Research ArticleProcess Monitoring and Fault Diagnosis forShell Rolling Production of Seamless Tube

Dong Xiao12 Xuyang Gao12 Jichun Wang3 and Yachun Mao4

1State Key Laboratory of Synthetical Automation for Process Industries Northeast University Shenyang 110004 China2Information Science and Engineering School Northeastern University Shenyang 110004 China3College of Science Liaoning Industry University Jinzhou 121000 China4College of Resources and Civil Engineering Northeastern University Shenyang 110004 China

Correspondence should be addressed to Dong Xiao xiaodongiseneueducn

Received 6 November 2014 Revised 9 January 2015 Accepted 14 January 2015

Academic Editor Sangmin Lee

Copyright copy 2015 Dong Xiao 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

Continuous rolling production process of seamless tube has many characteristics including multiperiod and strong nonlinearityand quickly changing dynamic characteristics It is difficult to build its mechanism model In this paper we divide productiondata into several subperiods by 119870-means clustering algorithm combined with production process then we establish a continuousrolling production monitoring and fault diagnosis model based on multistage MPCA method Simulation experiments show thatthe rolling production process monitoring and fault diagnosis model based on multistage MPCA method is effective and it has agood real-time performance high reliability and precision

1 Introduction

The deformation process of seamless tube production canbe summarized into three stages perforation extension andfinish rolling The main purpose of the perforation processis to perforate a solid round billet into a hollow shell Themain purpose of Elongator is to further reduce the crosssection and to make the shell improve on dimensionalaccuracy surface quality and organizational performanceAfter elongator rolling steel tube is called shell whichrequires further molding in the finishing mill in order toachieve the requirements of he finished tube [1] Continuousrolling mill is an elongator which has the highest productionefficiency and superior product quality So it has been widelyapplied to the big steel mills Online monitoring of oper-ating parameters can effectively avoid accidents eliminateequipment damage save a lot of maintenance costs increaserunning time improve set utilization and reduce spare partsinventory and time Process monitoring and fault diagnosishas a very important practical significance for safe productionand scientific maintenance [2]

Reference [3] analyzed the causes of roll sticking steel andgave several methods to avoid sticking steel but did not give asticking steel monitoring method Reference [4] introducedthe rolling steel tube transverse wall and longitudinal wallthickness error monitoring method but it needs to introduceexpensive measuring instrument Reference [5] introducedthe fault diagnosis methods for DC motor of rolling rollReference [6] introduced a continuous rolling mill onlinemonitoring system based on virtual instrument technologybut it monitored the variables separately and did not considercorrelation between the variables which made some deep-seated faults difficult to bemonitored In the area of industrialprocess significant researches have been done for onlineprocess monitoring and fault diagnosis [7ndash12] Multivariatestatistical analysis techniques such as principal componentanalysis (PCA) [13ndash15] and partial least squares (PLS) [16ndash18]have long been used for detection and diagnosis of abnormaloperating situations in many industrial processes

Continuous rolling process is a batch process which hastypical multi-period and dynamic multivariate characteris-tics According to multi-period characteristics of continuous

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 219710 12 pageshttpdxdoiorg1011552015219710

2 Mathematical Problems in Engineering

Displacement

21 7 8

Tim

e

Bite

stag

eSt

able

rolli

ng

stage

Stee

l le

avin

g sta

ge

Perio

d

0A

C

Dd

c

B

c998400

d998400

D998400

t1

t2

t3

A998400 aa998400

B998400

C998400

b998400 b

Figure 1 Time and displacement of rolling tube process

rolling process it can be divided into bite stage stable rollingstage and steel leaving stage The production data has thefollowing characteristics

(1) Large Amount of Data High Dimension and Strong Cou-pling Seamless tube rolling process is a computer supervisoryand computer control industrial process It need regularlyacquire system state variables and equipment status to be usedfor display and control Each cycle of seamless tube rollingproduction process will produce tens of thousands of processdata After accumulating the amount of data is enormousAt the same time the behavior of the rolling process is theoutcome of combining action of many variable factors whichhave a strong coupling relationship

