Aircraft Engine Gas Path Fault Diagnosis Based on Hybrid PSO－TWSVM

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Jiangsu Province Key Laboratory of Aerospace Power Systems，College of Energy and Power Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P．R．China

0 Introduction

The aircraft engine is an important equipment of an aircraft，and it is the power source of the aircraft．Modern aircraft engines work in a very harsh environment with high temperature，high pressure，strong vibration，and variable load，etc．，so it is very easy to damage aircraft engine components．Therefore，it is imperative for fault diagnosis of aircraft engines．According to the statistical data，most of faults occur in gas paths［1］and gas path faults are often potential and difficult to judge［2］，so researching on fault diagnosis of gas paths is important．

Classical support vector machine（SVM）algorithm has been introduced to complete aircraft engine gas path fault diagnosis，and a lot of papers［3－5］have reported it．Classical SVM has a－chieved good classification accuracy in fault diagnosis of aircraft engines．In this paper，a new SVM，i．e．，twin support vector machine（TWSVM）［6］is introduced，and it is a new development of SVM theory．It is a novel SVM based on nonparallel support hyper－plane and its performance is mature．Compared with the traditional SVM，the size of TWSVM quadratic programming problem is quarter of that of SVM［6－7］，so the operation velocity of TWSVM is accelerated．In addition，TWSVM also shows higher ability in dealing with imbalanced problems．

This paper introduces TWSVM algorithm to complete aircraft engine gas path fault diagnosis，uses a new optimization algorithm termed as hybrid particle swarm optimization （HPSO）［8］to optimize parameters of TWSVM and adopts a mixed kernel function to improve performance of TWSVM．

1 Aircraft Engine Gas Path Fault Diagnosis

The modern aircraft turbofan engine is generally composed of air inlet，fan，high pressure compressor，main combustion chamber，high pressure turbines，low pressure turbines，afterburner，nozzle，etc．Their faults are known as aircraft engine gas path faults，and if any of them occurs，the performance of the whole machine will decline．

In the working process of aircraft engine，there are various faults，such as oxygen corrosion，damage collided by foreign objects on surface of engine parts and blade fracture because of collision，and so on．These faults will cause damage to the gas path components of aircraft engine，and structure damage can lead to the saltation of engine performance parameters，such as pressure，temperature of high and low compressor import and export，efficiency of high and low compressors，speed of high and low pressure rotor and fuel flow rate．The principle of aircraft engine gas path fault diagnosis is to deduce backwards and find out engine component states，and finally find locations of engine faults based on the changes of engine part performance parameters．The usual method of aircraft engine gas path fault diagnosis is that according to multiple state variables of gas path components，comprehensively analyze these variables and make an evaluation so as to deduce the performance of this engine．

Aircraft engine faults mainly include three aspects：

（1）Engine structure component damage．

（2）The engine system or some components have lost their original function．

（3）Performance degradation exceeds the design requirements of an engine．

At present，methods of performance status monitoring and fault diagnosis for aircraft engines are divided into the following classes：（1）The status monitoring technique based on the small deviation linearization fault equation；（2）Fault diagnosis methods based on the artificial intelli－gence algorithms；（3）Fault diagnosis methods based on knowledge learning rules．The SVM algorithm for aircraft engine fault diagnosis belongs to machine learning that is a part of artificial intelligence algorithms．

In this paper，the TWSVM algorithm is introduced to carry out the fault diagnosis of aircraft engine．Because directly obtaining test data and fault data of engines is difficult，this paper uses gas turbine simulation program（GSP）software for modeling and simulation of the PW4056 engine developed by Pratt ＆ Amp Group［9］．Then according to the thermodynamic parameters of each section in different working conditions，the influence matrix of fault diagnosis is generated by establishing an influence matrix equation．

Fig．1shows the principle of using HPSOTWSVM to carry out fault diagnosis，in which training data are used to train TWSVM classification model and HPSO is used to optimize parameters of TWSVM．Then，the test accuracy of this classifier is obtained．

