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1.School of Automation，Shenyang Aerospace University，Shenyang 110136，P.R.China；2.Wind Tunnel Equipment Research and Development Department，AVIC Aerodynamics Research Institute，Shenyang 110034，P.R.China；3.Aviation Key Laboratory of Science and Technology on Fault Diagnosis and Health Management，Shanghai 201601，P.R.China

（Received 15 June 2020；revised 20 August 2020；accepted 10 October 2020）

Abstract:The health status of aero engines is very important to the flight safety.However，it is difficult for aero engines to make an effective fault diagnosis due to its complex structure and poor working environment.Therefore，an effective fault diagnosis method for aero engines based on the gravitational search algorithm and the stack autoencoder（GSA-SAE）is proposed，and the fault diagnosis technology of a turbofan engine is studied.Firstly，the data of 17 parameters，including total inlet air temperature，high-pressure rotor speed，low-pressure rotor speed，turbine pressure ratio，total inlet air temperature of high-pressure compressor and outlet air pressure of high-pressure compressor and so on，are preprocessed，and the fault diagnosis model architecture of SAE is constructed.In order to solve the problem that the best diagnosis effect cannot be obtained due to manually setting the number of neurons in each hidden layer of SAE network，a GSA optimization algorithm for the SAE network is proposed to find and obtain the optimal number of neurons in each hidden layer of SAE network.Furthermore，an optimal fault diagnosis model based on GSA-SAE is established for aero engines.Finally，the effectiveness of the optimal GSA-SAE fault diagnosis model is demonstrated using the practical data of aero engines.The results illustrate that the proposed fault diagnosis method effectively solves the problem of the poor fault diagnosis result because of manually setting the number of neurons in each hidden layer of SAE network，and has good fault diagnosis efficiency.The fault diagnosis accuracy of the GSA-SAE model reaches 98.222%，which is significantly higher than that of SAE，the general regression neural network（GRNN）and the back propagation（BP）network fault diagnosis models.

Key words：aero engines；fault diagnosis；optimization algorithm of gravitational search algorithm（GSA）；stack autoencoder（SAE）network

0 Introduction

Aero engines are very important for aircraft.As the core system of air engines，the gas path system has been the focus of the fault diagnosis technology［1-2］.Traditional intelligent fault diagnosis methods built the fault diagnosis model of rotating machinery on the basis of neural network（BP），which incline to cause local minimum problems due to different structure selections［3］.And the general regression neural network（GRNN）algorithm has been studied because of its advantages in approximation ability and fault tolerance［4］.Gai et al.［5］used the GRNN method to carry out fault diagnose of auxiliary inverter of rail transit train.However，it is difficult to determine the smoothing factor and weak adaptive ability for the GRNN method，which may limit its further application.Therefore，as a breakthrough in the field of neural network，deep learning has attracted much attention in recent years because it can well reveal the inherent laws of complex information.

Stack autoencoder（SAE）is a type of the deep learning networks and has the advance ability of modeling，generalization and function expression，so it can solve the problems of traditional methods in feature extraction and health status recognition.Lyu［6］applied the SAE method to fault diagnosis.However，the number of neurons hidden layer is manual and random，which is difficult to get the best diagnosis result.In this paper，an effective fault diagnosis method based on the gravitational search algorithm（GSA）-SAE is proposed to solve the fault diagnosis problem for a turbofan engine.The core idea of the method is to use the GSA optimization algorithm to optimize the number of neurons in each hidden layer of SAE network.According to the optimal neuron number，an effective fault diagnosis model based on GSA-SAE method is built to improve the accuracy of fault diagnosis for aero engines，thus avoiding the influence of manual parameter selection on model.

1 Methodology

1.1 SAE network

SAE network is a deep network model composed of multi-layer sparse autoencoder and regression classification layer［7］.The sparsity constraint is added to the sparse self-encoder，which makes the SAE network have the capabilities of excellent training learning and feature extraction.

（1）Sparse autoencoder

Sparse autoencoding is to add sparse constraints to neurons in the hidden layer.The autoencoder is a nonlinear feature extraction model，including input layer，hidden layer and output layer.The hidden layer is used to encode the input from the input layer and reconstruct the input in the output layer by decoding［6，8-10］.The structure of autoencoder is shown in Fig.1.