(2) Industrial Noise andUncertainty Because systemworks incomplex environments the output signal of electronic sens-ing devices is susceptible to noise And it is also vulnerable tothe uncertainties

(3) Multimode Data is a reflection of the system statechanging The data are not only in normal working statebut also in various kinds of abnormal state and fault stateThe former is the main part The latter has a relatively smallamount of data but it is indispensable in the knowledgediscovery and data mining

(4) Staircase Distribution In the biting stage the tube entersrolls from the first to the eighth and in the steel leaving stagethe situation is in turn So the data is staircase distribution

In this paper by using 119870-means clustering algorithmcombinedwith production process we establish a continuousrolling production process monitoring and fault diagnosismodel based onmultistageMPCAmethod Firstly accordingto the production process we divide production data intothree subperiods including bite stage stable rolling stage andsteel leaving stage Secondly we use clustering algorithm tofurther divide those larger changing stages A satisfactory

effect is difficult to be obtained if using clustering algorithmto classify the production data alone We classify it combinedwith production process Finally we get a satisfactory moni-toring result Online monitoring and fault diagnosis systemnot only can online monitor equipment but also can carryout fault alarm and diagnosis timely During the monitoringprocess it does not require the dedicated testers and doesnot need professional and technical personnel to make ananalysis and judgment

2 Analysis of Factors Affecting the ContinuousRolling Production

In order to build the monitoring and fault diagnosis modelwe analyze the influencing factors for continuous rollingproduction firstly

As shown in Figure 1 the continuous rolling process canbe divided into three stages

(1) Bite StageAs shown in Figure 1 the head of steel tubemoves from point a1015840 to point b1015840 and the tail movesfrom point A1015840 to point B1015840

(2) Stable Rolling Stage As shown in Figure 1 the head ofsteel tube moves from point b1015840 to point c1015840 and the tailmoves from point B1015840 to point C1015840

(3) Steel Leaving Stage As shown in Figure 1 the head ofsteel tube moves from point c1015840 to point d1015840 and the tailmoves from point C1015840 to point D1015840

Factors affecting seamless steel rolling production processare mainly the following roller rotational speed roller inputcurrent and roller torque

(1) Roller Rotational Speed In continuous rolling process theroller rotational speed is very important If the speed is toofast the surface of shell cannot be completely eliminated

Mathematical Problems in Engineering 3

Batches

Time

Variable1

1

X

K

J

J

2J KJ

I

I

X

Figure 2 Arrangement and decomposition of a three-way array byMPCA

In addition if the speed is too slow the consumption ofelectricity and other energy will increase

(2) Roller Torque The roller torque is one of the controlledvariables It is controlled by the input current The torquedirectly affects the quality of steel tube If torque is too small itcannot completely remove the blank of tube surface If torqueis too large shell will deform under the pressure Reasonablycontrolled torque of the rolling process is an important factorwhich cannot be ignored In order to save energy resourcesduring the wait state the torque will be reduced When thetube is rolled the torque increases When the tube leaves theroll the torque decreases

(3) Roller Input Current Roller current is one of the mostimportant control variables in the process of rolling It isan important factor to control torque and rotational speedBecause the sensitivity of the current is higher when thepierced tube enters the stand mill current increases quicklyThen the roller rotational speed and roller torque increase inorder to roll smoothly Current and torque are proportionalto the relationship The current increases when the steel tubegets into the rollThen it tends to a stable stateWhen the tubeleaves the roll current drops rapidly So correctly controllingthe size and direction of current is an important part of theprocess of rolling and the current is themain control variablein the rolling process

3 Multiway Principal Component Analysis

Batch processes are repetitive production process Theirdata sets have one more dimension than the continuousproduction process data set We can use three-dimensionaldata matrix 119883 = (119868 times 119869 times 119870) instead of the batch processdata collection where the three dimensions 119868 119869 and 119870respectively represent the batch number of samples numberof process variables and the number of measuring points ineach operation [19 20] MPCA will unfold119883 = (119868 times 119869 times119870) in

such a way as to put each of its vertical slices (119868 times 119869) side byside to the right which start with the one corresponding tothe first time interval The resulting two-dimensional matrixhas dimensions119883(119868 times 119869119870) [21 22] (Figure 2)

