fault diagnosis method based on gap metric data preprocessing...

Research ArticleFault Diagnosis Method Based on Gap Metric DataPreprocessing and Principal Component Analysis

ZihanWang ChenglinWen Xiaoming Xu and Siyu Ji

School of Automation Hangzhou Dianzi University Hangzhou 310018 China

Correspondence should be addressed to Chenglin Wen wenclhdueducn

Received 22 February 2018 Accepted 11 April 2018 Published 17 May 2018

Academic Editor Zhijie Zhou

Copyright copy 2018 Zihan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Principal component analysis (PCA) is widely used in fault diagnosis Because the traditional data preprocessing method ignoresthe correlation between different variables in the system the feature extraction is not accurate In order to solve it this paperproposes a kind of data preprocessing method based on the Gap metric to improve the performance of PCA in fault diagnosis Fordifferent types of faults the original dataset transformation through Gap metric can reflect the correlation of different variablesof the system in high-dimensional space so as to model more accurately Finally the feasibility and effectiveness of the proposedmethod are verified through simulation

1 Introduction

As the complexity of industrial manufacturing systemsincreases the correlation between variables in the systembecomes more complex and these variables contain impor-tant information about the status of the system Therefore itis an important issue that fault detection and diagnosis of thesystem are through the information of these variables

However in industrial manufacturing systems becauseof the different dimensions of system variables it is usuallynecessary to preprocess the data to standardize the dataIn traditional data preprocessing methods ignoring theinfluence of dimension on the correlation between systemvariables leads to the lack of correlation of system variablesafter data preprocessing which makes it difficult to extractthe representative principal components Therefore main-taining the correlation between system variables is the key todata preprocessing

In order to solve this problem many studies have beenmadeWen et al proposed a method called Relative PrincipleComponent Analysis (RPCA) [1] it introduces analyzing anddetermining the importance of each component to the priorinformation of the system giving the corresponding weightof each component of the system and establishing relative

principal component model Literature [2] proposed a faultdiagnosis method based on information incremental matrixBased on this Yuan et al proposed a relative transformationof information incremental matrix fault diagnosis method[3] which can effectively detect variables that play an impor-tant role in the system Because of their smaller absolutevalue and less absolute changes small changes of theseimportant variables usually play a very crucial role in thesystem Xu andWen proposed a fault diagnosismethod basedon information entropy and relative principal componentanalysis [4] in high-dimensional system the high correlationof system variables leads to the model not being able toselect the representative principal componentsThe approachgiven by them is to use information entropy to measure theuncertainty of variables and calculate the information gain ofvariables According to the different degrees of importance ofvariables relatively transform the data to get a more accuratedata model Jiao et al proposed a method for simulationmodel validation based on Theilrsquos inequality coefficient andprincipal component analysis [5] it is based on the TICmodel given a model of the differences in position and trendbetween the simulated output and the reference output thereis a correlation between the two differences using PCA toobtain the verification results Kangling et al proposed a

HindawiJournal of Control Science and EngineeringVolume 2018 Article ID 1025353 9 pageshttpsdoiorg10115520181025353

2 Journal of Control Science and Engineering

fault diagnosis study based on adaptive partition PCA [6] Inorder to solve the inaccuratemodeling problem the diagnosismodel can be automatically updated and adjusted so as toimprove the model matching and the accuracy of diagnosisresults

The Gap metric is proved to be more suitable for mea-suring the distance between two linear systems than thenorm-based ones [7 8] and the effect of dimension on eachvariable can be reflected in the Riemannian space when datapreprocessing is performed The gap metric is widely used inthe study of the uncertainty and robustness of the feedbacksystem Tryphon proposed a method that can easily calculatethe gap metric [9] which improves the practicality of the gapmetric Literature [10] proposed the concept of ]-gap metricin the frequency domain and then extended it to nonlinearsystems [11] Ebadollahi and Saki proposed that in the mul-timodel predictive control in order to track the maximumpower point without losing its control performance the gapmetric method is used to divide the entire area of the partialload system into a corresponding linear model This methodensures the stability of the original closed-loop system [12]Konghuayrob and Kaitwanidvilai used V-gap to measurethe distance between two linear systems [13] They usedlow-order controllers with similar dynamic characteristicsto replace traditional high-level and complex controllersand both controllers have similar dynamic characteristicsand robustness For multilinear model control of nonlinearsystems Du proposed a weighting method based on 1120575Gapmetric [14]The Gapmetric method was used to calculate theweighting function of the local controller combination Thevalidity of the method was verified by the CSTR system