Fig.1 Schematic diagram of autoencoder

The autoencoder is a typical symmetric neural network with unsupervised learning.Fig.1 shows a three-layer autoencoder with a hidden layer.The training of the autoencoder includes the encoding process and the decoding process.The encoding process is to transform the high-dimensional features of the input data into the low-dimensional features of the hidden layer through the activation function in Eq.（1）.In the decoding process，the feature representation of the hidden layer is reconstructed by the activation function as the output target through Eq.（2）.

whereWis the weight matrix from the input layer to the hidden layer，bthe hidden layer threshold，σthe sigmoid activation function，WTthe weight matrix from the hidden layer to the output layer，andb′the output layer threshold.

In order to prevent the autoencoder from the over fitting data，we can add a sparse constraint in the network so that the average activation valueρjof hidden layer neuronjis close to 0 andρis the sparse parameter.In order to restrainρjdeviating fromρ，KL divergence can be selected to limit，shown as

wheresis the total number of hidden layer neurons，jthej-th neuron of hidden layer，and 1≤j≤s.The overall loss function of sparse autoencoder is shown as follows

whereβis the sparse penalty coefficient，mthe total number of samples，nlthe number of network layers，slthe number of neurons in thellayer，andλthe weight attenuation coefficient.

（2）Regression classification layer

Softmax regression classification layer is an extended form of softmax regression model，which can be used to solve multiple classification problems.And it is a supervised learning algorithm.Therefore，softmax regression is used to construct a classifier to classify the features extracted by SAE［11］in this paper.

The task of softmax regression classification layer is to make a classification according to corresponding functions.Assume that there is a training sample set and the fault type code of gas path is the number of fault type.For the input of test samples，their probabilities of belonging to each fault type can be calculated by softmax regression layer.

1.2 Gravitational search algorithm

GSA is an optimization algorithm based on the law of gravity and the Newton’s second law［12］.The optimal solution based on GSA can be found by the change of the position of particles in the population.That is to say，with the continuous iteration，particles will move continuously in the search space by the gravitational force between particles.When the particle moves to the optimal position，the optimal solution is found［13］.

In GSA，two variables（i.e.position and velocity）of particles are firstly initialized，and the position represents the solution of the problem［14］.For example，there areNparticles in the population，the position and velocity of thei-th particle（i.e.，individual）ind-dimensional space are expressed as follows

（1）Calculating mass of particles

The mass of particleican be defined as follows

where fiti(t)andMi(t)denote the fitness function value and mass of thei-th particle at thet-th iteration，respectively，best（t）and worst（t）the best fitness function value and the worst fitness function value of all particles in thet-th iteration.The specific definitions are as follows

（2）Calculating gravity

In thed-dimension，the gravitation of particlejto particleiis defined as follows［16］

whereMi(t)andMj(t)represent the mass of particlesiandjat timet，andthe position of thei-th andj-th particles in thed-dimension，Rij(t)represents the Euclidean distance between the particlesiandj，Rij(t)=|Xi(t)，Xj(t)|，εis a constant to prevent the denominator from being zero，G(t)the coefficient of universal gravitation at the time of iterationt，and the formula is as follows

whereG0andαrepresent two constants，tis the number of iterations of the current group，andTthe total number of iterations of the algorithm.

In thed-dimension，the resultant force on the particle can be written as follows

whererdenotes a random variable with uniform distribution between［0，1］，kbestrepresents the number of particles with the highest particle mass and decreases linearly with the number of iterations.The initial value isNand the final value is 1.

（3）Calculating acceleration

According to the Newton’s second law，the acceleration equation of particles in the second dimension is

（4）Updating speed and location

The mass，force，acceleration，velocity and position of particles in the population can be calculated according to Eqs.（17）and（18）.

Through the analysis of GSA algorithm，we can see that the particles in the population move under the attraction.The particles with smaller mass have longer steps under the same attraction，while the particles with larger mass have shorter steps.The longer step of particles is helpful to the global search.Meanwhile，it can prevent particles from falling local optimum.Instead，the shorter step of particles is helpful to the local search and can improve the convergence accuracy of the algorithm.Finally，the particles gradually converge to the optimal position to achieve the purpose of optimization.