After three-dimensional data matrix 119883 is expanded intotwo-dimensional data it will be decomposed into a seriesof principal components consisting of score vectors 119905

119895and

loading matrices 119875119895 together with a residual matrix 119864 by the

principles of PCA The MPCA model can be written as

119883 =

119896

sum

119895=1

119905119895otimes 119901119895+ 119864 (1)

where the score vectors 119905119895is related only to batches and the

loading matrices 119875119895is related to variables and their time

variation The noise or residual part 119864 is as small as possiblein a least squares sense

4 Establish a Multiperiod ContinuousRolling Production Process MonitoringModel Based on 119870-Means and MPCA

41 Collecting the Production Data Based on actual produc-tion data characteristics of seamless steel rolling process inthis paper we use ibaAnalyzer software to collect 20 dataunder normal production condition As shown in Table 1 weselect 24 production process indicator variables to establish amonitoring model In this paper the three-dimensional datamatrix is 119883(119868 times 119869 times 119870) The three dimensions respectivelyrepresent the batch number of samples number of processvariables and the number of measuring points in eachoperation (119868 = 20 119869 = 24 119870 = 400) In order to obtainvertical data slice 119883

119896(119868 times 119869) we cut the three-dimensional

matrix along the direction of the third dimension Thusduring a period we can get 400 time slice matrixes By usingPCA for the 400 two-dimensional time slice matrix we got400 load matrixes and got the whole model load matrix bytaking an average

42 Online Monitoring Based on PCA Method The wholePCA model can be defined as follows

119875lowast=1

400

400

sum

119896=1

119875119896 (2)

where 119896 is the number of load matrixThe number of principal components 119860lowast can be calcu-

lated by the cumulative contribution rate In this paper weset 119860lowast = 6

The cumulative contribution rate =sum119860lowast

119895=1120582lowast

119895

trace (119878lowast)ge 90 (3)

where 119878lowast = diag(1205821 1205822 120582

119869) is an eigenvalues diagonal

matrix of the matrix119883The whole PCA load matrix 119875lowast is divided into two parts

the main component space 119875lowast(24 times 6) and the residual

4 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40014

145

15

155

16

165

Time

T2

(a) 1198792 control limits

0 50 100 150 200 250 300 350 40020

40

60

80

100

120

140

Time

SPE

(b) SPE control limits

Figure 3 Seamless tube rolling process 1198792 control limits and SPE control limits

Table 1 Measured variables for seamless steel rolling productionprocess

Symbol Variables Unit1 1st roller speed rs2 1st roller current A3 1st roller torque Nlowastm4 2nd roller speed rs5 2nd roller current A6 2nd roller torque Nlowastm7 3rd roller speed rs8 3rd roller current A9 3rd roller torque Nlowastm10 4th roller speed rs11 4th roller current A12 4th roller torque Nlowastm13 5th roller speed rs14 5th roller current A15 5th roller torque Nlowastm16 6th roller speed rs17 6th roller current A18 6th roller torque Nlowastm19 7th roller speed rs20 7th roller current A21 7th roller torque Nlowastm22 8th roller speed rs23 8th roller current A24 8th roller torque Nlowastm

space 119875lowast

(24 times 18) Similarly eigenvalue diagonal matrix 119878lowast

is correspondingly divided into two parts 119878lowast and 119878lowast

= 119883 (119875lowast

)119879

119883 = (119875

lowast

)119879

119864 = 119883 minus119883

(4)

In current online monitoring and fault diagnosis judgingwhether statistics1198792 and SPE are over the limit is usually usedto determine whether faults happen The control limits of 1198792approximately obey 119865 distribution

119863 sim

119860lowast(1198992minus 1)

119899 (119899 minus 119860lowast)119865119860lowast119899minus119860lowast120572 (5)

where 119870 = 400 and 119899 is the number of samples data formodeling The control limits of 1198792 is shown in Figure 3(a)For residual subspace SPE

119896of the PCAmodel approximately

obey 1205942 distribution [23 24] at time 119896

SPE119896120572= 1198921198961205942

ℎ119896120572

119892119896=

V119896

2119898119896

ℎ119896=2 (119898119896)2

V119896

(6)

where 119892 is a constant ℎ is the freedom degree of the 1205942distribution and V

