Soft Computing (2019) 23:2445–2462https://doi.org/10.1007/s00500-017-2940-9

METHODOLOGIES AND APPL ICAT ION

A novel intelligent diagnosis method using optimal LS-SVMwithimproved PSO algorithm

Wu Deng1,2,3,4,5 · Rui Yao2 · Huimin Zhao1,3,4,5 · Xinhua Yang1,5 · Guangyu Li1

Published online: 24 November 2017© Springer-Verlag GmbH Germany, part of Springer Nature 2017

AbstractAiming at the problem that the most existing fault diagnosis methods could not effectively recognize the early faults in therotating machinery, the empirical mode decomposition, fuzzy information entropy, improved particle swarm optimizationalgorithm and least squares support vector machines are introduced into the fault diagnosis to propose a novel intelligentdiagnosis method, which is applied to diagnose the faults of the motor bearing in this paper. In the proposed method, thevibration signal is decomposed into a set of intrinsic mode functions (IMFs) by using empirical mode decomposition method.The fuzzy information entropy values of IMFs are calculated to reveal the intrinsic characteristics of the vibration signaland considered as feature vectors. Then the diversity mutation strategy, neighborhood mutation strategy, learning factorstrategy and inertia weight strategy for basic particle swarm optimization (PSO) algorithm are used to propose an improvedPSO algorithm. The improved PSO algorithm is used to optimize the parameters of least squares support vector machines(LS-SVM) in order to construct an optimal LS-SVM classifier, which is used to classify the fault. Finally, the proposedfault diagnosis method is fully evaluated by experiments and comparative studies for motor bearing. The experiment resultsindicate that the fuzzy information entropy can accurately and more completely extract the characteristics of the vibrationsignal. The improved PSO algorithm can effectively improve the classification accuracy of LS-SVM, and the proposed faultdiagnosis method outperforms the other mentioned methods in this paper and published in the literature. It provides a newmethod for fault diagnosis of rotating machinery.

Keywords Intelligent diagnosis · Feature extraction · Fuzzy information entropy · Multi-strategy · Particle swarmoptimization · Least squares support vector machines · Combinatorial optimization

Communicated by V. Loia.

B Huimin Zhaohm_zhao1977@126.com

1 Software Institute, Dalian Jiaotong University, Dalian116028, China

2 School of Electronics and Information Engineering, DalianJiaotong University, Dalian 116028, China

3 Sichuan Provincial Key Lab of Process Equipment andControl (Sichuan University of Science and Engineering),Zigong 64300, China

4 Traction Power State Key Laboratory of Southwest JiaotongUniversity, Chengdu 610031, China

5 Liaoning Key Laboratory of Welding and Reliability of RailTransportation Equipment, Dalian Jiaotong University,Dalian 116028, China

1 Introduction

With the development of science and technology and socialprogress, in order tomeet the requirements of improving pro-duction efficiency and reducing production costs in the largeindustrial enterprises, the mechanical equipment is devel-oping toward large-scale, continuous and highly integrateddirection, which leads to their structures becoming more andmore complex. So it will be more difficult to monitor anddiagnose the potential faults for the mechanical equipment.The reliability and safety of mechanical equipment directlyaffect the development level of high-speed railway, machin-ery industry, national defense, science and technology andother industries (Chandra and Sekhar 2016; Jung and Bae2015). If the mechanical equipment breaks down in a veryshort time, it will not only cause that the equipment is dam-aged and the production line is interrupted, but also causehuge economic losses. It may even endanger personal safety

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and cause environmental pollution, which will have a seriousimpact on society. At the same time, the faults of mechan-ical equipment take on the consistency and diversity, andthe most faults are long-term and longer development cycle.It is extremely critical to early discover and eliminate thefaults when the faults occur. Therefore, the research of faultdiagnosis technology for mechanical equipment is of greatsignificance.

Rotating machinery is an important branch of modernmachinery and equipment and plays a key and decisive roleinmetallurgy, manufacturing, transportation, power, militaryand other fields. Rolling bearing is an important componentof typical rotating machinery and one of the most vulnerableparts. Its running state directly determines the performanceof the whole motor. As the motor structure is complex andit often works under the high temperature and speed, themotor bearing is prone to various faults. According to theincomplete statistics, the total number of motor rolling bear-ing faults is about 44%. All kinds of air disaster, shipwreckand other malignant accidents occurred one after another dueto motor faults. For example, a plane carrying Brazil foot-ball players to Columbia encountered a crash due to motorfaults in November 2016. There were 77 passengers andcrew on board, and only 6 people were rescued. The acci-dent caused serious casualties and huge economic losses.Catastrophic accidents such as those caused by motor faultoccur frequently, which bring serious losses to society, andmake human beings to more andmore realize the importanceof fault diagnosis (Zhao et al. 2017). Therefore, the motorrolling bearing running state is timely understood, and thefault types and damage degree are grasped, and the effectivepreventive measures are proposed, which is of great practicalvalue.

In the process of the development of fault diagnosis andidentification ofmotor bearing,many experts and researchershave proposed a lot of different fault diagnosis methods(Wang et al. 2008; Ahmadi and Shadizadeh 2012; Deng et al.2017a;Bae2016;Zhao et al. 2011;Deng et al. 2017b), such aswavelet analysis, expert system, empirical mode decomposi-tion method, genetic algorithm, particle swarm optimizationalgorithm, support vectormachine. Thesemethods have theirown advantages and disadvantages in the actual applications.For example, the wavelet analysis can decompose the signalinto high and low frequencies, but it is not fine to decom-pose the high frequency. The empirical mode decompositionmethod can better decompose the signal, but there existsthe modal aliasing phenomenon. The expert system is notaffected by the surrounding environment in solving actualproblems, but it can only be applied in a fairly narrow field ofknowledge. The genetic algorithm has more variables, whenthe range is large or it does not give the range, and the conver-gence speed of the algorithm is decreased. The least squaressupport vector machine can solve the small-sample, nonlin-

ear and high-dimensional problems, but when it is used tosolve nonlinear problems, the selection of kernel functiondirectly affects the final classification result. The particleswarm optimization algorithm is simple and easy to imple-ment and has fast solving speed, but it is easy to fall into localextreme point. Therefore, in order to effectively diagnose thefault of motor bearing, the advantages of empirical modedecomposition, fuzzy information entropy, improved PSOalgorithm and least squares support vector machine are fullyintegrated in order to propose a novel intelligent diagnosismethod for motor bearing. The empirical mode decomposi-tion and fuzzy information entropy are used to effectivelyextract the fault features, and the improved PSO algorithm isused to optimize the parameters of the LS-SVM to proposean optimal LS-SVM classifier with the high classificationaccuracy. And the actual vibration signal of motor bearing isused to verify the effectiveness of proposed fault diagnosismethod.

The rest of this paper is organized as follows: Sect. 2discusses the related work. Section 3 introduces the PSOalgorithm and its improvements. Section 4 expatiates on anintelligent fault diagnosis method based on improved PSOalgorithm. Section 5 describes the realization of intelligentdiagnosis method and result analysis by a case. Finally, theconclusions are discussed in Sect. 6.

