring is still important to guarantee the safe operation of EVs. Fortunately, in EVs, the arrangement of sensors is more convenient, which makes the acquisition of monitoring information more extensive [7–10]. On the other hand, the computing and processing capabilities of EVs are also increasing, which provides a good basis for the deployment of maintenance measures based on monitoring information [11,12].

Vibration monitoring is an important mean to prevent mechanical downtime [13–15], but how to obtain the weak fault information from the original monitoring data has been a major challenge in the scientific community [16,17]. In terms of kinematics, the damage to mechanical components corresponds to the periodic impact characteristics (PICs) in the vibration signal [18,19]. A large number of studies have shown that the multi-source vibration and random noise in the vehicle have caused great difficulties in PIC extraction. In order to identify faults at a low signal-to-noise ratio (SNR), many signal processing methods were proposed [20].

At present, scholars are working in two directions. One is the new signal decomposition method, the other is the intelligent identification method of fault characteristics. In terms of signal decomposition, wavelet transform [21–23], EMD [24–26], sparse representation [27–30], and their latest variants have been used to extract key features from noisy observations [31–33]. In order to reduce the dependence of diagnosis results on the experience of supervisors, many feature evaluation indexes have been invented, which are mainly used to correctly identify the components related to mechanical faults from the decomposition results [34]. A typical example of these indexes is spectral kurtosis [35,36]. Although new indicators emerge in an endless stream, the common shortcoming is that they rely too much on the statistical characteristics of the signal itself and ignore the model information behind it. For example, when the monitoring signal is accompanied by strong sporadic impulses, even if artificial intelligence [37–39] based methods are used, the correct feature extraction results cannot be guaranteed in many engineering scenarios.

In this paper, a novel model-based approach, enhanced by sparse representation, is proposed for mechanical fault diagnosis in EV. In the signal decomposition, the fractal wavelet representation is used, which is an efficient signal decomposition tool with centralized multi-resolution ability. In feature recognition and selection, the complex harmonic characteristics of PICs in the wavelet envelope domain are used, and a sparse representation enhancement method based on an over-complete Fourier dictionary (OFD) is proposed. The method realizes the quantitative evaluation of the proportion of PICs in the signal. Through the above measures, the robustness of the proposed method to multi-component coupling signal and noise is greatly enhanced. The superiority and effectiveness of the proposed method are verified by numerical simulation and engineering experiments.

2  Centralized Multi-Resolution Analysis

Wavelet transform is an effective tool for the multi-scale decomposition of signals. However, the center frequency of each subspace of the classical wavelet transform is different. In this section, a novel fractal wavelet decomposition (FWD), based on a dual tree wavelet basis [40], is introduced. FWD, an enhancement of wavelet packet transform, is a wavelet decomposition method with spectral focusing capability. FWD can realize multi-resolution analysis around some fixed center frequencies. For the convenience of discussion, Support{⋅} and CF{⋅} are utilized to represent the theoretical spectral passband and center frequency of the wavelet packet, respectively.

2.1 Data Augmentation Methods

Translation sensitivity is a significant defect of classical discrete wavelet decomposition, which often results in false features in the decomposition results. Maximal overlap decomposition strategy can avoid this defect, but the computational efficiency is significantly reduced. Dual tree wavelet transform (DTWT), proposed by Kingsbury, achieves a good trade off between accuracy and efficiency and the merit of translation invariance (TI) is realized. The wavelet of DTWT is a complex-valued function, as follows.

ψC(t)=ψRe(t)+j⋅ψIm(t)(1)

where j is the imaginary number, defined as j=−1 and the two wavelet generators construct a Hilbert transform pair, which is given as:

ψIm(t)=Hilbert{ψRe(t)}(2)

2.2 Motor Fault Diagnosis Methods Based on Current Signal

Although DTWT can alleviate the distortion of TV to the extracted features, it cannot solve the problem of transition band feature extraction in dyadic wavelet subspace. To address this problem, centralized multiresolution (CMR) is proposed by Chen [33]. The essential idea of CMR is the construction of an implicit wavelet packet (IWP). Let {x(n)|n=1,…,N} be a digitized signal of length N and dyadic wavelet packets (DWPs) at j-stage decomposition be {dwpj,1,…,dwpj,2j} with

CF{dwpj,1}<CF{dwpj,2}<⋯<CF{dwpj,2j}(3)

IWPs can be generated using

iwpj−1,k(n)=dwpj,2k(n)+dwpj,2k+1(n)(4)

A (j+1)-stage DTWT can generate 2j−1 IWPs. The spectral support and center frequencies of WP and IWP are demonstrated in Table 1. The CF of the IWP is just located at the edge of the spectral passband of the DWP, which can make up for the requirement of transition band feature extraction.