In the principal component analysis method based on thetraditional data preprocessing when removing the originaldata dimension all are based on the European metric datapreprocessing method and some important information ofthe data will be ignored and often these data are importantvariables that contain slowly changing fault information Inthe method proposed in this article Gap metric can projectdata onRiemann spheres in Riemann space and can highlightthe information easily ignored in the European space Afterthe Gap metric data preprocessing method is adopted theeigenvalue feature vector decomposition is performed onthe processed data matrix and the principal componentis constructed to construct the principal component spaceaccording to the cumulative percent variance criterion sinceGapmetric can highlight the variable datawith small absolutechange and relatively large variation in the system variablesso we can extract the main component and the main com-ponent vector when constructing the principal componentspace By calculating the 1198792 statistical limit and the 119878119875119864statistical limit of the normal system dataset we detectwhether the 1198792 statistics and 119878119875119864 statistics of the test datasetexceed the limit to judge whether the system is faulty andthen the fault variables are separated by the contribution ofsystem variables to fault samples

The rest of this article is organized as follows In thesecond section we briefly review the PCA approach In thethird section we propose a kind of improved PCA data

preprocessing method which is data preprocessing methodbased on gap metric In the fourth section we set up asystem model and test the feasibility and effectiveness of theproposed method by different types of faults In the fifthsection we give a summary and future research direction

2 PCA Based on TraditionalData Preprocessing

The basic idea of PCA is to decompose multivariable samplespace into lower dimensional principal component subspacescomposed of principal components variables and a residualsubspace according to the historical data of process variablesAnd the statistics which can reflect the change of space areconstructed in these two subspaces Then the sample vectorsare respectively projected into two subspaces and computethe distance from the sample to the subspace The processmonitoring and fault detection are performed by comparingthe distance with the corresponding statistics

First model by PCA to variable space Select a set ofvariables under normal conditions as the original data 119909 isin119877119898 is a test sample that contains 119898 variables and eachvariable has 119899 independent samples Construct the originalmeasurement data matrix

Xn = [119909 (1) 119909 (2) 119909 (119899)] 119883119899 isin 119877119898times119899 (1)

Here each column of Xn represents a variable and each rowrepresents a sample Because the dimensions of the measuredvariables are different each column of the data matrix isnormalized Assuming that the normalized measurementdata matrix is Xlowast

Xlowast = Xn minus 119890 sdot 119902diag (1205901 1205902 120590119899) (2)

Here 119890 = (1 1 1)119879 isin 119877119898times1 119902 = (1199021 1199022 119902119899) isin 1198771times119899 119902is the average of119883 columns 119894 = 1 119899 diag (1205901 1205902 120590119899)is the variance matrix of119883119899

The covariance matrix S of Xlowast is

S = 1119899 minus 1X

lowast119879Xlowast (3)

The processing of thematrix S is generally eigenvalue decom-position according to the size of the eigenvalues arranged indescending orderThePCAmodel decomposesXlowast as follows

Xlowast = Xlowast + E = TPT + ET = XlowastP

(4)

Journal of Control Science and Engineering 3

105 1

0500

0

05

1

15

2

minus1 minus1

minus05 minus05

(a) Riemann ball

1

2

(c1)

(c2)

(c1 c2) (c1 c2)

(b) The relationship between 120575 and 120579

Figure 1 The geometric meaning of the gap metric

where Xlowast is the projection in the principal component spaceE is the projection in the residual space and P isin R119898times119860 is theload matrix which consists of the first 119860 eigenvectors of ST isin R119899times119860 is the scoring matrix the elements of T are calledthe primary variables and 119860 is the number of the primaryThe principal component space is the part of the modelingand the residual space is not the part of the modeling whichrepresents the noise and fault information in the data