2 Related work

For the fault diagnosis of rotating machinery, the faultresearch of rolling bearing is very important and difficult.The fault diagnosis technology for rolling bearings hasbeen developed abroad. With the development of diagnosismethod and the expanding application field, the reliabilityof fault diagnosis has been improved. A lot of fault diag-nosis methods have been proposed in the past few decades.Gustafsson and Tallian (1962) used the acceleration sensor tocollect signals for diagnosing the fault in 1962. Fast fouriertransformation (FFT) is proposed to study the vibration sig-nals in 1965, which laid the foundation for fault diagnosistechnology. In the late 1960s, SPM Instrument Companyin Sweden developed the shock pulse meter instrument. Itsprinciple is to measure the impact pulse amplitude of thedamaged rolling bearings under load. Therefore, this char-acteristic is used to diagnose the motor bearing fault andit can effectively diagnose the early damage fault of thebearing. Hrating in Boeing Company invented the resonancedemodulation analysis system in 1974. Compared with theshock pulse method, this technology can determine the faultand its location and severity. In 1980s, a series of analysismethods and processing technologies of neural network andexpert system and so on were combined in order to carryout intelligent monitoring and diagnosis of rolling bearings.

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Bently Nevada company has carried out a series of bear-ing fault diagnosis test and achieved better results. In recentyears, with the rapid development of modern signal process-ing technology, along with the continuous improvement inintelligent optimization algorithm, fault diagnosis technol-ogy has alsomade great progress. Li et al. (2000) proposed anapproach for motor rolling bearing fault diagnosis using neu-ral networks and time-/frequency-domain bearing vibrationanalysis by discussing bearing vibration frequency featuresof motor bearing fault diagnosis. Rubini and Meneghetti(2001) proposed the limits of the mentioned methodologiesby showing their application to bearings affected by differentpitting failures on the outer or inner race or a rolling elementand subjected to a very low radial load. Lou and Loparo(2004) proposed a new scheme for the diagnosis of local-ized defects in ball bearings based on the wavelet transformand neuro-fuzzy classification. Chiang et al. (2004) pro-posed a fault diagnosis method based on Fisher discriminantanalysis and support vector machines. Nandi et al. (2005)reviewed condition monitoring and fault diagnosis technolo-gies andmethods of electricalmotors, such as expert systems,neural networks, fuzzy logic-based systems. Purushothamet al. (2005) proposed a new multi-fault pattern recognitionmethod for detecting localized bearing defects based on dis-crete wavelet transform and hidden Markov model. Liu et al.(2006) proposed a fault diagnosis method based on empiri-cal mode decomposition and Hilbert spectrum for vibrationsignal analysis for localized gearbox fault diagnosis. Leeet al. (2006) proposed a hybrid fault diagnosis model basedon the signed digraph and support vector machine. Widodoand Yang (2007) proposed a survey of machine conditionmonitoring and fault diagnosis using support vector machinein order to summarize and review the recent research anddevelopments of SVM in machine condition monitoring anddiagnosis. Rai and Mohanty (2007) proposed a bearing faultdiagnosis method based on fast fourier transformation ofintrinsic mode functions fromHilbert–Huang transform pro-cess. Basir andYuan (2007) proposed a preliminary reviewofevidence theory and explained how the multi-sensor enginediagnosis problem can be framed in the context of this theory,in terms of faults frame of discernment, mass functions andthe rule for combining pieces of evidence. Lee et al. (2010)proposed a novel multivariate statistical process monitoringmethod based on modified independent component analysis.Kankar et al. (2011a) proposed a fault diagnosis method ofball bearings having localized defects on the various bearingcomponents using wavelet-based feature extraction. Kankaret al. (2011b) proposed a methodology for rolling elementbearings fault diagnosis using continuous wavelet transformand three machine learning techniques. Kadri et al. (2012)proposed a novel hybrid algorithm for fault diagnosis ofrotary kiln based on a binary ant colony and support vec-tor machine. Van et al. (2013) proposed an induction motor

fault diagnosis method based on Fourier–Bessel expansionand simplified fuzzy ARTMA. Pandya et al. (2014) proposeda fault diagnosis method of rolling element bearing basedon multinomial logistic regression and wavelet packet trans-form. Jaouher et al. (2015) proposed an automatic bearingfault diagnosis method based on empirical mode decompo-sition and artificial neural network. Rodriguez Ramos et al.(2017) proposed a novel fault diagnosis scheme based onfuzzy clustering techniques. The other algorithms and meth-ods are proposed to realize the fault diagnosis (Ahmadi et al.2015a,b; Ahmadi 2011; Oliveira et al. 2017; Ahmadi andBahadori 2015).

In our country, fault diagnosis technology started in 1980s.The fault diagnosis technologies and methods are contin-uously learned and innovated and have rapidly developed.They have mainly gone through three major stages. Thefirst stage is before 1980: The foreign advanced technolo-gies were introduced into China. The tape recorder was usedto record the vibration signals, and an FFT analyzer wasused to analyze in order to study some related fault prin-ciple and diagnosis methods, which were applied to actualengineering. The second stage is between 1980 and 1990: Alot of researchers have constantly studied and explored, indepth, new fault diagnosis technologies and methods. Theypay more attention to the intelligent signal processing. Thethird stage is from 1990s to now. A lot of innovation theoriesare proposed to comprehensively consider the signal param-eters and more timely and accurately monitor the faults.In recent years, there are many researchers to be engagedin studying the fault diagnosis technologies and methods.Their fault diagnosis theories and products have been widelyused in China. Nowadays, many scholars introduce intelli-gent algorithms to obtain satisfactory optimization results.Lin and Qu (2000) proposed a denoising method based onwavelet analysis to realize feature extraction for mechanicalvibration signals. Sun et al. (2004) proposed a fault diagnosismodel based on fuzzy Petri nets for electric power systems.Yu et al. (2005) proposed a fault diagnosis of roller bearingsbased on empirical mode decomposition method and Hilbertspectrum. Yu et al. (2006) proposed a roller bearing faultdiagnosis method based on empirical mode decompositionand energy entropy according to the non-stationary charac-teristics of roller bearing fault vibration signals. Hu et al.(2007) proposed a novel method for fault diagnosis based onan improved wavelet package transform, a distance evalua-tion technique and the support vectormachines ensemble. Feiand Zhang (2009) proposed a fault diagnosis method basedon support vector machine and genetic algorithm for powertransformer.Wu et al. (2011) proposed a newversion of fuzzywavelet support vector classifier machine to diagnose thenonlinear fuzzy fault system with multi-dimensional inputvariables. Shen et al. (2012) proposed a novel model for faultdiagnosis based on empiricalmode decomposition andmulti-

