Sparsity-Assisted Intelligent Condition Monitoring Method for Aero-engine Main Shaft Bearing

，，，，*，，2

1.State Key Laboratory for Manufacturing Systems Engineering，Xi’an Jiaotong University，Xi’an 710049，P.R.China；2.Science and Technology on Plasma Dynamics Laboratory，Air Force Engineering University，Xi’an 710038，P.R.China

（Received 15 July 2020；revised 10 August 2020；accepted 15 August 2020）

Abstract: Weak feature extraction is of great importance for condition monitoring and intelligent diagnosis of aeroengine. Aimed at achieving intelligent diagnosis of aero-engine main shaft bearing，an enhanced sparsity-assisted intelligent condition monitoring method is proposed in this paper. Through analyzing the weakness of convex sparse model，i.e. the tradeoff between noise reduction and feature reconstruction，this paper proposes an enhanced-sparsity nonconvex regularized convex model based on Moreau envelope to achieve weak feature extraction. Accordingly，a sparsity-assisted deep convolutional variational autoencoders network is proposed，which achieves the intelligent identification of fault state through training denoised normal data. Finally，the effectiveness of the proposed method is verified through aero-engine bearing run-to-failure experiment. The comparison results show that the proposed method is good at abnormal pattern recognition，showing a good potential for weak fault intelligent diagnosis of aero-engine main shaft bearings.

Key words:aero-engine main shaft bearing；intelligent condition monitoring；feature extraction；sparse model；variational autoencoders；deep learning

0 Introduction

Bearing condition monitoring is one of the most important parts in engine health management（EHM），which is a typical symbol of advanced aircraft engines. However，the fault diagnosis of aeroengine main shaft bearing is very difficult because of the complex engine structure and serious interference. The classical bearing condition monitoring methods mainly use specific indicators to identify the fault state，whose threshold value is set according to the manual experience. While the intelligent diagnosis method learns an effective intelligent monitoring model through incorporating intelligent algorithms and signal priors into the procedure of training samples［1-3］. Since aero-engines have high requirements for service safety，the failure state is a small probability event compared with the health state，and it is very difficult to capture effective failure data. Therefore，the key of intelligent diagnosis for aero-engine main shaft bearing is to train a model based on health state data and realize the intelligent identification of fault state.

High quality and low noise training samples are very important to improve the abnormal monitoring performance of intelligent methods. Filtering and sparsity-assisted fault diagnosis（SAFD）methods are two classical and advanced techniques for signal noise reduction and feature extraction. Filtering methods，such as wavelets［4］，empirical mode decomposition（EMD）［5］and spectral kurtosis［6］，usually convert original signal from time domain to a specific transform domain based on physical priors.Such methods use the different coefficient distribution in the transform domain of noise and fault feature to achieve noise reduction and fault feature extraction. The SAFD method［7-9］is to incorporate such priors into regularization terms to construct a sparse model，and obtain the optimum solution of constructed model based on convex optimization theory to achieve the goal of reducing noise and extracting features. Compared with the traditional method，the SAFD method takes fully use of the physical priors in the framework of convex optimization，thus showing a better denoising performance. The classical sparsity-assisted diagnosis model generally constructs a convex model based on a convex sparse penalty［10］，such as L1 norm penalty，which suppresses the amplitude of the estimated signal［11］.Therefore，both the traditional filtering-like feature extraction method and the classical SAFD method will suppress the denoised signal’s amplitude，i.e.without balancing the performance of noise reduction and feature reconstruction. Constructing a sparse model with less signal underestimation is a key to improve the performance of weak feature extraction and intelligent condition monitoring.

Due to the measured signal of aero-engine bearing is seriously corrupted by strong noise，an enhanced sparsity-assisted intelligent condition monitoring method is proposed in this paper to realize weak feature extraction and intelligent identification of bearing fault state. The remainder of this paper is organized as follows. The basic idea of SAFD method is briefly introduced in Section 1. Section 2 describes in detail the intelligent monitoring method based on sparsity-assisted deep convolution variational autoencoders network. Then，the proposed method is verified through aero-engine bearing runto-failure experiment in Section 3. Finally，conclusions are drawn in Section 4.

1 Sparsity-Assisted Fault Diagnosis Model

In order to obtain high quality and low noise training samples to support intelligent identification of fault abnormal state，the SAFD strategy for noise reduction and feature enhancement is firstly studied. The core idea of the SAFD method is to represent the fault feature signal with specific transformation or redundant dictionary based on signal sparse prior，and reconstruct the characteristic signal to realize the extraction and recognition of fault features. The classical sparsity-assisted fault diagnosis convex model，i.e. basis pursuit denoising problem，is shown as

whereyis the measured noisy signal，Dthe sparse representation dictionary，xthe sparse coefficient，λthe regularization p ara meter，and ‖ ?‖1the convex penalty constructed with L1 norm function.