The selection of the principal component number 119860 isbased on the Cumulative Percent Variance (CPV)This crite-rion determines the number of principal elements based onthe cumulative sum of percentages of principal componentsThe CPV represents the ratio of the data changes explainedby the first 119860 principal component to the total data changesTherefore the cumulative contribution rate CPV of the first119860 principal can be expressed as

CPV = sum119860

119894=1120582119894

sum119898119894=1120582119894 (5)

where 120582119894 is the eigenvalue of the covariance matrix S Ingeneral when the cumulative contribution rate reaches 85or more it is considered that the number of elements 119860contains enough information of the original data

3 Improved Data Preprocessing Method

In this section we propose a kind of data preprocessingmethod based on Gap metric

31 Gap Metric In the Riemann space 120593(1198881) and 120593(1198882)are used to represent the spherical projection of complexnumbers 1198881 and 1198882 on a three-dimensional Riemann ball with

diameter 1 and the chord between 1198881 and 1198882 is denoted by120575(1198881 1198882) then 120575(1198881 1198882) is defined by

120575 (1198881 1198882) = 1003817100381710038171003817120593 (1198881) minus 120593 (1198882)1003817100381710038171003817 =10038161003816100381610038161198881 minus 11988821003816100381610038161003816

radic1 + 11988812radic1 + 11988822 (6)

120579(1198881 1198882) is used to express the spherical distance between 1198881and 1198882 that is the arc length connecting 120593(1198881) and 120593(1198882) onthe Riemann ball then

120579 (1198881 1198882) = arcsin 120575 (1198881 1198882) = arcsin10038161003816100381610038161198881 minus 11988821003816100381610038161003816

radic1 + 11988812radic1 + 11988822 (7)

As can be seen from Figure 1 the shortest arc length onthe circle is obtained from a plane-cut Riemann spheredetermined by 3 points of the center of the ball 120593(1198881) and120593(1198882)

The character of the Gap metric in the control system hassimilar properties in the data space The nature of the Gapmetric in the data space is as follows

(1) Gap metric can be regarded as the distance char-acterization of data in Riemann space which is anextension to the traditional method based on infinitenorm metrics

(2) The value of the Gap metric is in the range of 0 to 1The smaller the value the closer the characteristics ofthe two datasets The larger the value the greater thedifference in the characteristics of the two datasets Ifthe Gap metric of two datasets is 0 then they containexactly the same characteristics


32 Gap Metric Data Preprocessing and Fault Diagnosis Setthe data observation matrix 119883119899 isin 119877119898times119899 of the multivariablesystem as

Xn =[[[[[[[

1199091 (1) 1199091 (2) sdot sdot sdot 1199091 (119899)1199092 (1) 1199092 (2) sdot sdot sdot 1199092 (119899) d

119909119898 (1) 119909119898 (2) sdot sdot sdot 119909119898 (119899)

]]]]]]]

(8)

Here the column vector 119909119894(119895) = [1199091(119895) 1199092(119895) 119909119898(119895)]119879119895 = 1 2 119899 represents the system variable and the row

vector represents sampled data at a sampling instant Thenpreprocessing the data matrix the mean vector of Xn is

b119899 = 1119898 l119898Xn (9)

where l119898 = [1 1 1] isin 1198771times119898Step 1 Project the original data onto the Riemann sphere andcalculate the gap metric for each variable and the matrix Xlowastis calculated

Xlowast =[[[[[[[[[

120575 (1199091 (1) b119899 (1)) 120575 (1199091 (2) b119899 (2)) sdot sdot sdot 120575 (1199091 (119899) b119899 (119899))120575 (1199092 (1) b119899 (1)) 120575 (1199092 (2) b119899 (2)) sdot sdot sdot 120575 (1199092 (119899) b119899 (119899))

d

120575 (119909119898 (1) b119899 (1)) 120575 (119909119898 (2) b119899 (2)) sdot sdot sdot 120575 (119909119898 (119899) b119899 (119899))

]]]]]]]]]

(10)

Here

120575 (119909119894 (119896) b119899 (119896)) = 1003817100381710038171003817120593 (119909119894 (119896)) minus 120593 (119887119899 (119896))1003817100381710038171003817=

1003816100381610038161003816119909119894 (119896) minus 119887119899 (119896)1003816100381610038161003816radic1 + (119909119894 (119896))2 sdot radic1 + (119887119899 (119896))2