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class transductive support vector machine, which is appliedto diagnose the faults of the gear reducer. Bin et al. (2012)proposed a new approach based on wavelet packet decom-position and empirical mode decomposition to extract faultfeature frequency and neural network for rotating machineryearly fault diagnosis. Lei et al. (2013) surveyed and summa-rized the recent research and development of EMD in faultdiagnosis of rotating machinery, providing comprehensivereferences for researchers concerning this topic and help-ing them identify further research topics. Chen et al. (2014)proposed a novel intelligent fault diagnosis model based onmulti-kernel support vector machine with chaotic particleswarm optimization for roller bearing fault diagnosis. Guet al. (2015) proposed an effective incremental support vec-tor ordinal regression formulation based on a sum-of-marginsstrategy. Pan et al. (2015) proposed an efficient motion esti-mation and disparity estimation algorithm for reducing thecomputational complexity. Gu and Sheng (2017) proposeda regularization path algorithm for ν-support vector classi-fication. Ma et al. (2016) proposed an efficient overlappingcommunity detection algorithm based on structural cluster-ing. Sun and Gu (2017) proposed a compressive sensingof piezoelectric sensor response signal for the structuralhealth monitoring. Gu et al. (2017) proposed a structuralminimax probability machine for constructing a margin clas-sifier. Liu et al. (2016) proposed a speculative approach forspatial-temporal efficiency with multi-objective optimiza-tion. Kong et al. (2016) proposed a belief propagation-basedoptimization method for solving task allocation problem.Zhang et al. (2015) proposed an intelligent fault diagno-sis of rotating machinery based on support vector machineand ant colony algorithm for synchronous feature selection.Fu et al. (2016) proposed an efficient multi-keyword fuzzyranked search scheme that is able to address the aforemen-tioned problems. Chu et al. (2016) proposed a FOA-SVMmodel by combining fruit fly optimization algorithm andsupport vector machine to realize the optimization of theSVM parameters. Wang et al. (2017) proposed a novelmulti-watermarking scheme based on hybrid multi-bit mul-tiplicative rules. Xue et al. (2017) proposed a self-adaptiveartificial bee colony algorithm based on the global best can-didate for solving global optimization problems. Wang et al.(2017) proposed a backpropagation neural network modelby using solar radiation to establish the relationship. Li etal. (2017) proposed a novel rolling bearing fault diagnosisstrategy based on improved multi-scale permutation entropy,Laplacian score, least squares support vector machine, andquantum-behaved particle swarm optimization. Rong et al.(2017) proposed a novel K+-isomorphismmethod to achieveK-anonymization state among subgraphs. Zhang et al. (2016)proposed a special model known as RELAX-RSMN witha totally unimodular constraint coefficient matrix to solvethe relaxed 0–1 ILP rapidly through linear programming.

vokelj et al. (2016) proposed a novel multivariate and multi-scale statistical process monitoring method in large slewingbearings. Yuan et al. (2016) proposed a new software-baseddetection approach using multi-scale local-phase quantityand principal component analysis. Zhang et al. (2017) pro-posed anoptimal cluster-basedmechanism for loadbalancingwith multiple mobile sinks. Chen et al. (2017) proposed animproved quaternion principal component analysis methodfor processing nonlinear quaternion signals. Xiong et al.(2017) proposed a novel reversible data hiding scheme usinginteger wavelet transform, histogram shifting and orthogo-nal decomposition. Hu et al. (2017) proposed an intelligentfault diagnosis method based on deep neural networks forthe high-speed train.

In summary, we can see that the traditional fault diag-nosis methods cannot effectively analyze the signal, whichleads to the inaccurate fault diagnosis results according tothe personal experiences. Intelligent optimization algorithmsmake up some shortcomings of the traditional fault diagnosismethod and improve the accuracy of fault diagnosis to a cer-tain extent. But there are still difficulties to meet the accuracydemandof fault diagnosis, and the intelligent algorithms haveslowconvergence speed and are easy to fall into localminima.At same time, some intelligent algorithms improve the opti-mization performance of intelligent algorithm by spendinglong running time. Therefore, how to use appropriate meanto introduce the intelligent algorithms into the fault diagno-sis and construct new fault diagnosis models and methods inorder to further improve the accuracy of fault diagnosis formotor bearing, is a problem that needs to be studied deeply.

3 PSO algorithm and its improvements

3.1 PSO algorithm

The PSO algorithm (Kennedy and Eberhart 1995; Chen et al.2017) is a population-based search algorithm based on thesimulation of the social behavior of birds within a flock. Inthe PSO algorithm, individuals, referred to as particles, are“flown” through hyperdimensional search space. The parti-cles’ positions within the search space are changed based onthe social-psychological tendency of individuals in order todelete the success of other individuals. The particle changingwithin the swarm is influenced by the experience or knowl-edge. The consequence of modeling for this social behavioris that the search is processed in order to return toward pre-viously successful regions in the search space. Namely, thevelocity (v) and position (x) of each particle will be changedaccording to the following expressions:

vi j (t + 1) = wvi j (t) + c1r1(pBi j (t) − xi j (t)

)

+ c2r2(gBi j (t) − xi j (t)

) ; (1)

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xi j (t + 1) = xi j (t) + vi j (t + 1) , (2)

where vi j (t + 1) is the velocity of particle i at iteration jand xi j (t + 1) is the position of particle i at iteration j . w

is inertia weight to be employed to control the impact ofthe previous history of the velocity. t denotes the iterationnumber, c1 is the cognition learning factor, c2 is the sociallearning factor, and r1 and r2 are random numbers in [0,1] fordenoting remembrance ability. Generally, the value of eachcomponent in V can be clamped to the range [− Vmax,Vmax]for controlling excessive roaming of the particle outside thesearch space. The PSO algorithm is terminated with maxi-mal generations or the best position of particle in the swarm,which cannot be improved further after a sufficiently largenumber of generations. Therefore, the PSO algorithm hasshown its robustness and efficacy in solving complex opti-mization problems.

3.2 The improvements of PSO algorithm

3.2.1 The diversity mutation strategy

In the random initialization stage of PSO algorithm, the pop-ulation diversity is higher. But with the advance of evolution,the difference of particles is reduced and the population diver-sity is decreased, which fall into the local optimum and showthe premature phenomenon. Therefore, the diversity muta-tion strategy is used to improve the basic PSO algorithm.The advantages of the evolutionary algorithm and the basicPSO algorithm are combined in order to improve the searchperformance. The diversity expression of PSO algorithm isdescribed as follows:

dt = 1

nδ

n∑

i=1

√√√√n∑

j=1

(xti j − xtj

)2, (3)

where δ is the length of the longest diagonal line in the searchspace.

In order to keep the population diversity, when there isdt < dlow, the followed mutation operation is performed inorder to make the population to jump away the local extremepoint.

{ptj = ptj + ρ × δ × ζ

ptg, j = ptg, j + ρ × δ × ζ,(4)

where ζ ∼ N (0, 1) and ρ is a specified parameter. ρ ≥10dlow is proposed to meet the requirement of dt < dlowafter the mutation is executed.

3.2.2 The neighborhoodmutation strategy

The particle closely moves to the optimal position of popula-tion and gradually gathers to a smaller area in the evolutionprocess. This will reduce the diversity and search ability ofpopulation. In order to improve the search efficiency of thePSO algorithm, the optimal individual of population is ran-domly mutated in the generational reducing neighborhoodrange to locally fine search. If the fitness value of new individ-ual is increased by using the neighborhoodmutation strategy,the global optimal individual is replaced by new individual.Otherwise, the individual is randomly replaced according tothe certain probability. Set the variable Y and is mutated toget Y ′. The calculation formula is described as follows:

Y ′ = Y + Rk(2r4 − 1); (5)

Rk = (R − R) × (k − k) × k−1 + R, (6)

where Rk is the radius of neighborhood search of kth iter-ation, and R and R, respectively, are the upper bound andlower bound of radius of neighborhood search. r4 is a uni-form random number on [0,1].