Eq.（1）is the synthesis model of the basis pursuit de-noising（BPDN）problem，while its analysis model［12-13］is shown as

whereAdenotes the sparse transform.

The traditional SAFD method cannot take into account the two important properties of signal processing，namely noise reduction and signal reconstruction. Taking the BPDN problem （shown in Eq.（1）and Eq.（2））as an example，parameterλbalances the data fidelity and sparse regularization term. The largerλis，the less the sparse coefficient is and the less the noise component is，which means the better noise reduction performance. But at the same time，the more suppressed the amplitude ofxis，the smaller the energy of reconstructed feature signal is，which means worse reconstruction performance.

In order to overcome this problem，an enhanced sparsity-assisted fault diagnosis model is proposed， namely nonconvex regularized convex sparse model. This model makes fully use of the good amplitude-preserving performance of nonconvex penalties and the advantage of convex model，i.e. easy to be solved，thus combining the performance of noise reduction and feature reconstruction.

The relationship among L0 norm，L1 norm and minimax concave（MC）penalty is shown in Fig.1. L0 norm is the definition of sparsity and means the number of nonzero elements，which will make the sparse model constructed with L0 norm be nonconvex. L1 norm is a convex function closest to the L0 norm，which can make the constructed sparse model be convex，but will underestimate the amplitude of feature signal. MC penalty［14］is a nonconvex function，which is between L0 and L1 norms. By setting appropriate parameters， the sparse model constructed with MC penalty can be convex，thus making it convenient to solve the model with the convex optimization method. This kind of nonconvex penalty provides a possible way to realize signal noise reduction and feature reconstruction simultaneously.

Fig.1 Relationship among MC，‖ x ‖0 and‖ x ‖1

2 Sparsity-Assisted Intelligent Condition Monitoring Method

Motivated by the construction idea of MC penalty，this section proposes a kind of nonconvex penalty construction method， and builds a convex sparse model based on such nonconvex penalty. Accordingly，an intelligent condition monitoring method based on sparsity-assisted deep convolution variational autoencoders network is proposed in this seciton.

2.1 Nonconvex penalty construction based on Moreau envelope

Through analyzing the definition of MC penalty，the relationship among MC function，Huber function and Moreau envelope is firstly studied in this section. Then a nonconvex penalty construction method based on generalized Moreau envelope is proposed.

Supposing thatφ:R →R，s:R →R andf:RN→R，we show the definition of three functions in Eqs.（3）—（6）.

The MC penalty in scalar form can be defined as

The definition of Huber function in scalar form is described as

It can be also written as another form

The Moreau envelope is defined as

Based on the above definition formula，it can be found that the three functions have the following relationships.

（1） Huber function is a special case of the

Morea u function whenf(v)=‖?‖1.（2）MC penalty can be constructed with the Huber function

Inspired by this relationship，the scalar MC can be extended to multivariate form，thus forming a nonconvex penalty construction strategy based on generalized Moreau envelope.

SupposeΓ0(RN) denotes the set of proper lower semi-continuous convex functions RN→R ∪{+ ∞}，fMB:RN→R andB∈RM×N. The generalized Moreau envelope is defined as

Iff(v)=‖ ?‖1，we have the generalized Huber function，shown as

Accordingly，the generalized MC（GMC）penalty can be deduced as

2.2 Enhanced sparsity-assisted fault diagnosis（ESAFD）convex model

Assuming that feature signal is sparse in the transform domainA，an enhanced sparsity-assisted fault diagnosis model can be constructed by combining proposed GMC penalty and data fidelity term，shown as

thenFB(x)is a convex function.

Proof:WriteFB(x)as

whereg(x，v) is affine inx，which is convex. Since L1 norm is convex，the objective functionFB(x) is convex ifATA-λBTBis positive semidefinite，which is equivalent to Eq.（12）.

The design of parameterBdetermines not only the convexity of the model，but also the way of GMC implementing regularizations. In order to design a betterB，we can incorporate the physical prior of the processed signal into the design procedure.Since the objective function of the proposed model is convex，which is a saddle-point problem，according to the convex optimization theory，the forwardbackward splitting algorithm can be used to obtain the optimum solution.