(11)

Step 2 Decompose eigenvalues and eigenvectors of Xlowastselect pivotal elements based onCumulative PercentVariance(CPV) and construct principal component space and resid-ual space

Step 3 Calculate the statistical limit of 1198792 based on theprincipal component space of normal data and calculate the119878119875119864 statistical limit through the residual space

Step 4 Set the test data matrix Yn isin 119877119898times119899 as

Yn =[[[[[[[[[


119910119898 (1) 119910119898 (2) sdot sdot sdot 119910119898 (119899)

]]]]]]]]]

(12)

Preprocess the test dataset Yn with Gap metric method to getYlowast

Ylowast =

[[[[[[[[[[[[


d


]]]]]]]]]]]]

(13)

Here b119899 is the mean vector of Xn

Step 5 Calculate the 1198792 statistic and 119878119875119864 statistic separatelyand detect whether the statistic exceeds the statistical limit ofnormal data If the statistic exceeds the statistical limit it willbe faulty Otherwise it will be normal

Step 6 Fault variables can be separated by the contributionof system variables to the fault in the residual space

The physical meaning of Gap metric is the chord pitchof the data projected on the Riemann sphere through aspherical surface which can highlight the impact of the


Table 1 Detection result

Constant deviation error PCA InEnPCA GAPPCA1198792 SPE 1198792 SPE 1198792 SPE

Misdiagnosis 8 7 7 8 16 8Rate of misdiagnosis 0010 0009 0009 0010 0020 0010Omissive judgement 54 0 66 0 8 0Rate of omissive judgment 0270 0 0330 0 0040 0

PCA statistics diagram


100 200 300 400 500 600 700 800 900 10000Time

0

50

100

SPE

stat

istic

0

20

40

60

T2

statis

tic

T2 statistic95 statistic limit

100 200 300 400 500 600 700 800 900 10000Time

SPE statistic95 statistic limit

(a) Fault detection results of PCA

1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te

Contribution rate(b) Contribution plot of PCA

Figure 2 PCA with constant deviation fault in 801sim1000 samples

relative changes of the data on its ownThe transformed datadoes not ignore variables that have small absolute changes butrelatively large transformations Frequently these variablesalso contain important information

4 Simulation

In order to verify the effectiveness of the proposed methodconsidering random variables and their linear combinations6 system variables are constructed as follows

x1 = 01 times randn (1 119899) x2 = 02 times randn (1 119899) x3 = 03 times randn (1 119899) x4 = minus131199091 + 021199092 + 081199093 + 01 times randn (1 119899) x5 = 1199092 minus 031199093 + 01 times randn (1 119899) x6 = 1199091 + 1199094 + 01 times randn (1 119899)

(14)

Among them randn(1 119899) is a random sequence of 1 row and119873 columns generated byMATLAB First choose 1000normalsamples to establish the PCAmodel then select 1000 samplesas the test data and introduce the constant deviation error ofmagnitude 3 to the last 200 samples of the variable x6 in thetest data and detect them respectively with the model 119878119875119864statistics and Hotellingrsquos 1198792 statistics are used as indicatorsto measure the number of misinformation and the missingnumber of PCA

As shown in Table 1 the average misinformation and thenumber of misstatements of the PCA InEnPCA and GAP-PCA methods are obtained through 10 simulation statistics

In the test 1sim800 times exceeds the control limit formisdiagnosis 801sim1000 times below the control limit foromissive judgement It can be seen from Figures 2ndash4 that themisdiagnosis rate of PCA InEnPCA and GAPPCA is nothigher than 2 However in the fault omission detectionT2 statistics of traditional PCA and InEnPCA show a largenumber of fault omissions The omissive judgement rate ofT2 in PCA and InEnPCA is 27 and 33 respectivelywhile that of GAPPCA is only 4 this shows that after the