3.2.3 The improvement of learning factors

(1) Linear change of learning factor

In the basic PSO algorithm, the c1 and c2 parameters aregiven in advance according to the experience. But their val-ues [0,4] will reduce the self-learning ability of particles. Inthis paper, the value range of parameters c1 and c2 is given.The initial value and final value are c1 ∈ (2.75, 1.25) andc2 ∈ (0.5, 2.25), respectively. The learning factor functionexpression of linear change is descried as follows:

c1 = c1max + (c1min − c1max) × t/T; (7)

c2 = c2max + (c2min − c2max) × t/T, (8)

where c1max and c2max are the initial values of c1 and c2,c1min and c2min are the final values of c1 and c2, T is themaximum number of iterations, and t is the current numberof iteration.

(2) Anticosine change of learning factor

The characteristics of anticosine strategy are to accelerate thechanges of c1 and c2 in order to quickly perform local searchin the initial stage. It is amore ideal strategy to linearly set thevalues of c1 and c2 later. The initial value and final value ofc1 and c2 are c1 ∈ (2.75, 1.25) and c2 ∈ (0.5, 2.25), respec-tively. The acceleration factors of anticosine are obtainedaccording to the following expressions:

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c1 = c1min + (c1max − c1min)

∗(1 − arccos ((− 2t/T + 1)/π)); (9)

c2 = c2max + (c2max − c2min)

∗(1 − arccos ((− 2t/T + 1)/π)). (10)

(3) Arctangent change of learning factor

The research results discovered that the PSO algorithm canmake particles to leap thewhole search space asmuch as pos-sible, in order to obtain the diversity of particles in the initialstage of the search and converge to the global optimal solu-tion with fast speed in the end of the search under the idealcondition. By analyzing the influence of the change of learn-ing factors, the arctangent function is used to dynamicallyadjust the parameters c1 and c2 in order to better balance theglobal search and the local search. The arctangent functionexpression is described as follows:

c1 = c1max − (c1max − c1min) × (a tan (20 × t/T − e)

+ a tan (e)) /h; (11)

c2 = c2max − (c2max − c2min) × (a tan (20 × t/T − e)

+ a tan (e)) /h, (12)

where e is the adjustment coefficient, e = 6. h =a tan (20 − e) + a tan (e).

3.2.4 The improvement of inertia weight

(1) Linear adjustment of w method

The research results of Shi and Eberhart are used in thispaper. The characteristics of linearly decreasing w take onstronger search ability in the early iterations. It can searchlarger solution space and has the ability to find new areascontinuously. But in later iterations, the algorithm will con-verge to a better area to accelerate the convergence speed.The expression of the linear adjustment of inertia weight isdescribed as follows:

w = wmax − (wmax − wmin) ∗ t/T , (13)

where wmax and wmin are the maximum inertia weight andthe minimum inertia weight:wmax = 0.85 andwmin = 0.1. tis the current iteration number, andT is themaximumnumberof iterations.

(2) Quadratic form adjustment of w method

By linear adjustment of w, the larger inertial weight cansearch the larger area at the initial stage of iteration, whereasthe smaller inertia weights can perform better local search

in the later iterations. And the quadratic form adjustment ofw method is a nonlinear decreasing method, which makesthe inertia weight to fall slowly in the initial iteration anddecrease faster in the later iteration in order to more accu-rately search. The updating formula of the quadratic formadjustment of w is described as follows:

w = wmax − (wmax − wmin) ∗ t2/T 2. (14)

(3) S-shaped function adjustment of w method

The inertia weight w is declined by using S-shaped func-tion in order to ensure that the population can keep a highsearch speed in the initial search, decline the search speed inthe middle search to easily converge to the global optimum,and keep a certain speed to finally converge to the optimalsolution in the last search. The inertia weight expression ofS-shaped function is described as follows:

w = (wmax − wmin) / (1 + exp (2 × o × t/T − o))+wmin,

(15)

where o is the control factor to adjust the speed and is set too = 13 here.

4 A novel intelligent diagnosis method andmodel

The fault diagnosis is to select appropriate technology ormethod to determine the fault type, the fault severity, thefault location, and so on. In recent years, many experts andresearchers have proposed a lot of different fault diagnosismethods, which are used to solve many fault diagnosis prob-lems and obtain better application results. But some of theexisting fault diagnosis methods could not effectively recog-nize the early faults in the rotating machinery. The EMDmethod can better decompose the signal, but there existsthe modal aliasing phenomenon. The LS-SVM can solvethe small-sample, nonlinear and high-dimensional problems,but when it is used to solve nonlinear problems, the selec-tion of kernel function directly affects the final classificationresult. The PSO algorithm is simple and easy to imple-ment and has fast solving speed, but it is easy to fall intolocal extreme point. Therefore, in order to improve the faultdiagnosis accuracy for motor bearing, the advantages of theEMD, fuzzy information entropy, improved PSO algorithmand LS-SVM are introduced into the fault diagnosis in orderto propose a novel intelligent diagnosis method, which isapplied to diagnose the faults of the motor bearing in thispaper. Firstly, the EMD method based on the direct extrac-tion of the energy associatedwith various intrinsic time scales

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Original vibration signal

Select parameters of EMD

Extract the IMFs components

Select parameters of fuzzy information entropy

Calculate the entropy values

Normalization processing

Obtain the feature values

Feature extraction

Initialize parameters of PSO

Improvements of PSO algorithm

Obtain improved PSO algorithm

Select kernel function of LS-SVM

Train the LS-SVM model

Construct optimal LS-SVM classier

Construct a classier

Input testing sample data

Diagnose faults

Obtain Diagnosis results

Fault diagnosis

Fig. 1 Flow of the proposed intelligent diagnosis method

is used to decompose the vibration signals into a series ofIMFs and residual signal. And the fuzzy information entropymeasuring the complexity of time series based on the con-cept of approximate entropy and sample entropy is used toeffectively extract the features of vibration signal, which areregarded as input feature vectors. Then the diversity muta-tion strategy, neighborhoodmutation strategy, learning factorstrategy and inertia weight strategy for basic PSO algorithmare used to propose an improved PSO algorithm, which isused to optimize the parameters of the LS-SVM in order topropose an optimal LS-SVM classifier with the high classi-fication accuracy. A novel intelligent diagnosis method isproposed to realize the fault diagnosis of motor bearing.

The flow of the intelligent diagnosis method is shown inFig. 1.