Fig．1 Simple flow chart of HPSO－TWSVM fault diagnosis

2 Basic Principle of TWSVM

TWSVM was proposed by Jayadeva et al．based on the generalized eigenvalue SVM（GESVM）in 2007［6］．Both of TWSVM and GESVM change the model structure from two parallel support hyper planes to a pair of nonparallel hyper planes，which extends SVM to a wider and more complex field．So TWSVM can process some data distribution styles that the SVM is difficult to deal with．Especially，the size of convex quadratic programming optimization problems of TWSVM is only one fourth of that of SVM′s，and it improves training speed in theory．

TWSVM，like SVM，was originally designed for a binary classification，and its basic principles are as follows．

Assuming that the training sample set is

In this paper，the nonlinear TWSVM is used，so its brief introduction is given here．Since the linear TWSVM could not be extended to the nonlinear case directly like standard SVM，we need kernel functions for the nonlinear case．The kernel－generated plane is

Two primal problems of nonlinear TWSVM are

and

whereC＝ ［A；B］∈ Rl×n．

After solving their corresponding dual problems

and

where

The solutions of Eq．（2，3）are obtained by

Thus a new test point is predicted to its own class by

The classical SVM algorithm is based on the principle of the distance maximum between two kinds of sample points to find all support vector points，and then find a classification hyper plane in middle of the two classes．The innovation of TWSVM is to find a relatively independent support hyper plane for either class．Thus，TWSVM has a new model structure and some new abilities．But a good TWSVM classifier is heavily relies on parameter optimization，so a fine parameter optimization algorithm is important．In this situation，the HPSO is introduced．

3 TWSVM Based on HPSO

3．1 Characteristics and principles of HPSO

The particle swarm optimization（PSO）is proposed by an American electrical engineer Eberhart and a social psychologist Kenndy in 1995［10］．This algorithm is one of common methods for parameter optimization of SVM．The key of this method is how to ensure that particles land at the optimal solution．In order to achieve this goal，the PSO algorithm cleverly simulates bird foraging behaviors，and forms a model about social and individual nature．By regulating two weight coefficients of personality and society，particles land in objective［10－11］．

The mathematical description of the PSO algorithm is as follows：assume an－dimensional search space，mparticles compose of a particle swarmwherexi＝xi，1，xi，2，…，xi，n（）Tis a position of particlei，and its velocity isThe individual optimal position of particleiispbestiand it can be described aspbesti＝pbesti，1，pbesti，2，…，pbesti，n

（）T．The global op

timal position of this swarm isgbest＝gbest1，gbest2，…，gbestn

（）T． After particle swarm finds the two optimal positions in the current iteration，they will update their own velocity and position according to the following equations

wherec1，c2are acceleration constants；rand（）generates a random number in （0，1）；andare the velocity and position of particleiat dimensiondand iterationk；pbesis the individual optimal position of particleiat dimensiondand iterationk；gbestis the global optimal position of particle swarm at dimensiondand iterationk，andωis the inertia weight．

From the above particle evolution equations，it can be find thatc1andc2adjust the step size of aparticle to flight forward its own best position and the global best position，respectively．For the PSO algorithm，exploration is that particles leave the original optimized track by a larger extent．Then exploitation is that particles search more detail on the original track．Ref．［11］pointed out thatωwas a proportional parameter for exploration and exploitation of the PSO and the influence ofωon algorithm performance was researched．From experimental results，it can be find that a largerωis good for the algorithm to jump out of the local optimum，and a smallerωis good for the local deep optimization，in other words，the converging speed of the algorithm is accelerated．

However，the performance of the classical PSO algorithm needs to be improved to deal with the optimization problems with high complexity．Recently，evolutionary algorithms，traditional optimization algorithms or other techniques are used to improve the PSO．Among the improved methods，HPSO is one of the most effective algorithms．The improvement principle is that algorithms enhance global exploring ability by enriching the diversity of particles or enhance the convergent speed and accuracy by improving the PSO local exploitation ability．There are two general mixed strategies：（1）Use other optimization techniques to self－adaptively adjust shrinkage factor／inertia weight／acceleration constant，etc．（2）Combine PSO with other evolutionary algorithms or other techniques．HPSO belongs to the first kind．The hybrid concept is from the genetic algorithm（GA）．In each time of iteration，according to the hybridization rate，the algorithm puts a part of particles into the hybrid pool，particles in this pool arise randomly pairwise hybridization to produce progeny particles with the same number compared with their father particles，and then progeny particles replace parent particles．The position of the progeny is obtained by the crossover of the parent positions

wherexnis the position of progeny，xmthe position of father particles，andithe random number in （0，1）．