InEnPCA statistics diagram


100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic

(a) Fault detection results of InEnPCA

001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005

(b) Contribution plot of InEnPCA

Figure 3 InEnPCA with constant deviation fault in 801sim1000 samples

GAPPCA statistics diagram


100 200 300 400 500 600 700 800 900 10000Time

0

20

40

80

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic

(a) Fault detection results of GAPPCA

1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002

(b) Contribution plot of GAPPCA

Figure 4 GAPPCA with constant deviation fault in 801sim1000 samples

data is preprocessed by gap metric the principal componentmodel established containsmost of the key information of thesystem thus improving the accuracy of fault detection In thefault diagnosis it can be seen from the contribution graph of

the fault that the contribution rate of the fault variable 119909 inGAPPCA is more easily distinguished from the contributionof other variables than PCA and InEnPCA because systemvariables are preprocessed for gap metric projection on a







100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te


Figure 5 PCA with microfaults detection in 801sim1000 samples

Riemann sphere the correlation between system variablescan be better reflected

The application of traditional PCA method in faultdiagnosis is based on the absolute distance between samplesas the criterion for fault detection and fault diagnosisHowever in real systems the occurrence of microfaults oftenchanges very little Therefore the detection of minor faultsis extremely necessary Literature [15] combines the PCAtechnique with the univariate exponential weighted-slidingaveraging for the correlation of variables in the chemicalprocess In view of the small deviation between normal andmicrofaults [16] combines probability distribution metricand Kullback-Leibler measure to quantify residuals betweenpotential scores and reference scores and proposed PCAalgorithm control limits for small faults Literature [17] is ananalysis model based on literature [16] The authors [17] givean approach where Kullback-Leibler divergence (KLD) wasapplied to the principal component variables obtained afterprocess dimensionality reduction using PCA This methodwas used to diagnose microfaults

In order to verify the detection accuracy in the case of amicrofault a slight fault with a slowly increasing rate of 01is introduced into the last 200 sample points of the variablex6 in the test data Use this model to test the fault diagnosisperformance of the three methods


It can be seen from Figures 5ndash7 that in the detection ofmicrofaults due to the small changes in system variablesthe traditional PCAmethod cannot extract the representativeof the main element as can be seen from Figure 6 PCAcannot reflect the system fault status during fault detectionand diagnosis Because the relative change of microfaults isbigger than absolute change GAPPCA can extract represen-tative principal component variables for small changes ofsystem variables and the detection results are better thanthe traditional PCA In fault diagnosis InEnPCA reflectsthe information gain of system variables and thus has goodperformance in fault diagnosis Besides since the gap metric




100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005


Figure 6 InEnPCA with microfaults detection in 801sim1000 samples



100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002


Figure 7 GAPPCA with microfaults detection in 801sim1000 samples


reflects the correlation of system variables in the Riemannianspace better than those in the European space when usingthe contribution graph to separate the fault variables wecan see that the contribution rate of fault variable x6 in thecontribution graph of GAPPCA is remarkable

To sum up compared with the traditional PCA pre-processing method in European space the preprocessingmethod based on the gapmetric can better reflect the relevantinformation between the variables and the faults that occurto some small but important variables can be diagnosedaccurately

5 Conclusions

In this paper we propose a fault diagnosis method basedon Gap metric data preprocessing and PCA When somevariables in the system play an important role and theabsolute value of the variables themselves is small theycannot detect the smaller faults The proposed method candetect the source of the fault more accurately and reduce therate of misdiagnosis and rate of omissive judgement

However there are still some problems to be studiedin this paper In the PCA fault diagnosis method based onthe Gap metric data preprocessing there will be a problemthat the projection overlaps The fault information reflectedby the fault samples may be projected after the Gap metricprocessing becomes normal Within the region how toseparate fault information and normal information in high-dimensional space is the focus of further research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation (NNSF) of China under Grants nos U150920361333005 61490701 61573137 and 61673160

References

[1] C L Wen J Hu T Z Wang and Z Chen ldquoRelative PCAwith applications of data compression and fault diagnosisrdquoActaAutomatica Sinica vol 34 no 9 pp 1128ndash1139 2008

[2] C L Wen and Y C Hu ldquoFault diagnosis based on theinformation incremental matrixrdquo Training of research workersin the medical sciences World Health Organization vol 38 no5 pp 832ndash840 2012

[3] T Yuan J Hu and C Wen ldquoFault diagnosis based on relativetransformation of information increment matrixrdquo Journal ofCentral South University vol 44 no 1 pp 238ndash242 2013