5 Realization of intelligent diagnosismethod and result analysis

5.1 Experimental environment and data sources

Rolling bearings are prone to inner race fault, outer racefault, rolling element fault and so on during the work. Sothe vibration data of rolling bearing from Bearing DataCenter of Case Western Reserve University are used tovalidate the effectiveness of the proposed intelligent diag-nosis method (http://csegroups.case.edu/bearingdatacenter/home). The 6205-2RS 6 JEM SKF deep-groove ball bear-ing is employed in the experiment. The motor with 1.5 KWis connected to a dynamometer and a torque sensor by aself-aligning coupling. The vibration signals were collectedfrom an accelerometer mounted on the radial vertical direc-tion of the motor. The fault type in the experiment is crackfault. The vibration signals were measured under no load (0HP) at a rotation speed of 1797 r/min. The bearing vibra-tion signal was sampled at the frequency of 12,000 Hz, andthe duration of each vibration signal was 10 s. The fre-quency shift is 29.95 Hz. The inner race fault, outer racefault and rolling element fault were introduced to the testbearings by using electro-discharge machining method. Thenormal vibration signal, inner race fault vibration signal,outer race fault vibration signal and rolling element faultvibration signal are collected. They are original vibrationsignals without pretreatment. The fault diameter of 0.007inches is selected as the case here, which is used to showtime-domain waveform of fault vibration signal. The time-domain waveforms of vibration signals are shown in Figs. 2,3, 4 and 5. The time-domain waveform of normal vibrationsignal is shown in Fig. 2. The time-domain waveform of faultvibration signal for outer race fault is shown in Fig. 3, thetime-domainwaveformof fault vibration signal for inner racefault is shown in Fig. 4, and the time-domain waveform offault vibration signal for rolling element fault is shown inFig. 5.

It is shown in Figs. 2, 3, 4 and 5 that the time-domainwave-form represents the dynamic signal relationship by using thetime axis, which reflects the change characteristics of thesignal amplitude with time. When the rolling bearings worknormally, the amplitudes of vibration signals vary little ineach time interval. When the rolling bearings work abnor-mally, the amplitude of vibration signals will increase ordecrease in each time interval.

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0 200 400 600 800 1000 1200 1400 1600 1800 2000-0.5

0

0.5

t/s

Am

plitu

de/ µ

m

Fig. 2 Time-domain waveform of normal vibration signal

Fig. 3 Time-domain waveformof vibration signal for outer racefault

0 200 400 600 800 1000 1200 1400 1600 1800 2000-4

-2

0

2

4

t/s

Am

plitu

de/µ

m

0 200 400 600 800 1000 1200 1400 1600 1800 2000-4

-2

0

2

4

t/s

Am

plitu

de/µ

m

Fig. 4 Time-domain waveform of vibration signal for inner race fault

0 200 400 600 800 1000 1200 1400 1600 1800 2000-0.5

0

0.5

t/s

Am

plitu

de/µ

m

Fig. 5 Time-domain waveform of vibration signal for rolling element fault

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sign

al

Empirical Mode Decomposition

imf1

imf2

imf3

imf4

imf5

imf6

imf7

res.

Time(s)

Fig. 6 Decomposition result of the normal vibration signal

5.2 Feature extraction based on EMD and fuzzyinformation entropy

5.2.1 Vibration signal decomposition based on EMDmethod

Vibration signal is an important carrier of the motor statusinformation, which contains a large number of fault informa-tion in the motor. Therefore, the vibration signal feature isan important index to evaluate the running state of the motor.The empirical mode decomposition (EMD) is an adaptivedecomposition technique,which is based on the direct extrac-tion of the energy associatedwith various intrinsic time scalesin order to generate a collection of intrinsic mode functions(IMFs). The EMD can decompose the complicated signalinto a definite number of high-frequency and low-frequencycomponents. According to the definition of EMD and theprocess of signal processing, it can be seen that the EMDmethod is used to decompose the original signal from highfrequency to low frequency. The EMD method is used todecompose the original signals of normal vibration signal,inner race fault, outer race fault and rolling element fault toobtain the intrinsic mode functions, shown in Figs. 5, 6, 7, 8and 9.

It can be seen from Figs. 6, 7, 8 and 9, the EMD methoddecomposes the original signal from the high frequency tolow frequency into a series of IMFcomponents, and each IMFcomponent is independent of each other and contains differ-ent frequency components. Therefore, the vibration signaldecomposition method based on EMD can gradually decom-pose the local characteristics of the original signal of rolling

bearing under different times and scales. And the decom-posed sequence has more regularity than the original signalof rolling bearing and provides a basis for efficient computingof fuzzy entropy value of each IMF component.

5.2.2 Fault feature extraction of motor bearings based onfuzzy information entropy

A non-probability entropy based on the form of fuzzy settheory is defined, which considers the overall measurementof state uncertainty. It can be taken as a measurement of irrel-evant information with random experiment. The uncertaintyof this entropy comes from the inside, which is useful. There-fore, it gives a measurement of the fuzziness degree of thestate, and it can also be regarded as an average intrinsic infor-mation when the decision is made. The quantification of thefuzzy degree of fuzzy sets is an important aspect of fuzzy settheory.

Definition 1 Set a substance (information, energy) system,and the states of each element are {x1, x2, . . . , xn}. Theprobabilities of random occurrence for each element arep(x1), p(x2), . . . , p(xn), and there is

∑ni=1 p(xi ) = 1, i =

1, 2, 3, . . . , n. If the information quantity of the determinacyprobability of the state xi is taken as I (xi ), then there isI (xi ) = − loga P(xi ). Usually, there is a = 2 or e. Themathematical expectation of the information quantity of thesystem states is taken as the information entropy. That is,

H(xi ) =n∑

i=1

p(xi )I (xi ) = −n∑

i=1

p(xi ) loga p(xi ). (16)

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2454 W. Deng et al.

sign

al

Empirical Mode Decomposition

imf1

imf2

imf3

imf4

imf5

imf6

imf7

imf8

imf9

res.

Time(s)

Fig. 7 Decomposition result of the outer race fault vibration signal

sign

al

Empirical Mode Decomposition

imf1

imf2

imf3

imf4

imf5

imf6

imf7

imf8

imf9

imf1

0re

s.

Time(s)

Fig. 8 Decomposition result of the inner race fault vibration signal

Kaufmann and Delaca extended the information entropyto fuzzy sets in the 1973∼1974, which is called fuzzy infor-mation entropy.

Definition 2 In the fuzzy event xi j , ui j expresses the sub-ordinate degree of the j th event index of the i th event.The probability is p(xi j ) = ∫

�ui j dp = E(ui j ), and the

fuzzy information entropy of the i th event is described asfollows:

H(xi j ) = −n∑

j=1

u(xi j )p(xi j ) loga p(xi j ). (17)

The fuzzy information entropy is used to extract the faultfeatures of motor bearing in this paper. This technology isused to calculate the entropy values of the obtained IMFcomponents. TheEMDmethod is used to decompose the nor-mal vibration signal, inner race fault vibration signal, outerrace fault vibration signal and rolling element fault vibra-tion signal of motor bearing into a series of intrinsic mode

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A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm 2455

sign

al

Empirical Mode Decomposition

imf1

imf2

imf3

imf4

imf5

imf6

imf7

imf8

imf9

res.