2.3 Fractal Wavelet Decomposition

As shown in Eq. (4), the wavelet generators of IWP are constructed based on those of WPs. Therefore, the property of TI can be preserved. The distribution of IWPs as the scale of analysis deepens is shown in Fig. 1. There are IWP sets in which the IWPs share an identical CF and their spectral resolutions are constantly refined. For example, CF{iwp0,1}=CF{iwpj,2j−1} and Support{iwpj+1,2j}=Support{iwpj,2j−1}/2 hold or j∈Z+ (see Fig. 2).

Figure 1: Centralized multiresolution analysis

Figure 2: CMR provided by set of IWPs with identical CFs

3  Sparse Fourier Decomposition for Cycle-Stationary Process

3.1 Fundamentals of Sparse Representation

Compared with the classical basis expansion method, the sparse representation (SR) allows the addition of other optimization constraints, which can better suppress the monitoring noise. For a discrete signal {x(n)}, the ℓ1−norm norm, ℓ2−norm, and ∞−norm are expressed as below:

||x||1=∑i=0N|x(n)|(5)

||x||22=∑i=0N|x(n)|2(6)

||x||∞=max1≤n≤N,n∈Z|x(n)|(7)

Let w be the representation coefficient vector to be solved, a typical optimization problem of SR is formulated as

arg minx||w||1s.t.y=Aw(8)

where the matrix AN×K (N≪K) is a redundant dictionary with predetermined atoms and y is the observation signal. As stated above, the existence of noise is inevitable in the condition monitoring of EVs, a more feasible problem P1ε can be formulated as

P1ε:arg minw||y−Aw||22+λ||w||1(9)

where λ is the Lagrangian parameter. This kind of problem is called the basis pursuit problem in the literature.

3.2 Sparse Fourier Decomposition (SFD)

In fast Fourier transform (FFT), an orthonormal basis is used for decomposing the input signal. The spectral interval for adjacent sinusoidal atoms is Δf=fs/N. The basis for FFT can be expressed as

A=[ϕ1ϕ2…ϕN]N×N(10)

The column vector ϕi=exp⁡(j⋅2π(i−1)n/N), in which 1≤n≤N, is a complex-valued sinusoidal atom. The Picket fence effect will occur for signals that are sampled at non-integer periods. In order to overcome this disadvantage, an over-complete Fourier dictionary (OFD), shown in Eq. (11), is proposed to represent the signal.

ARFFT=[exp⁡(j2πNmn)]N×M(11)

where m=R−1k(0≤k≤RN−1) and R is a positive integer. Equivalently AR is an OFD with redundancy R. In this paper, to solve the P1ε problem, the strategy of split augmented Lagrangian shrinkage algorithm (SALSA) can be employed.

w^=arg minw12||y−ARFFTw||22+||λ⊙w||1(12)

wopt=arg minw,u12||y−ARFFTw||22+||λ⊙u||1s.t.u−w=0(13)

On the basis of augmented Lagrangian theory, the above problem has an equivalent matrix form, given as:

arg minz1,,z212||y−ARFFTz1||22+||λ⊙z2||1s.t.Cz−b=0(14)

where C=[I−I], b=0, and z=[z1z2].

3.3 Numerical Implementation of SFD

Let H be the complex conjugate of a matrix and the thresholding function Soft_Thres(x,TV) be defined as in Eq. (15), the SFD algorithm can be summarized as in Table 2. Although iterations are employed in the algorithm, practices have proved that the whole algorithm can be completed in tens to hundreds of microseconds for signals with less than 10000 samples.

{y=max([|x|−TV,0])y=xy/(y+TV)(15)

4  Performance of SFD in Representation of Harmonic Component

When localized damage occurs in mechanical parts, periodic impacts are often generated in the monitoring signal, which causes multiple harmonics in the envelope spectrum. In order to evaluate the amount of PIC in the signal, it is necessary to calculate the sum of the energy of each harmonic.

4.1 SFD of Simple Harmonic Wave without Noises

A sinusoidal signal y(t)=Accos⁡(2πfct+π/6), in which Ac=1 and fc=(500+0.21)Hz, is synthesized as the dynamic signal without measurement noise. The sampling rate and the sampling number are set as 1000 Hz and 1000. The signal y(t) in the time domain and spectral domain are shown in Fig. 3. Because this signal is not a positive periodic sampled, there is a significant picket fence effect in the FFT spectrum.