2.3 Sparsity-assisted deep convolutional variational autoencoders network monitoring model

It is a semi-supervised learning task to learn the discriminant model based on the health state data and to detect the fault state. In terms of the task requirement，variational autoencoders show a good application potential for its strong compression and signal reconstruction property.

A deep convolutional variational autoencoders network is constructed in this paper by combining the structure of deep convolutional neural network and variational autoencoders. The model contains two sub-modules，named encoderfθ(μ，σ|x) and decodergφ(x|z)，wherexdenotes input data andzdenotes compressed code. It is supposed that all variablesziare independent distributed as normal distribution with the mean ofμiand the variance ofσi. In the updating procedure，the model learns to establish an effective mappinggφ(|fθ(μ，σ|x))between the original signalxand the reconstructed signal. The mean square error loss function‖x-‖is used to guide the reconstruction process. It is also assumed that each variablezihas a standard normal prior distributionp(zi|x)～N(0，1). The Kullback-Leibler（KL）divergence between the conditional posterior distribution and the prior distribution is used as a loss function to constrain their similarities.The loss function of whole model can be expressed as

wheredrepresents the dimension of compressed code，andωis introduced to adjust the weight between reconstruction error and distribution constraint.

The encoders contain five layers of convolutional blocks with channel size of 32，64，128，256，and 512，respectively. Each block represents a combination of 2D-convolutional layer，batch normalization layer and leaky ReLU layer. After obtaining the latent codes of eachμiandσifrom the encoders，the resampling process is implemented by sampling eachzifrom a normal distribution with the parameters ofμiandσi. The decoders have the similar structure of encoders with the convolutional layer replaced by transposed convolutional layer and arranged in reverse order.

Combining the denoised feature signal of ESAFD and deep convolution variational autoencoders network，a sparsity-assisted intelligent monitoring method is proposed in this paper. Its algorithm diagram is shown in Fig.2. Firstly，the denoised feature signal is extracted from original signal by the ESAFD method，and then the denoised signal is used as the input to train the deep convolution variational autoencoders network. The reconstruction error of variational autoencoders is utilized as the bearing condition monitoring index. If the reconstruction error is above the threshold value，the tested signal is different from the trained model corresponding to health status，meaning the abnormal status of the tested bearing occurs. Then the early warning is activated timely，thus achieving intelligent identification of fault state.

3 Experiment Verification

In this section，we verify the proposed method using an aero-engine bearing run-to-failure experiment. The test rig consists of several parts，such as mechanical system，drive and lubrication system，loading system，and electrical control system，as shown in Fig.3. The load system applies 11 kN radial load and 2 kN axial load to the test bearing，respectively. Vibration signals are collected by two accelerometers mounted in the horizontal and vertical directions of the bearing seat，with a sampling frequency of 20 kHz.

Fig.2 Sparsity-assisted intelligent monitoring model for aero-engine main shaft bearing

Fig.3 Bearing test rig

The test bearing is an H7018C angular contact ball bearing，with a pitch diameter of 117 mm，ball number of 27，ball diameter of 11.12 mm and a contact angle of 15° . The rotating speed is about 6 000 r/min. After about 156.3 working hours，the vibration signal of the bearing increases sharply and obvious periodic impact appears. Then we stop the test rig，and find a spall on the inner race with an area of 3 mm2，shown in Fig.4. According to parameters of bearing and rotating speed，the ball passing frequency through inner raceway is about 1 475 Hz.

Fig.4 Spall on the bearing inner race

3.1 Verificatin 1:ESAFD for weak feature extraction

We verify the performance by comparing proposed ESAFD method with L1-based SAFD method. Both methods are applied to analyze the bearing vibration signal in the weak fault stage of the experiment，shown in Fig.5. In this experiment，we set the TQWT［15］parameters toQ=1，r=5，J=10.The parameterλof L1-based SAFD method is chosen to preserve 2% of the sparse coefficients to extract a significant fault impact characteristic. In order to better compare the performance，we setλof ESAFD as the same value. For display convenience，the 100 Hz rotating speed is recorded asfr，and the 1 475 Hz characteristic frequency is recorded asf.