Time(s)

Fig. 9 Decomposition result of the rolling element fault vibration signal

Table 1 Normalized values ofnormal vibration signal

No. M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11

1 0.019 0.025 0.001 0.012 0.008 0.017 0 0 0 1 1

2 0.036 0.039 0.017 0.003 0.047 0.031 0.051 0 0 1 1

3 0.004 0.035 0.001 0.003 0.021 0 0.031 0 1 1 1

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

40 0.078 0.042 0.004 0.004 0.003 0.041 0.062 0 0 1 1

Table 2 Normalized values ofinner race fault vibration signal

No. M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11

1 0.402 0.231 0.114 0.154 0.215 0.322 0.454 0.251 0 0 0

2 0.757 0.617 0.266 0.301 0.283 0.302 0.471 0.138 0.064 0.22 1

3 0.548 0.644 0.613 0.651 0.385 0.806 0.878 0.375 0.2 0 0

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

40 0.711 0.257 0.195 0.419 0.203 0.396 0.363 0.251 0 0 1

functions. Each vibration signal can obtain 40 sets of featurevalues. The feature values of motor bearing have differentdimension, and the numerical values exhibit a very big differ-ence; thus, the feature value of horizontal direct comparisonhas a certain degree of difficulty. Therefore, the feature val-ues of the training samples and test samples are normalizedin order to eliminate the influence of different dimensionsand take on the comparability among all feature values. Inthis paper, each 40 sets of feature values from the normalvibration signal, inner race fault vibration signal, outer racefault vibration signal and rolling element fault vibration sig-

nal of motor bearing is normalized between 0 and 1. Thenormalized values of normal vibration signal are shown inTable1, the normalized values of inner race fault vibrationsignal are shown in Table2, the normalized values of outerrace fault vibration signal are shown in Table3, and the nor-malized values of rolling element fault vibration signal areshown in Table4. In these tables, the M1 represents the cor-responding entropy values of IMF1, the M2 represents thecorresponding entropy values of IMF2, ...., M10 representsthe corresponding entropy values of IMF10, the M11 repre-sents the corresponding entropy values of residual signal.

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2456 W. Deng et al.

Table 3 Normalized values ofouter race fault vibration signal

No. M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11

1 0.243 0.611 0.039 0.201 0.238 0.166 0.291 0.124 0 0 1

2 0.452 0.255 0.035 0.541 0.165 0.046 0.121 0.125 0 1 1

3 0.363 0.757 0.068 0.132 0.177 0.189 0.213 0.251 0 0 0

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

40 0.153 0.487 0.068 0.059 0.147 0.115 0.152 0.125 0 0 1

Table 4 Normalized values ofrolling element fault vibrationsignal

No. M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11

1 0.182 0.028 0.013 0.019 0.056 0.051 0.091 0 0 0 1

2 0.276 0.032 0.018 0.016 0.058 0.023 0.062 0 0 0 1

3 0.113 0.088 0.019 0.056 0.073 0.04 0.090 0 0 0 1

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

40 0.227 0.082 0.034 0.023 0.052 0.041 0.061 0 0 1 1

5.3 Optimized LS-SVM classifier

5.3.1 LS-SVM

TheLS-SVMis amachine learning algorithmbased on statis-tical theory, which has the unique superiority in dealing withthe small-sample learning problem. It can effectively avoidthe local optimal solutionof the neural network andovercomethe curse of dimensionality. The LS-SVM is an improvedSVM in essence. The best characteristic of LS-SVM is tomodify the inequality constraint in the SVM to the equalityconstraint, and the training error square is used to replace theslack variable in order to transform quadratic programmingproblem into the linear equation problem for greatly improv-ing the speed and accuracy of model parameters. For thenonlinear sample set of arbitrary known inputs and outputs,the LS-SVM model can be constructed by finding suitablenonlinear transformation in following expression:

f (x) =l∑

i=1

αi K (x, xT ) + b. (18)

5.3.2 Select the kernel function for LS-SVM

In the LS-SVM, kernel function plays an important role. Thekernel idea of kernel function is to use kernel function tomaplinearly inseparable samples into high-dimensional space andsolve the curse of dimensionality. The mapped samples arelinearly separable. Then a classification plane is constructedin the high-dimensional space in order to separate the samplesof two classes. Because the structure of feature space is com-pletely determined by kernel function, it is very important toselect the kernel function for designing a classifier. Different

kernel functions make classifier performance to vary greatlyand affect classification effect. There are several commonlyused kernel functions such as linear kernel function, poly-nomial kernel function, radial basis kernel function, sigmoidkernel function. In this paper, the kernel function with theleast error and the prior knowledge by experts is used toselect the kernel function, and then the radial basis kernelfunction (RBF) is selected in here. The kernel function isdescribed as follows:

K (x, xi ) = exp(−γ ‖x − xi‖2), (19)

where x is a m-dimensional input vector and xi is the centerof the i th RBF and has the same dimension as x . γ is theparameter of RBF kernel function.

5.3.3 The optimization idea for LS-SVM parameters

The performance of LS-SVM classifier depends on whetherit can accurately predict the unknown data. In the LS-SVMmodel, regularization parameter γ and kernel parameter σ

are key parameters. The regularization parameter γ deter-mines the model error and the generalization ability, andthe kernel parameter σ reflects the complexity of trainingsample data distribution in high-dimensional feature space.The regularization parameter γ and kernel parameter σ aretwo parameters of LS-SVM model, which must be adjusted.Because the regularization parameter γ and kernel parame-ter σ are used as a whole, the parameter values will directlydetermine the training and generalization performance ofthe LS-SVM. However, two parameters are not necessar-ily related to the performance of LS-SVM in the theory.Therefore, in the application, the values of the regularizationparameter γ and kernel parameter σ become amain problem,

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A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm 2457

and there are no feasible methods to adjust the parametersin the actual operation. The majority is to use the trial anderror according to the usual experience. But this method istime-consuming, and the result is far from the optimal value.In the actual application, it is urgent to propose a practicalmethod to optimize and select the multi-parameter problems.Therefore, it is of great significance to study the regulariza-tion parameter γ and kernel parameter σ of LS-SVM in orderto improve the classification performance of LS-SVM.

The regularization parameter γ and kernel parameter σ

of LS-SVM have great influence on classification accuracy.However, it is difficult to determine the appropriate valuesof LS-SVM in advance. Therefore, the improved PSO algo-rithm with high optimization performance is used to finda set of combination values of LS-SVM. The optimizationidea of LS-SVM based on the improved PSO algorithm isdescribed as follows. The particle dimension of the improvedPSO algorithm is determined, and the position and velocityare initialized. Then the fitness value of each particle is cal-culated, the optimal position of each particle is saved, andthe best fitness value for all particles and the best positionin the population are also saved. According to the speedand the position update formula, the position and velocityare updated. The obtained best positions are compared withthe set range; if the obtained best position is out of range,the position of particle is updated. The fitness value of theupdated particle is calculated, and the obtained fitness valueis compared with the historical best fitness value and the bestposition in order to select the individual optimal value andglobal optimal value. The obtained global optimal value isassigned to the regularization parameter γ and kernel param-eter σ in order to construct a LS-SVM classification model,which can improve the classification accuracy of LS-SVM.

The performance evaluation of LS-SVM parameters isan important work. The classification interval between twoclasses of samples is used as the index of generalizationability of LS-SVM. But different kernel parameters gener-ate different feature spaces, which result in the fact that theinterval between them are not comparable. At present, thewidely used k-cross-validation method is one of the gener-alization error calculation methods. The specific steps aredescribed as follows. The training set is divided into k sub-sets of the same size and disjoint s1, . . . , si , . . . , sk . k trainingand testing are executed, i = 1, . . . , k. One subset si for eachtime is used to test, and other k − 1 subsets are used to trainthe classifier to obtain the number of training points li thataremisclassified. In thisway, l1, . . . , lk are obtained and eachsubset in the whole training set is predicted once. The cross-validation method obtains an estimated value of error rate(∑k

i=1 li )/l, which is called the k-cross-validation error. Inthis paper, k = 10 cross-validation method is used to selectGauss radius.