Velocity of progeny can be computed by

wherevmis the velocity of father particles，andvnthe velocity of progeny．

Fig．2is the 3－D graph of one of universal standard functions named Rastrigin．The minimum of Rastrigin is 0．

Fig．2 3－D graph of Rastrigin function

In the paper，Rastrigin is used to test the performance of PSO and HPSO briefly．From this simple test，we could find that HPSO algorithm can find the global optimal value in almost the same time．Through many experiments，we find that HPSO has better ability on finding global optimal value than PSO （Fig．3），though they are stochastic algorithms．HPSO needs more time to compute，but when the dimension of a problem is big enough，this disadvantage can be ignored．

Fig．3 Performance comparison of PSO and HPSO

3．2 Selection of kernel function

The essence of improving performance of TWSVM algorithm is to improve its learning ability and generalization ability．Eenhancing the performance of its kernel function is an important way to improve the performance of TWSVM algorithm．We often use the Gauss kernel function as the kernel function，which has a strong learning ability but relatively weak generalization ability．So the mixed kernel（MK）function［12］which can balance the generalization ability and learning ability well is introduced．

In this section，a simple polynomial kernel function and a Gaussian radial basis function are used to construct the mixed kernel function

whereγandθare the proportions of the Gaussian radial basis function and the polynomial kernel function，respectively．In order to ensure that the mixed kernel function does not change the rationality of the original mapping space，generally make 0≤γ，θ≤1andγ＋θ＝1．At the same time，makethereby the mixed kernel function is

The basic information of data sets used in the paper is shown in Table 1．

Table 1 Data sets

In these experiments，c1＝c2＝0．01are fixed and then the grid search method is used to find optimal parameters．Variable parameters of MKTWSVM areγ，sand that of TWSVM iss．The first two groups of comparative experiments use 5－fold cross validation to estimate accuracy，and the last group about Vehicle data uses a single accuracy rate to estimate accuracy for reducing the amount of calculation．In addition，the second contrast group uses the 1vrest［13］multi－class classification algorithm，and the accuracy is not high for the third group after using this method，so the 1v 1multi－class classification algorithm is used．

Results of these experiments are shown in Table 2．Through the experimental verification，we can see that the mixed kernel function plays a significant role in performance improvement．

Table 2 Comparison results of MK－TWSVM and TWSVM

3．3 Training algorithm of TWSVM

This paper introduces a fast calculation method named successive over relaxation （SOR）［14］as follow

Aiming at a quadratic programming problem above，SOR is written as

whereQis the quadratic matrix of convex quadratic programming （Eq．（15）），andLandEare composed of elements ofQ．Lis the strict lower triangular matrix，and elements located in lower left of the main diagonal correspond to the elements of same positions inQ．Eis the diagonal matrix，and diagonal elements are in one－to－one correspondence to the diagonal elements ofQ．t∈（0 ，2）．This algorithm starts with anyα0∈ Rn，and through Eq．（16）unknown variables can be solved by iteration methods．The corresponding numerical solutions of convex quadratic program－ming problems of TWSVM are obtained until‖αi＋1－αi‖is enough small．

In addition，（）＃is an operator

In particular，Eq．（16）is not the final iteration equation and there must be noαi＋1in right of the equation．Only in this way，it can be running on computers．

4 Gas Path Fault Diagnosis Based on HPSO－TWSVM

After obtaining the corresponding classification rules and the fault data，aircraft engine fault diagnosis becomes a pattern recognition problem．In this paper，the corresponding classification rules are introduced［9］and the method of resample is used to enrich data sets．After then，TWSVM with a mixed kernel function is used to carry out this pattern recognition problem．In this process，TWSVM classification model is a fitness function for HPSO．Through choosing suitable parameters for HPSO algorithm，an optimal classification model of HPSO－TWSVM is built．

Procedures of gas path fault diagnosis based on HPSO－TWSVM are as follows：

Step 1 Obtain the corresponding classification rules of gas path fault diagnosis and divide the fault data set into a training set and a test set according to a certain proportion．