Figure 3: (a) Time domain waveform and (b) FFT spectrum of the synthesized noise-free signal

Applying the SFD algorithm on the synthesized signal by setting R=10 and NITR=100, the associated spectrum is shown in Fig. 4. Only three spectral lines (250.1, 250.2 and 250.3 Hz) with a frequency close to the actual 250.21 Hz have large amplitudes. The amplitudes of other frequencies in the range (245, 255) are smaller than 10−3. The reason for this phenomenon is that SFD is described as a P1ε problem, which makes most of the linear representation coefficients non-zero.

Figure 4: SFD spectrum of the synthesized signal

In contrast to the FFT spectrum where the energy of the harmonic components leaks in the entire frequency domain, the energy of the signal in the SFD spectrum is compressed in a narrow band with a bandwidth of only 0.2 Hz. An approximation signal y~(t) can be reconstructed using three spectral lines. The energy ratio of y~(t) to y(t) is 99.89%, and the related quantization error is ||y(t)−y^(t)||∞=0.0275.

In order to demonstrate the performance of SFD, the spectrum is compared with the FFT spectrum and the FFT spectrum with Hanning window. As shown in Fig. 5, the bandwidth of the main lobe of the SFD spectrum is the smallest, and the decay rate is the fastest near the main lobe. On the other hand, it is found that the amplitude of the side lobe in the SFD spectrum is only about 1/1000 of that in the FFT spectrum. This shows that the SFD spectrum, based on the redundant Fourier dictionary, has a good sparse representation ability for the harmonic components.

Figure 5: Comparison of SFD spectrum, FFT spectrum and windowed spectrum of signal

To analyze the impact of the redundancy of ARFFT on sparse representation results, different redundancy values were tested. The SFD spectra with different values of redundancy are shown in Fig. 6. In the case of R=2, there are some side-lobes with large energy in the SFD spectrum. With the increase of dictionary redundancy, the attenuation rate of side-lobe is accelerated, while the energy occupied by the main-lobe becomes more prominent. On the other hand, the width of the main-lobe also decreases with the increase of redundancy, and the number of spectral lines representing harmonic components alone does not decrease. For example, when R=50, the number of main-lobe lines is 5.

Figure 6: SFD spectra with different values of redundancy

4.2 SFD of Noisy Harmonic Component

In order to test the ability of SFD spectra to characterize noisy harmonic components, white noise is added to the simulation signal in the formula. The time domain waveform of a noisy signal yn(t) with SNR=10dB is shown in Fig. 7. Because of the noise, the characteristic information of the harmonic wave cannot be well identified.

Figure 7: (a) Time domain waveform and (b) FFT spectrum of the noisy signal

The SFD spectrum, generated by the proposed method, is shown in Fig. 8. Due to the existence of white noise, there are some prominent energy concentration regions in the SFD spectrum (Fig. 8a). However, the spectral component of the harmonic component is still dominant, and there are only three spectral lines in the main lobe (Fig. 8b). Comparing the spectral lines of the main-lobes in Figs. 4 and 8b, they are almost the same. An approximation signal y~n(t) can be reconstructed using the spectral lines in the main-lobe. The related quantization error between y~n(t) and y(t) is ||y(t)−y^n(t)||∞=0.0542. The above analysis shows that the presence of noise does not affect the effectiveness of the SFD method. That is, the harmonic components can still be sparsely represented.

Figure 8: (a) FFT spectrum with (b) zoom-in plot of the noisy signal

4.3 SFD of Periodic Impact Characteristics

In this subsection, the performance of SFD on PICs is validated. In the time domain, a typical PIC can be modeled as

Pic(t)=∑i=1NIimp(t−iT)(16)

where NI is the number of impulses in the signal and T is the interval between adjacent impulses. The impulse in the PIC can be expressed as

imp(t)=e−βtsin⁡(2πfrest)(17)

where β>0 is the decaying rate and fres is the ringing frequency of the impulse. Let β=60, fres=160+π, T=0.9, a synthesized PIC and its noisy version (SNR=20dB) are shown in Fig. 9. The SFD spectra of the two simulated signals are shown in Fig. 10. The bandwidth of the main-lobes of each harmonic component is still very narrow, and the side lobes are rapidly attenuated. It can be concluded from the above results that SFD still has a good sparse representation ability for cycle-stationary processes such as PIC.

Figure 9: Time domain waveform of the simulated noisy signal

Figure 10: Comparison of PSD spectrum and FFT spectrum of the noisy signal

5  Proposed Fault Diagnosis Approach

In the condition monitoring of EV, the PIC caused by the local damage of mechanical parts can be regarded as a multi-harmonic signal with noise in the envelope demodulation spectrum. Combined with FWD and SFD introduced in this paper, an intelligent fault diagnosis method is proposed. Taking the mechanical transmission chain and fault frequency and speed as prior knowledge, the procedure of the algorithm is as below. For a wavelet packet wp(t), either a DWP or an IWP, the compound impulsiveness indicator can be defined as below:

IMP{wp(t)}=sgn(PICER{wp(t),fc}−TPIC)⋅Kurt{wp(t)}(18)

where the operator Kurt{⋅} calculates the kurtosis value of the input signal, PICER{wp(t),fc} calculate the energy proportion of PICs at the frequency fc, and sgn(⋅) outputs one for positive input and zero otherwise. The optimal wavelet packet is selected based on the maximization of the IMP indicator.