The specific steps of optimizing LS-SVM parametersbased on the improved PSO algorithm are described as fol-lows:

Step 1 Set the size of the particle swarm and determine theinitial position and speed of the particle swarm.

Step 2 According to the objective function of the optimiza-tion problem, the fitness function is defined. Andthe fitness values of all particles are calculated. (Theaccuracy of the training set in the CV sense is usedas the fitness function value in the PSO algorithm.)

Step 3 Calculate the fitness value of each particle in orderto obtain the extreme value of each particle and thebest position of each particle. The best positions ofall particles are compared to obtain the best positionof all particles.

Step 4 The positions and velocity values of each particle areupdated by two extremes.

Step 5 Determinewhether the condition ismeet; if the condi-tion is notmet, return to Step 2. If the condition ismet,the iteration is terminated and the optimal parameteris obtained.

Step 6 The obtained optimal regularization parameter γ andkernel parameter σ are plugged into the LS-SVMmodel to train this model.

Step 7 The test data are used to validate the obtained optimalLS-SVM model.

Step 8 Output the tested result.

5.4 Fault diagnosis results

The extracted 160 feature vectors of the motor bearing areused as the experimental data. These feature vectors include40 normal feature vectors, 40 inner race fault feature vectors,40 outer race fault feature vectors and 40 rolling elementfault feature vectors. For each feature vector, the randomlyselected 20 samples from 40 fault feature vectors are usedas training samples and the remaining 20 samples are usedas test samples. The experiment environment is described:the Pentium IV, 2.40GHz, 2.0GB RAM, Windows 7 andMATLAB 2010b. The initial parameters of the algorithmare selected after thorough testing. In the simulation exper-iments, the alternative values were tested and modified forsome functions in order to obtain the most reasonable ini-tial values of these parameters. These selected values of theparameters take on the optimal solution and the most reason-able running time of these algorithms to efficiently completethe problem solving. So the selected values of these param-eters are described as follows: population size (M = 20),iteration (Tmax = 200), initial inertia weight (w = 0.8),max velocity (V = 80), and initial learn factor c1 = c2 = 2.In the LS-SVM model, the kernel function uses RBF ker-nel function, and the regularization parameter γ and kernel

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Table 5 Experiment results by using different methods

Methods Time 1 2 3 4 5 6 7 8 9 10 Avg.

PLM Accuracy (%) 73.75 73.75 76.25 87.5 76.25 87.5 73.75 81.25 76.25 81.25 78.75

Iterations 3 13 8 6 3 7 1 5 7 6 5.9

LCPLSM Accuracy (%) 83.75 80 72.5 88.75 80 80 87.5 87.5 87.5 87.5 83.5

Iterations 3 9 5 14 3 2 4 1 16 29 8.6

LCWPLM Accuracy (%) 90 87.5 90 87.5 87.5 88.75 90 90 80 87.5 87.88

Iterations 1 4 163 10 68 25 1 7 12 2 29.3

LCQWPLM Accuracy (%) 88.75 87.5 87.5 90 87.5 90 88.75 80 90 87.5 87.75

Iterations 1 149 5 57 143 4 158 36 132 1 68.6

LCSWPLM Accuracy (%) 88.75 87.5 80 90 88.75 90 90 90 90 88.75 88.38

Iterations 16 136 5 3 99 25 1 82 3 107 47.7

ACLWPLM Accuracy (%) 85 80 85 88.75 85 76.25 80 87.5 90 86.25 84.38

Iterations 1 6 119 76 3 176 183 136 118 3 82.1

ACQWPLM Accuracy (%) 78.75 90 88.75 88.75 90 90 87.5 90 87.5 88.75 88.00

Iterations 7 2 104 145 17 8 3 1 92 6 38.5

ACSWPLM Accuracy (%) 88.75 88.75 90 90 90 90 87.5 88.75 87.5 88.75 89.00

Iterations 105 3 7 41 1 57 108 95 1 162 58

ATLWPLM Accuracy (%) 88.75 87.5 87.5 88.75 90 87.5 90 87.5 88.75 88.75 88.50

Iterations 105 90 136 101 16 28 71 2 1 90 64

ATQWPLM Accuracy (%) 87.5 80 90 87.5 90 81.25 90 77.5 90 90 86.38

Iterations 7 5 1 24 1 2 6 4 2 14 6.6

ATSWPLM Accuracy (%) 88.75 90 90 90 87.5 90 90 88.75 90 90 89.50

Iterations 161 134 1 1 2 61 1 40 5 2 40.8

Bold values indicate better diagnosis accuracy

parameter σ need to be optimized. The initial values of LS-SVM are described as follows: the regularization parameterγ ∈ [0.1, 100] and kernel parameter σ ∈ [0.01, 1000].The experiments were carried out for 10 consecutive sim-ulations, and the average is used as the final classificationaccuracy. Here, the PLM method is based on PSO algorithmand LS-SVM, the LCPLSM method is based on improvedPSO algorithm by linear change of learning factor and LS-SVM, the LCWPLM method is based on improved PSOalgorithm by linear change of learning factor and quadraticform adjustment of w method and LS-SVM, the LCQW-PLM method is based on improved PSO algorithm by linearchange of learning factor and quadratic form adjustment ofw method and LS-SVM, the LCSWPLM method is basedon improved PSO algorithm by linear change of learningfactor and S-shaped function adjustment of w method andLS-SVM, theACLWPLMmethod is based on improved PSOalgorithm by anticosine change of learning factor and lin-ear adjustment of w method and LS-SVM, the ACQWPLMmethod is based on improved PSO algorithm by anticosinechange of learning factor and quadratic form adjustment ofw method and LS-SVM, the ACSWPLM method is basedon improved PSO algorithm by anticosine change of learn-ing factor and S-shaped function adjustment ofwmethod and

LS-SVM, the ATLWPLMmethod is based on improved PSOalgorithm by arctangent change of learning factor and lin-ear adjustment of w method and LS-SVM, the ATQWPLMmethod is based on improved PSO algorithm by arctangentchange of learning factor and quadratic form adjustmentof w method and LS-SVM, and the ATSWPLM methodis based on improved PSO algorithm by arctangent changeof learning factor and S-shaped function adjustment of w

method and LS-SVM. The experiment results are shown inTable5.