[4] X Xu and C Wen ldquoFault Diagnosis Method Based on Infor-mation Entropy and Relative Principal Component AnalysisrdquoJournal of Control Science and Engineering pp 1ndash8 2017

[5] S Jiao W Li and M Yang ldquoA Method for Simulation ModelValidation Based onTheils Inequality Coefficient and PrincipalComponent Analysisrdquo Communications in Computer Informa-tion Science pp 402-126 2013

[6] Kangling Zhengshun and J Liang ldquoAdaptive partitioning PCAmodel for improving fault detection and isolationrdquo ChineseJournal of Chemical Engineering vol 23 no 6 article no 235pp 981ndash991 2015

[7] A K El-Sakkary ldquoThe gap metric robustness of stabilizationof feedback systemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 30 no 3 pp240ndash247 1985

[8] T T Georgiou and M C Smith ldquoOptimal robustness in thegap metricrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 35 no 6 pp 673ndash6861990

[9] T T Georgiou ldquoOn the computation of the gap metricrdquo inProceedings of the 27th IEEEConference onDecision andControlpp 1360-1361 December 1988

[10] G Vinnicombe ldquoFrequency Domain Uncertainty and theGraph Topologyrdquo IEEE Transactions on Automatic Control vol38 no 9 pp 1371ndash1383 1993

[11] G Vinnicombe ldquov-gap distance for uncertain and nonlinearsystemsrdquo in Proceedings of the The 38th IEEE Conference onDecision and Control (CDC) pp 2557ndash2562 December 1999

[12] S Ebadollahi and S Saki ldquoWind Turbine Torque OscillationReduction Using Soft Switching Multiple Model PredictiveControl Based on the GapMetric and Kalman Filter EstimatorrdquoIEEE Transactions on Industrial Electronics vol 99 pp 1-1 2017

[13] P Konghuayrob and S Kaitwanidvilai ldquoLow order robust ]-gapmetric Hinfin loop shaping controller synthesis based on particleswarm optimizationrdquo International Journal of Innovative Com-puting Information and Control vol 13 no 6 pp 1777ndash17892017

[14] J Du ldquoComparison of two gap-metric-based weighting meth-ods in multilinear model controlrdquo in Proceedings of the 36thChinese Control Conference CCC 2017 pp 834ndash837 China July2017

[15] E G Zhi-Qiang J C Yang and Z H Song ldquoesearch and Appli-cation of Small Shifts Detection Method Based on MEWMA-PCArdquo Information amp Control vol 35 no 9 pp 650ndash656 2007

[16] J Harmouche C Delpha and D Diallo ldquoIncipient faultdetection and diagnosis based on Kullback-Leibler divergenceusing Principal ComponentAnalysismdashpart Irdquo Signal Processingvol 94 no 1 pp 278ndash287 2014

[17] J Harmouche C Delpha and D Diallo ldquoIncipient faultdetection and diagnosis based on Kullback-Leibler divergenceusing principal component analysis Part IIrdquo Signal Processingvol 109 pp 334ndash344 2015

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fault diagnosis study based on adaptive partition PCA [6] Inorder to solve the inaccuratemodeling problem the diagnosismodel can be automatically updated and adjusted so as toimprove the model matching and the accuracy of diagnosisresults

The Gap metric is proved to be more suitable for mea-suring the distance between two linear systems than thenorm-based ones [7 8] and the effect of dimension on eachvariable can be reflected in the Riemannian space when datapreprocessing is performed The gap metric is widely used inthe study of the uncertainty and robustness of the feedbacksystem Tryphon proposed a method that can easily calculatethe gap metric [9] which improves the practicality of the gapmetric Literature [10] proposed the concept of ]-gap metricin the frequency domain and then extended it to nonlinearsystems [11] Ebadollahi and Saki proposed that in the mul-timodel predictive control in order to track the maximumpower point without losing its control performance the gapmetric method is used to divide the entire area of the partialload system into a corresponding linear model This methodensures the stability of the original closed-loop system [12]Konghuayrob and Kaitwanidvilai used V-gap to measurethe distance between two linear systems [13] They usedlow-order controllers with similar dynamic characteristicsto replace traditional high-level and complex controllersand both controllers have similar dynamic characteristicsand robustness For multilinear model control of nonlinearsystems Du proposed a weighting method based on 1120575Gapmetric [14]The Gapmetric method was used to calculate theweighting function of the local controller combination Thevalidity of the method was verified by the CSTR system