2 Fault Diagnosis Scheme of Aero Engines

2.1 Fault diagnosis scheme based on GSA-SAE

A special data sensor network is used to collect the relevant parameters of the gas path system of a turbofan engine.The data are standardized and preprocessed，which are divided into training sample set and test sample set.The GSA optimization algorithm is used to optimize and determine the number of neurons nodes in each hidden layer of SAE network to obtain the optimal SAE network，thus avoiding the impact of manual setting the number of neurons nodes in SAE network.In this way，a fault diagnosis model of the gas path system of a turbofan engine is established using the training sample set.Furthermore，the fault diagnosis technology is studied，and its scheme of the gas path system based on GSA-SAE is shown in Fig.2.

Fig.2 Fault diagnosis scheme of an aero engine based on GSA-SAE

2.2 Optimization of SAE network based on GSA algorithm

The core idea of GSA optimization algorithm is：Firstly，the number of neurons in input layer and output layer of SAE network is determined；Secondly，the initial population is generated randomly，that is，the number of neurons in each hidden layer of SAE network is randomly initialized.On this basis，the number of neurons in each hidden layer of SAE network is optimized by a series of operations such as constantly updating the speed and position of particles until the optimization conditions are met.Therefore，the number of neurons in each hidden layer of SAE network can be finally determined.

The end conditions of the optimization process are that：Whether the number of iterations reaches the set value or whether the value of fitness function meets the requirements.If the optimization condition is satisfied，the search is stopped and the optimization algorithm is finished.The optimal number of neurons in each hidden layer of SAE network can be obtained.

The specific process of GSA algorithm to optimize SAE network is shown in Fig.3.

Fig.3 Process of optimizing the number of neurons in each hidden layer of SAE network

3 Fault Diagnosis Steps

The above fault diagnosis strategies are used to diagnose the gas path system of a turbofan engine.The specific steps are as follows：

Step 1Standardize and preprocess the collected data.

Step 2Divide the standardized data into training data set and test data set.

Step 3Set the number of neurons in the input layer and output layer of SAE network and determine the structure of SAE network according to the actual needs.

Step 4Initialize the number of neurons in each hidden layer of SAE network.

Step 5Optimize the number of neurons in each hidden layer of SAE network using GSA algorithm，and determine the number of neurons in each hidden layer of SAE network.

Step 6Establish the optimal fault diagnosis model based on SAE network using training sample set data.

Step 7Take the test data set as the input of the optimal SAE network，obtain and analyze the diagnosis results.

4 Experimental Verification

4.1 Data collection

In order to verify the effectiveness of the method proposed in this paper，the monitoring data of 17 parameters of the turbofan engine in flight status are used as data samples in experiment，including total inlet air temperature，high-pressure rotor speed，low-pressure rotor speed，turbine pressure ratio，total inlet air temperature of high-pressure compressor，outlet air pressure of high-pressure compressor，total gas temperature after low pressure turbine combustion，oil pressure difference，oil return temperature，high pressure speed increase rate，cabin pressure，adjustable blade angle of compressor，engine casing vibration，nozzle throat diameter，adjustable blade angle of fan，afterburner connotation of fuel flow measurement valve displacement，and afterburner fuel flow measurement valve displacement.

Due to the different dimensions of the parameters，it is necessary to carry out standardization and preprocess in order to improve the accuracy of fault diagnosis.

Select 560 samples as the training set（including 320 normal samples，140 samples with afterburner fault，and 100 samples with fake breath fault）and 225 samples as test set（including 128 normal samples，57 samples with afterburner fault，and 40 samples with fake breath fault）.The number of training samples and test samples for each state is shown in Table 1.