6  Case Study of Fault Diagnosis

6.1 Descriptions of the Experiment

To verify the effectiveness of the proposed approach, a case study using actual signals from engineering experiments, is investigated. The tested mechanical part is a roller element bearing with slight peeling on the outer race. Specifications of the test bearing are shown in Table 3. This test bearing was removed from a certain type of electric drive vehicle. It provides mechanical support for the drive shaft of the AC motor and works under heavy load. In this test, the bearing was placed in a hydraulically driven loading device. Schematic diagrams of the test set-up are shown in Fig. 11.

Figure 11: Schematic diagrams of the experimental set-up

6.2 Descriptions of the Experiment

The time domain waveform and the FFT spectrum of a record of vibration signals are shown in Fig. 12. It can be seen from the figure that there is a lot of noise, which complicates the identification of fault features. The proposed method is applied to the acceleration signal. In the fault diagnosis algorithm, fc=57.8 Hz, TPIC=0.5. The evaluated impulsiveness values of the decomposed wavelet subspaces are shown in Fig. 13. An optimal wavelet subspace is selected. It is an implicit wavelet packet. The central frequency and the theoretical passband are 400 Hz and [200,600] Hz. The kurtosis value of this IWP is 4.633. The associated time domain waveform and its envelope spectrum are shown in Fig. 14. In the time domain waveform, it can be found that the frequency of the periodic impact is very close to the ball pass frequency of outer-race (BPFO, Fig. 14a), and the energy proportion of the PIC component in the envelope spectrum is 0.68 (see Fig. 14b).

Figure 12: (a) Time domain waveform and (b) FFT spectrum of the noisy signal

Figure 13: The impulsiveness values of the decomposed wavelet subspaces by the proposed method

Figure 14: (a) Time domain waveform and (b) PSD spectrum of extracted by the proposed method

6.3 Comparisons

If the indicator of PER is not calculated in the algorithm, the processing results are shown in Fig. 15. The central frequency and the theoretical passband of the selected wavelet subspace are 5750 Hz and [5750,5800] Hz. The kurtosis value of the extracted feature is 9.00. Although the subspace extracted by the comparison method is significantly larger than that of the method proposed in this paper, it is not a periodic impact feature to characterize the failure of mechanical parts. In the envelope spectrum, even if the PSD algorithm proposed in this paper is used, there is no energy concentration region characterizing the harmonic components. The PER indicator of this wavelet subspace is calculated as 0.17. This value is significantly less than 0.5, so the wavelet subspace is identified as a non-periodic impact component and filtered out in the method proposed in this paper.

Figure 15: (a) Time domain waveform and (b) PSD spectrum of extracted features by the comparison method

7  Discussion on the Sharp Resolution of SFD Spectrum

From the materials given above, it is known that the SFD spectrum has an extremely high resolution. This is quite different from the classical windowed spectral analysis. According to the Heisenberg uncertainty principle, the main lobe resolution and the side lobe attenuation rate cannot be improved simultaneously. The SFD spectrum proposed in this paper is based on the principle of sparse representation and does not depend on the window function, which can ensure a very high resolution of the main lobe while accelerating the rate of side lobe attenuation. Nevertheless, it is also found that such improvements are limited and still cannot completely break through the constraints of the Heisenberg uncertainty principle.

8  Conclusions

In this paper, the problem of PICs extraction is studied, which is the core problem in the mechanical fault diagnosis of electric vehicles. In order to improve the accuracy and robustness of fault feature identification, statistical information and model information in the monitoring signal were combined comprehensively. A sparse Fourier decomposition method based on OFD is proposed, which realizes the quantitative evaluation of the energy proportion of fault feature components on the envelope spectrum in signal time-frequency representation. This model information plays an important role in eliminating the interference of measurement noise in the analysis signal. The effectiveness of the proposed sparsity-enhanced model-based fault diagnosis method is demonstrated by numerical simulations and case studies.

Funding Statement: This research is supported financially by the National Natural Science Foundation of China (Grant No. 51805398), the Natural Science Basic Research Program of Shaanxi (Grant No. 2023-JC-YB-289), the Project of Youth Talent Lift Program of Shaanxi University Association for Science and Technology (Grant No. 20200408), the Fundamental Research Funds for the Central Universities (Grant No. JB211303).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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