In the principal component analysis method based on thetraditional data preprocessing when removing the originaldata dimension all are based on the European metric datapreprocessing method and some important information ofthe data will be ignored and often these data are importantvariables that contain slowly changing fault information Inthe method proposed in this article Gap metric can projectdata onRiemann spheres in Riemann space and can highlightthe information easily ignored in the European space Afterthe Gap metric data preprocessing method is adopted theeigenvalue feature vector decomposition is performed onthe processed data matrix and the principal componentis constructed to construct the principal component spaceaccording to the cumulative percent variance criterion sinceGapmetric can highlight the variable datawith small absolutechange and relatively large variation in the system variablesso we can extract the main component and the main com-ponent vector when constructing the principal componentspace By calculating the 1198792 statistical limit and the 119878119875119864statistical limit of the normal system dataset we detectwhether the 1198792 statistics and 119878119875119864 statistics of the test datasetexceed the limit to judge whether the system is faulty andthen the fault variables are separated by the contribution ofsystem variables to fault samples

The rest of this article is organized as follows In thesecond section we briefly review the PCA approach In thethird section we propose a kind of improved PCA data

preprocessing method which is data preprocessing methodbased on gap metric In the fourth section we set up asystem model and test the feasibility and effectiveness of theproposed method by different types of faults In the fifthsection we give a summary and future research direction

2 PCA Based on TraditionalData Preprocessing

The basic idea of PCA is to decompose multivariable samplespace into lower dimensional principal component subspacescomposed of principal components variables and a residualsubspace according to the historical data of process variablesAnd the statistics which can reflect the change of space areconstructed in these two subspaces Then the sample vectorsare respectively projected into two subspaces and computethe distance from the sample to the subspace The processmonitoring and fault detection are performed by comparingthe distance with the corresponding statistics

First model by PCA to variable space Select a set ofvariables under normal conditions as the original data 119909 isin119877119898 is a test sample that contains 119898 variables and eachvariable has 119899 independent samples Construct the originalmeasurement data matrix

Xn = [119909 (1) 119909 (2) 119909 (119899)] 119883119899 isin 119877119898times119899 (1)

Here each column of Xn represents a variable and each rowrepresents a sample Because the dimensions of the measuredvariables are different each column of the data matrix isnormalized Assuming that the normalized measurementdata matrix is Xlowast

Xlowast = Xn minus 119890 sdot 119902diag (1205901 1205902 120590119899) (2)

Here 119890 = (1 1 1)119879 isin 119877119898times1 119902 = (1199021 1199022 119902119899) isin 1198771times119899 119902is the average of119883 columns 119894 = 1 119899 diag (1205901 1205902 120590119899)is the variance matrix of119883119899

The covariance matrix S of Xlowast is

S = 1119899 minus 1X

lowast119879Xlowast (3)

The processing of thematrix S is generally eigenvalue decom-position according to the size of the eigenvalues arranged indescending orderThePCAmodel decomposesXlowast as follows

Xlowast = Xlowast + E = TPT + ET = XlowastP

(4)


105 1

0500

0

05

1

15

2

minus1 minus1

minus05 minus05

(a) Riemann ball

1

2

(c1)

(c2)

(c1 c2) (c1 c2)





CPV = sum119860

119894=1120582119894

sum119898119894=1120582119894 (5)






120575 (1198881 1198882) = 1003817100381710038171003817120593 (1198881) minus 120593 (1198882)1003817100381710038171003817 =10038161003816100381610038161198881 minus 11988821003816100381610038161003816

radic1 + 11988812radic1 + 11988822 (6)



radic1 + 11988812radic1 + 11988822 (7)







Xn =[[[[[[[


119909119898 (1) 119909119898 (2) sdot sdot sdot 119909119898 (119899)

]]]]]]]

(8)



b119899 = 1119898 l119898Xn (9)


Xlowast =[[[[[[[[[


d


]]]]]]]]]

(10)