As shown in Table 5, Figs. 10 and 11, the average accu-racy rate of the ATSWPLM method based on improvedPSO algorithm by arctangent change of learning factor andS-shaped function adjustment of w method and LS-SVMfor motor bearing is 89.50%, which is the best accuracyamong the PLM method, LCPLSM method, LCWPLMmethod, LCQWPLM method, LCSWPLM method, ACLW-PLM method, ACQWPLM method, ACSWPLM method,ATLWPLM method, ATQWPLM method and ATSWPLMmethod. The diagnosis accuracy of the ATSWPLM methodimproves 10.75% than the diagnosis accuracy of the PLMmethod. The average number of iterations of the ATSW-PLMmethod for fault diagnosis is 40.8. The average numberof iterations of the PLM method for fault diagnosis is 5.9,

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A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm 2459

78.75

83.5

87.8887.75 88.38

84.38

88 89 88.586.38

89.5

72

75

78

81

84

87

90

93

PLM

LCPLSM

LCWPLM

LCQW

PLM

LCSW

PLM

ACLW

PLM

ACQW

PLM

ACSW

PLM

ATLW

PLM

ATQW

PLM

ATSW

PLM

Methods

Accuracy(%)

Fig. 10 Comparison results of the diagnosis accuracy by using different methods

5.9 8.6

29.3

68.6

47.7

82.1

38.5

58 64

6.6

40.8

0

20

40

60

80

100

PLM

LCPLSM

LCWPLM

LCQW

PLM

LCSW

PLM

ACLW

PLM

ACQW

PLM

ACSW

PLM

ATLW

PLM

ATQW

PLM

ATSW

PLM

Methods

Iterations

Fig. 11 Comparison results of the iterations by using different methods

which is the least iterations among these fault diagnosismeth-ods.

5.5 Result analysis and comparison

The diversity mutation strategy, the neighborhood muta-tion strategy, the improvement of learning factors and theimprovement of inertia weight are used to improve the basicPSO algorithm in order to propose an improved PSO algo-rithm, which is used to optimize the regularization parameterγ and kernel parameter σ of LS-SVM. In order to illus-trate the effectiveness of the improved PSO algorithm, thebasic PSOalgorithm is selected to optimize the regularizationparameter γ and kernel parameter σ of LS-SVM. The accu-racy analysis and comparison results are shown in Table6.

As shown in Table6, all improved PSO algorithms canfind the optimal parameters of LS-SVM for fault classifica-tion. And the obtained average diagnosis accuracy based onoptimized LS-SVM using the improved PSO algorithms isbetter than the average diagnosis accuracy based on the opti-mized LS-SVMusing the basic PSO algorithms. The averagediagnosis accuracy based on the ACLWPLMmethod is only84.38%. Although the best diagnosis accuracy based on theACLWPLMmethod reached 90%, the stability of theACLW-PLM classifier is poor, which results in the low averagediagnosis accuracy. The average diagnosis accuracy basedon the ATSWPLMmethod is 89.50% and is the best averagediagnosis accuracy among thesemethods. Therefore, an opti-

Table 6 Accuracy analysis and comparison results

No. Methods Average rate (%) Running time (s) Iteration

1 PLM 78.75 25.38 5.9

2 LCPLSM 83.50 27.79 8.6

3 LCWPLM 87.88 25.92 29.3

4 LCQWPLM 87.75 27.18 68.6

5 LCSWPLM 88.38 27.29 47.7

6 ACLWPLM 84.38 27.11 82.1

7 ACQWPLM 88.00 27.38 38.5

8 ACSWPLM 89.00 27.29 58

9 ATLWPLM 88.50 26.72 64

10 ATQWPLM 86.38 27.79 6.6

11 ATSWPLM 89.50 28.52 40.8

Bold values indicate better diagnosis accuracy

mal LS-SVM classifier should take on good stability, whichis key to find the optimal combination values of LS-SVMparameters by using improved PSO algorithm in order toobtain the higher classification accuracy. In the ATSWPLMmethod, the diversity mutation strategy, the neighborhoodmutation strategy, the neighborhood mutation strategy, arc-tangent change of learning factor strategy and S-shapedfunction adjustment of w strategy are used to improve thebasic PSO algorithm, and then the improved PSO algorithmcan obtain best optimization performance for optimizing the

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parameters of LS-SVM and obtain the optimal combinationvalues of LS-SVM parameters. Therefore, the ATSWPLMmethod can effectively classify the faults of motor bearing,and the running time of the ATSWPLM method is 28.52 s.Compared with other improved PSO algorithms based ondifferent strategies, the ATSWPLMmethod takes longer run-ning time, but the classification accuracy is still the best,which is 89.50%. From the test results of 11 methods, wecan see that the accuracy rate of 90% has 7 times, and thechange of the accuracy rate is smaller, which fully demon-strates that the ATSWPLM takes on the better validity andstability.

6 Conclusion

With the development of science and technology and socialprogress, the mechanical equipment is developing towardlarge-scale, continuous and highly integrated direction,which leads to their structures becoming more and morecomplex. It will be more difficult to monitor and diagnosethe potential faults. In this paper, for the existing problemsof conventional fault diagnosis methods, a novel intelligentdiagnosis method based on combining the advantages of theEMD, fuzzy information entropy, improved PSO algorithmand LS-SVM is proposed to diagnose the faults of the motorbearing. In the proposed fault diagnosis method, the EMDmethod is used to decompose the vibration signals into aseries of IMFs and residual signal, fuzzy information entropyis used to effectively extract the features of vibration signal,and multiple strategies are used to improve PSO algorithmin order to optimize the parameters of the LS-SVM. Then anoptimal LS-SVM classifier with the high classification accu-racy is constructed and a novel intelligent diagnosis methodis obtained. In order to evaluate the effectiveness of the pro-posed fault diagnosis method, the actual vibration signal ofmotor bearing is used to test and analyze. The average accu-racy rate of the ATSWPLMmethod based on improved PSOalgorithm by arctangent change of learning factor and S-shaped function adjustment of w method and LS-SVM formotor bearing is 89.50%, which is the best accuracy amongthe 11 tested methods. The diagnosis accuracy of the ATSW-PLM method improves 10.75% than the diagnosis accuracyof the PLM method. And the accuracy rate of 90% for theATSWPLM method occurs 7 times, and the change of theaccuracy rate of other times is smaller, which fully demon-strates that the ATSWPLM takes on the better validity andstability. Therefore, the experiment results indicate that thefuzzy information entropy can accurately and more com-pletely extract the characteristics of the vibration signal. Theimproved PSO algorithm can effectively improve the classifi-cation accuracy of LS-SVM, and the proposed fault diagnosis

method outperforms other mentioned methods. It provides anew method for fault diagnosis of rotating machinery.

Due to the complexity of fault diagnosis, the collectedvibration signals contain a large number of interfering sig-nals. At the same time, the improved PSO algorithms takelonger running time. Therefore, how to reduce the time com-plexity of the improved PSO algorithm and eliminate a largenumber of interfering signals is the future work.

Acknowledgements The authors would like to thank all the reviewersfor their constructive comments. This research was supported by theNational Natural Science Foundation of China (51475065, 51605068,61771087, U1433124), Open Project Program of Sichuan ProvincialKey Lab of Process Equipment and Control (GK201613), Open ProjectProgram of the Traction Power State Key Laboratory of SouthwestJiaotong University (TPL1705), Natural Science Foundation of Liaon-ing Province (2015020013, 20170540126, 20170540125), and Scienceand Technology Project of Liaoning Provincial Department of Educa-tion (JDL2016030). The program for the initialization, study, training,and simulation of the proposed algorithm in this article was written withthe toolbox of MATLAB 2010b produced by the MathWorks, Inc.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflicts ofinterest.

Ethical standard This article does not contain any studies with humanparticipants performed by any of the authors.

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