119896and 119898

119896are respectively the mean and

variance of the square prediction error at time 119896 SPE controllimits are shown in Figure 3(b)

In order to monitor the rolling process first we needto obtain the measured data of new production periodstandardize the new data calculate the main component andthe prediction error of the data by formula (7) and checkwhether the 1198792 and SPE are beyond their own control limitIf the statistic exceeds the control limit it indicates that afault may occur at the time We now should analyze possiblecauses of the failure by variable 1198792 and SPE time-varyingcontribution plots and exclude or isolate the faults Consider

119905 = 119909119875lowast

119890 = 119909 (119868 minus 119875lowast

(119875lowast

)119879

)

1198792= 119905119879(119878lowast

)minus1

119905

SPE = 119890119890119879

(7)

Mathematical Problems in Engineering 5

0 50 100 150 200 250 300 350 40002468

1012141618

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 4 1198792 and SPE plot for MPCA monitoring results of the normal rolling process

As shown in Figure 4 under normal conditions moni-toring plot of SPE has an obvious alarm phenomenon duringthe first 50 sampling times and the last 50 sampling timesWecan explain the reason for alarm in the process of monitoringcombining with the rolling process The first 50 samples arein the bite stage where current speed and torque suddenlychange due to steel tube enter The last 50 samples are inthe steel leaving stage where speed current and torquedecrease sharply because of tube leaving Because of suddenchanges of variables standardized data still remain largedeviations Therefore PCA monitoring model appears alarmphenomenon So it is necessary to establish a multistageMPCA monitoring model

43 Build a Multistage MPCA Monitoring Model accordingto Production Process According to the process rollingproduction process can be divided into three stages bitestage stable rolling stage and steel leaving stage Then thispaper established a multistage MPCA monitoring modelaccording to the three stages

As shown in Figure 5 the situation of the steel leavingstage has improved But alarm phenomenon still exists andthe monitoring effect still needs to be improved So thebite stage and steel leaving stage should be further divided119870-means and Fuzzy 119862-means (FCM) clustering algorithmare commonly used FCM algorithm does not consider anyinformation related to the image space continuity so it ishighly sensitive to noise However 119870-means algorithm issimple and fast Particularly when dealing with the large datasets it has a very high efficiency So this paper chooses 119870-means clustering algorithm as a segmentation method

44 119870-Means and Multistage MPCA Combined to Build aMonitoring Model 119870-means algorithm is to cluster 119899 objectsbased on attributes into 119870(119870 lt 119899) partitions It assigns eachobject to the cluster which has the nearest center The centeris defined as the average of all the objects in the cluster whichstarts from a set of random initial centers The main steps of119870-means clustering algorithm is as follows

(1) Set up the cluster number 119870(2) Directly generate119870 random points as cluster centers(3) Assign each other points to the nearest cluster center(4) Recalculate the new cluster centers after new points

are clustered into the clusters(5) Repeat 3 and 4 until cluster centers do not change

According to process and the 119870-means algorithm forsegmentation number of principal components retained ineach substage PCA model is 119860lowast

119888 which can be obtained by

formula (3) The whole PCA load matrix 119875lowast119888is divided into

two parts the main component space 119875lowast119888and the residual

space 119875lowast

119888 which can be obtained by formula (4) Similarly

eigenvalue diagonalmatrix 119878lowast119888is correspondingly divided into

two parts 119878lowast119888and 119878

lowast

119888 SPE control limits can be obtained by

formula (4) 1198792 control limits is defined as follows

119863119896sim

119860lowast

119888(1198992

stage 119888 minus 1)

119899stage 119888 (119899stage 119888 minus 119860lowast

119888)

119865119860lowast

119888119899stage 119888minus119860

lowast

119888120572 (8)

As shown in Figure 6 when the number of stages is 7(bite stage is divided into three stages no segmentation stablerolling stage and steel leaving stage is divided into threestages) 1198792 and SPE are beyond their own control limits Themodel has a good performance in monitoring