Here

120575 (119909119894 (119896) b119899 (119896)) = 1003817100381710038171003817120593 (119909119894 (119896)) minus 120593 (119887119899 (119896))1003817100381710038171003817=


(11)




Yn =[[[[[[[[[


119910119898 (1) 119910119898 (2) sdot sdot sdot 119910119898 (119899)

]]]]]]]]]

(12)


Ylowast =

[[[[[[[[[[[[


d


]]]]]]]]]]]]

(13)











100 200 300 400 500 600 700 800 900 10000Time

0

50

100

SPE

stat

istic

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te




4 Simulation



(14)







100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

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SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

80

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002











100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te











100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



105 1

0500

0

05

1

15

2

minus1 minus1

minus05 minus05

(a) Riemann ball

1

2

(c1)

(c2)

(c1 c2) (c1 c2)





CPV = sum119860

119894=1120582119894

sum119898119894=1120582119894 (5)






120575 (1198881 1198882) = 1003817100381710038171003817120593 (1198881) minus 120593 (1198882)1003817100381710038171003817 =10038161003816100381610038161198881 minus 11988821003816100381610038161003816

radic1 + 11988812radic1 + 11988822 (6)



radic1 + 11988812radic1 + 11988822 (7)







Xn =[[[[[[[


119909119898 (1) 119909119898 (2) sdot sdot sdot 119909119898 (119899)

]]]]]]]

(8)



b119899 = 1119898 l119898Xn (9)


Xlowast =[[[[[[[[[


d


]]]]]]]]]

(10)

Here

120575 (119909119894 (119896) b119899 (119896)) = 1003817100381710038171003817120593 (119909119894 (119896)) minus 120593 (119887119899 (119896))1003817100381710038171003817=


(11)




Yn =[[[[[[[[[


119910119898 (1) 119910119898 (2) sdot sdot sdot 119910119898 (119899)

]]]]]]]]]

(12)


Ylowast =

[[[[[[[[[[[[


d


]]]]]]]]]]]]

(13)











100 200 300 400 500 600 700 800 900 10000Time

0

50

100

SPE

stat

istic

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te




4 Simulation



(14)







100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

80

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002











100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te











100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Xn =[[[[[[[


119909119898 (1) 119909119898 (2) sdot sdot sdot 119909119898 (119899)

]]]]]]]

(8)



b119899 = 1119898 l119898Xn (9)


Xlowast =[[[[[[[[[


d


]]]]]]]]]

(10)

Here

120575 (119909119894 (119896) b119899 (119896)) = 1003817100381710038171003817120593 (119909119894 (119896)) minus 120593 (119887119899 (119896))1003817100381710038171003817=


(11)




Yn =[[[[[[[[[


119910119898 (1) 119910119898 (2) sdot sdot sdot 119910119898 (119899)

]]]]]]]]]

(12)


Ylowast =

[[[[[[[[[[[[


d


]]]]]]]]]]]]

(13)











100 200 300 400 500 600 700 800 900 10000Time

0

50

100

SPE

stat

istic

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te




4 Simulation



(14)







100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

80

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002











100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te











100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








100 200 300 400 500 600 700 800 900 10000Time

0

50

100

SPE

stat

istic

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te




4 Simulation



(14)







100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

80

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002











100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te











100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

80

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002











100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te











100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








100 200 300 400 500 600 700 800 900 10000Time

SPE

stat

istic

0

10

20

30

0

10

20

30

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time



1 2 3 4 5 6variable

0

001

002

003

004

005

006

007

008

Con

trib

utio

n ra

te











100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





100 200 300 400 500 600 700 800 900 10000Time

0

20

40

60

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

2

4

6

8

SPE

stat

istic


001

002

1 2 3 4 5 6variable

0

0025

Con

trib

utio

n ra

te

Contribution rate

0015

0005





100 200 300 400 500 600 700 800 900 10000Time

0

50

100

T2

statis

tic


100 200 300 400 500 600 700 800 900 10000Time


0

10

20

30

SPE

stat

istic


1 2 3 4 5 6variable

Contribution rate

001

0

0012

Con

trib

utio

n ra

te

0006

0004

0008

0002






5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





5 Conclusions



Data Availability




Acknowledgments


References




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


fault diagnosis method based on gap metric data preprocessing...

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