Table 1 The number of training samples and testing samples for each state of aero engines

4.2 Verification and result analysis

In order to verify the effectiveness of the proposed method and simplify the complexity of the network，a stack autoencoding network with two hidden layers is designed.The number of neurons in the two-layer hidden layer of SAE network model is optimized by the GSA optimization algorithm.The optimal numbers of neurons in the two hidden layers of SAE network model are 47 and 47，respectively.Other parameters of SAE network model are set as follows：The maximum number of iterations is set to 1 000，the learning rate is set to 0.01，the sparsity parameter is 0.1，the weight attenuation coefficient is 0.002，and the weight of sparse penalty term is 3.The network structure parameters of GSA-SAE model are shown in Table 2.

Table 2 Structure parameters of GSA-SAE network model

According to the above parameters，the optimal fault diagnosis model of GSA-SAE can be obtained.And 225 test samples are used for fault diagnosis test study.

To compare the results of different methods，SAE network，GRNN and BP models are built using the same test set，respectively.The engine states corresponding to the output layer neurons of each diagnosis network model are shown in Table 3.

Table 3 Aero engine states corresponding to neurons in output layer of fault diagnosis model

The same test set data are used as the input of GSA-SAE，SAE，GRNN and BP network fault diagnosis models.The different diagnosis results are shown in Table 4.

Table 4 Accuracy of fault diagnosis based on different models

The main parameters of different fault diagnosis models are set as follows：

（1）SAE model：The number of neurons in the input layer is 17，the number of neurons in the first hidden layer is 70，the number of neurons in the second hidden layer is 70，and the number of neurons in the output layer is 3.

（2）GRNN model：The number of neurons in the input layer is 17 and the number of neurons in the output layer is 3.The value of smoothing factor is 0.14.

（3）BP model：The number of neurons in the input layer is 17 and the number of neurons in the hidden layer is 35.The activation function of the hidden layer is the Sigmoid function，and the number of output neurons is 3.

From Table 4，it can be seen that the GSA-SAE model has the best diagnosis results with an accuracy of 98.222%.It is obviously higher than SAE，GRNN and BP network model，which verifies the effectiveness of the GSA-SAE method.

5 Conclusions

The accuracy of fault diagnosis for aero engines is not high in practical application.In this paper，an effective fault diagnosis method based on GSA-SAE is proposed for a turbofan engine.Firstly，the monitoring data of 17 parameters are preprocessed.After preprocessing，the data of 17 parameters are divided into training sample set and test sample set.Then，the SAE fault diagnosis model is constructed.In order to solve the problem of manual setting the number of neurons in each hidden layer of SAE network，this paper proposes a GSA optimization algorithm to optimize the SAE network to find and obtain the optimal number of neurons in each hidden layer of SAE network.On this basis，the optimal fault diagnosis model of aero engines based on GSA-SAE is established.The validity of the optimal GSA-SAE model is verified using collected aero engine data.In this paper，the same training sample set and test sample set are used to build different fault diagnosis models and carry out simulations in order to show the advantages of the proposed GSA-SAE method.The results show that the accuracy of the proposed GSA-SAE fault diagnosis method is significantly higher than that of SAE，GRNN and BP network models.Based on the proposed method，the problem that the number of neurons in each hidden layer of SAE network usually depends on manual setting has been well solved，which effectively improves the performance of fault diagnosis.

AcknowledgementsThe work was supported by the National Natural Science Foundation of China（No.51605309）and the Aeronautical Science Foundation of China（Nos.201933054002，20163354004）.

AuthorProf.CUI Jianguo received the M.S.and Ph.D.degrees in 1995 and 2006，respectively.Now，he is a member of the academic committee of“AVIC Key Laboratory of Science and Technology on Fault Diagnosis and Health Management”and Shenyang Aerospace University，a member of Liaoning Province Aerospace Association，and a science and technology expert in Shenyang province.His research direction consists of aircraft/engine fault diagnosis and prediction，integrated health management，health status assessment and trend analysis，intelligent information collection and processing technology，etc.

Author contributionsProf.CUI Jianguo designed the technical solution，and revised the article.Ms.TIAN Yan designed the fault diagnosis method and wrote the first draft of the article.Mr.CUI Xiao，Dr.TANG Xiaochu，Mr.WANG Jinglin，Dr.JIANG Liying and Dr.YU Mingyue tested and verified the fault diagnosis method.All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.