Step 2 Use the training set to train MKTWSVM classification model．

Step 3 Use HPSO to find the optimal parameters of MK－TWSVM．Set the position and velocity of each particle randomly．

Step 4 Calculate the fitness value of each particle．If the fitness function value is better，update the best individual position of the particle，i．e．pbest．Compare allpbestand find the global optimal positiongbest．

Step 5 Choose a certain number of particles according to hybrid probability，and put them into the hybrid pool．The particles in the pool are randomly paired to produce offspring with the same amount of their father generation particles．The position and velocity of the offspring are computed by Eqs．（11，12）and updated by Eqs．（9，10）．

Step 6 When this algorithm meets stop conditions，stop the search and give the output．Or return to Step 5to continue．

Step 7 After obtaining the optimal classification model，use the test data set to get test accuracy and running time．

Fig．4gives the flow chart of HPSO－TWSVM algorithm．

5 Experiments

According to the procedure of obtaining the aircraft fault analysis rules in Section 2and the procedure of gas path fault diagnosis based on HPSO－TWSVM in Section 4，experiments are carried out．In this paper，the decision rules for fault diagnosis in Ref．［9］are adopted．In Ref．［9］，the decision rules for fault diagnosis are obtained in conditions：flight heightH＝10 700m，atmospheric pressurep0＝0．237 23bar，atmospheric temperatureT0＝218．6K，atmospheric densityρ0＝0．378 06kg／m3，Mach numberMa＝0．39，thrustFN＝47．01kN．All programs in this paper are edited on MATLAB 2014aand computer configurations are Intel Core i5－4200MCPU with dominant frequency 2．5GHz，RAM 8GB，a 64 bit operate system of Windows 10．

We choose all kinds of fault data in Ref．［9］and generate randomly samples obeying the normal distribution by using the variance and the mean of every original sample according to correlation methods of resampling［15］．Its formula is

wherexiis any sample，L＝0．1is a constant，σfaultis the standard deviation ofxi，and randngenerates random number obeying the normal distribution．According to this method，we generate 200 samples，100of which are used as the training samples，and the other 100as test samples．The 5－fold cross validation is used in training，the average of which is adopted as the training accura－cy，and the time of the whole training process is defined as the running time．

Fig．4 Flow chart of HPSO－TWSVM algorithm

Fig．5shows the parameter convergence of different algorithms in a simple experiment，and these two algorithms can find the best values within two steps．These experiments give us a guide and then we can use this result to complete the following research．

Fig．5 Parameter convergence

The following experiments are all carried out under the condition of mixed kernel function as Eq．（14）．The first and second experiments are carried out in two different multiple classification methods．In last two experiments，we use PSO and HPSO respectively to optimize parameters of MK－TWSVM．In order to reduce the amount of computation，we set two penalty factors are equal，i．e．，c1＝c2，and fix the mixed kernel function parameterγ＝0．5．Accordingly，we need to optimize only two parameters．In order to better contrast the two optimization methods，the common parameters of the two algorithms are set as follows：the number of population is 10，two acceleration constants are all 2．05［11，16］，initial inertia weight is 0．8，the maximum times of iteration is 2，and the dimension of any particle position is 2．

The experimental results are shown in Table 3．In dealing with the multi－class classification problem，1v1method has a higher accuracy．Because every two classes of samples need a binary classifier and all of the samples requireis the total number of categories）binary classifiers，and it needs to design more complex voting algorithm，the computation needs more time．In the case of adequate computing power，we usually choose the 1v1multiple classification algorithm in 2—4groups．However，parameter optimization methods used in 2—4 groups are different．They are respectively grid search，PSO and HPSO．Experimental results show that HPSO can find the global optimum，and PSO can find a group of local best parameters．In addition，grid search algorithm also finds the best parameters，but it excessively relies on experience．

The experiment that has four parameters to optimize is carried out，and results are shown in Table 4．From the results we can see that HPSO uses much less time in the case that there are more parameters need to be optimized，because it can intelligently find out the best parameters．