Fig.5 Original signal and its spectrum

Fig.6 Denoised result by ESAFD method

Fig.7 Denoised result by L1-based SAFD

The results of ESAFD and L1-based SAFD are shown in Fig.6 and Fig.7，respectively. The comparison results of denoised signal show that the amplitude of the fault feature signal extracted by L1-based SAFD method is small，while the ESAFD method extracts the bearing fault signal with high amplitude when reducing noise. The zoomedin comparison results show that the ESAFD method can effectively extract the group sparse structure of transients，due to the periodic change of the load applying to bearing inner race. The comparison results of square envelope spectrum show that the ESAFD method is superior to the L1-based SAFD method，which can significantly extract the rotating frequency，characteristic frequency of bearing fault（1 475 Hz）and its double and triple frequency component，thus realizing the effective identification of bearing inner-race fault. Therefore，compared with L1-based SAFD method，the proposed ESAFD method can give better consideration to both noise reduction and feature reconstruction，showing a stronger ability of extracting weak features.

3.2 Verificatin 2:Sparsity-assisted intelligent monitoring method for fault status identification

The performance verification of the intelligent monitoring model is carried out using the aero-engine bearing run-to-failure experiment. The curve of root-mean-square（RMS）value in the run-to-failure experiment is shown in Fig.8. It is found that the RMS index jumps slightly at about 139.8 h，corresponding to the bearing initial fault. At about 147.6 h，the RMS index jumps with a larger value，meaning the extension stage of bearing fault. At about 156.3 h，the test ends. In order to verify the performance of intelligent identification for bearing status，the data of the former 72 h are selected as the training set，and the data after 72 h are selected as the test set. We use 139.8 h as threshold value to divide the test samples into two groups，i.e. the sample collected before 139.8 h belongs to bearing healthy state，while the others correspond to bearing failure state.

Fig.8 RMS of the vibration in the run-to-failure experiment

During the implementation of the algorithm，the data are firstly processed by the ESAFD model to obtain the denoised signal. In order to better compare results，the spectrum of original signal and denoised signal are taken as the input of the autoencoders respectively.After 200 training iterations，the reconstruction error of the deep convolutional variational autoencoders is used as the monitoring index.The probability density function（PDF）curves of the reconstruction error based on the original signal and denoised signal are shown in Fig.9. It can be seen that the intelligent monitoring method based on denoised signal behaves better，i.e. the distribution of health test set matchs the normal training set，while the PDF of fault test set is different from the traning set，meaning the monitoring accuracy of the denoised signal outperforms the original signal. The denoised signals of the ESAFD method can significantly strengthen the fault feature information，and distinguish the distribution of health state data from abonormal state data under the reconstruction error index，which helps achieve the precise identification of bearing fault status.

Fig.9 PDF for reconstruction signal of deep convolutional variational autoencoders

In order to compare the performance of anomaly monitoring，the 98%，99% and 100% quantiles of reconstruction error based on the training sample are selected as the anomaly threshold. Samples with reconstruction error higher than the threshold will be judged as abnormal state. Model accuracy（Acc），health detection rate（HDR），fault detection rate（FDR）and false alarm rate（FAR）are selected to quantitatively evaluate the performance. In this paper，true positiveNTPcan be defined as the number of health samples identified as healthy state.Similarly，the definitions of true negativeNTN，false positiveNFPand false negativeNFNcan be obtained.The calculation formulas of several indexes are as follows

Comparison results in Table 1 are obatined by 10 randomized repeat experiments on deep convolutional variational autoencoders. It can be seen that compared with original signal-based method，the sparsity-assisted intelligent monitoring model makes significant improvement in Acc，HDR，FDR and FAR. The average improvements of these four indicators are 5.6%，2.3%，20.3% and 7.3%，respectively. It is proved that the proposed intelligent monitoring method can identify anomaly state sensitively and stably，which effectively support the intelligent identification of aero-engine bearing weak fault state.

Table 1 Comparison results between original signalbased and denoised signal-based intelligent monitoring %

4 Conclusions

Aimed at achieving the intelligent monitoring of aero-engine main shaft bearing，this paper incorporates sparse denoising into the training procedure of deep learning to propose a new intelligent condtion monitoring method. The conclusions are drawn as follows:

（1）This paper establishes an enhanced sparsity-assisted fault diangois convex model based on constructed nonconvex penalty and convexity conditon. The main advantage of ESAFD is that it can preserve the amplitude of bearing signal while reducing noise，showing a good ability of extracting weak features from strong noisy signal.

（2）An intelligent condition monitoring method based on sparsity-assisted deep convolutional variational autoencoders network is proposed in this paper. The main advantage of the proposed method is that it makes fully use of sparse denoised feature signal and obtains a sensitive and stable training model，thus realizing accurate identification of abnormal state.

（3）The effectiveness and feasibility of the proposed method is verified through aero-engine bearing run-to-failure experiment. It is proved that the proposed method is good at abnormal pattern recognition，showing a good potential for weak fault intelligent diagnosis of aero-engine main shaft bearings.