Table 3 Comparison of experimental results of different algorithms

Table 4 Comparison results of TWSVM and HPSO－TWSVM

6 Conclusions

（1）The feasibility and effectiveness of TWSVM in aircraft engine fault diagnosis are verified through the experiments．This is a new exploration for SVM technology in aircraft engine fault diagnosis．

（2）It is proved that HPSO has better optimization performance than PSO though it needs slightly more time．HPSO is easier than PSO to find the global optimum．

（3）A mixed kernel function can improve the performance of a kernel function so as to ensure that the TWSVM algorithm can better balance the generalization ability and the learning ability．

（4）The 1v1multiple classification algorithm［17］of TWSVM has better training accuracy．

Acknowledgement

This work is supported by the Fundamental Research Funds for the Central Universities（No．NS2016027）．

［1］ WANG Xiuyan，LI Cuifang，GAO Mingyang，et al．Aircraft engines gas path fault diagnosis based on SVM and SNN ［J］．Journal of Aerospace Power，2014，29（10）：2493－2498．（in Chinese）

［2］ LI Yibo，ZHANG Guangming，JIANG Liying．Research status of aero engine gas path fault diagnosis technology［J］．Gas Turbine Technology，2009，22（3）：10－15．（in Chinese）

［3］ ZHAO Yongping，SUN Jianguo．Fast online approximation for hard support vector regression and its application to analytical redundancy for aeroengines［J］．Chinese Journal of Aeronautics，2010，23（2）：145－152．

［4］ SHI Hong，WANG Jing．Fault diagnosis of aeroengine sensor based on support vector machine［C］／／2011International Conference on Measuring Technology and Mechatronics Automation （ICMTMA）．Shanghai：IEEE Computer Society，2011：186－189．

［5］ CAI Kailong，XIE Shousheng，YANG Wei，et al．Fault diagnosis and adaptive reconfiguration control for sensors in aeroengine based on improved least squares support vector machine［J］．Journal of Aerospace Power，2008，23（6）：1118－1126．（in Chinese）

［6］ KHEMCHANDANI R J，CHANDRA S．Twin support vector machines for pattern classification ［J］．IEEE Transactions on Pattern Analysis and Machine Intelligence，2007，29（5）：905－910．

［7］ NIE Panpan，ZANG Lei，LIU Leilei．Based on the multiple class classification algorithm of support vector machine application in intrusion detection ［J］．Journal of Computer Applications，2013，33（2）：426－429．（in Chinese）

［8］ WEN Zheng．Master MATLAB intelligent algorithms［M］．Beijing：Tsinghua University Press，2015：129－132．（in Chinese）

［9］ PEHG Shuhong．Research on aero engine gas path fault diagnosis technology ［D］．Shanghai：Shanghai Jiao Tong University，2012．（in Chinese）

［10］KENNEDY J，EBERHART R C．Particle swarm optimization［C］／／Proceedings of IEEE International Conference on Neural Networks．Perth：IEEE，1995：1942－1948．

［11］TIAN Yubo．Particle swarm optimization algorithm and electromagnetic applications ［M］．Beijing：Science Press，2014：21－27．（in Chinese）

［12］WU Fulin．Research on model selection of twin support vector machines［D］．Xuzhou：China University of Mining and Technology，2015．（in Chinese）

［13］SHAO Yuanhai，ZHANG Chunhua，WANG Xiaobo，et al．Improvements on twin support vector machines［J］．IEEE Trans Neural Nets，2011，22（6）：962－968．

［14］WANG Zhen，CHEN Jin，QIN Ming．Non－parallel planes support vector machine for multi－class classification［J］．Logistics Systems and Intelligent Management，2010，1（3）：581－585．

［15］BI Hua，LIANG Hongli，WANG Yu．Resampling methods and machine learning［J］．Chinese Journal of Computers，2009，32（5）：862－877．（in Chinese）

［16］ZHANG Yuchen，DU Zhonghua，DAI Wei．Design of ballistic consistency based on least squares support vector machine and particle swarm optimization ［J］．Transactions of Nanjing University of Aeronautics and Astronautics，2015，32（5）：549－554．

［17］YE Fei，GONG Jian，YANG Wang．Webshell black box test based on support vector machine［J］．Journal of Nanjing University of Aeronautics and Astronautics，2015，47（6）：924－930．（in Chinese）