According to the contrast of the foregoing analysis wecan easily conclude that seamless tube continuous rollingproduction process has too many characteristics includ-ing multiperiod strong nonlinearity and quickly changingdynamic characteristics It is hard tomonitor production pro-cess by the traditional MPCA method In this paper we usemultistage MPCA method according to production processand clustering algorithm This method can solve nonlinearproblem of seamless tube rolling production process andimprove the accuracy of online monitoring At the sametime the method has a strong guiding significance for theproduction

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40002468

101214161820

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 5 1198792 and SPE plot for multistage MPCA monitoring results of the normal rolling process

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 6 1198792 and SPE plot for 7 stages monitoring results of the normal rolling process

Through the experiment we find that the number ofsegments is larger and the precision is higher But at thesame time it easily causes misjudgment In this paper afterseveral experiments we finally decided to select 119870 = 7which both can be very good for rolling production processmonitoring and avoid misjudgment Compared with thesituation without segmentation segmentation model canmore accurately judge running state of rolling process whichhas a positive meaning for actual production

5 Fault Diagnosis

By monitoring the test method based on statistic can onlymonitorwhether faults occur and the approximate time of theoccurrence but it could not determine the source of the faultThe method of contribution plot provides possibilities ofdetermining the fault sources It can reflect the contributionto the statistics from variables at each moment

For the main component and residual subspace there aretwo contribution plots that can be used for fault diagnosismdash1198792 contribution plot and SPE contribution plot

The contribution to the 119886th principal component 119905119886from

119895th process variable 119909119895can be defined as follows

119862119905119886119909119895

=

119909119895119901119895119886

119905119886

(119886 = 1 119860 119895 = 1 119898) (9)

The contribution to the statistic SPE from the 119895th processvariables is

119862SPE119909119895

=

(119909119895minus 119909119895)2

SPE

(10)

In order to verify the performance of the multistagemonitoring model this paper introduces two typical faultdata for monitoring and fault diagnosis

Fault 1 1st roller speed fault from75th to 125th sampling timethe roller speed is 0

Fault 2 1st roller current fault from 70th to 130th samplingtime the roller current is 0

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Displacement

21 7 8

Tim

e

Bite

stag

eSt

able

rolli

ng

stage

Stee

l le

avin

g sta

ge

Perio

d

0A

C

Dd

c

B

c998400

d998400

D998400

t1

t2

t3

A998400 aa998400

B998400

C998400

b998400 b

Figure 1 Time and displacement of rolling tube process

rolling process it can be divided into bite stage stable rollingstage and steel leaving stage The production data has thefollowing characteristics

(1) Large Amount of Data High Dimension and Strong Cou-pling Seamless tube rolling process is a computer supervisoryand computer control industrial process It need regularlyacquire system state variables and equipment status to be usedfor display and control Each cycle of seamless tube rollingproduction process will produce tens of thousands of processdata After accumulating the amount of data is enormousAt the same time the behavior of the rolling process is theoutcome of combining action of many variable factors whichhave a strong coupling relationship

(2) Industrial Noise andUncertainty Because systemworks incomplex environments the output signal of electronic sens-ing devices is susceptible to noise And it is also vulnerable tothe uncertainties

(3) Multimode Data is a reflection of the system statechanging The data are not only in normal working statebut also in various kinds of abnormal state and fault stateThe former is the main part The latter has a relatively smallamount of data but it is indispensable in the knowledgediscovery and data mining

(4) Staircase Distribution In the biting stage the tube entersrolls from the first to the eighth and in the steel leaving stagethe situation is in turn So the data is staircase distribution

In this paper by using 119870-means clustering algorithmcombinedwith production process we establish a continuousrolling production process monitoring and fault diagnosismodel based onmultistageMPCAmethod Firstly accordingto the production process we divide production data intothree subperiods including bite stage stable rolling stage andsteel leaving stage Secondly we use clustering algorithm tofurther divide those larger changing stages A satisfactory

effect is difficult to be obtained if using clustering algorithmto classify the production data alone We classify it combinedwith production process Finally we get a satisfactory moni-toring result Online monitoring and fault diagnosis systemnot only can online monitor equipment but also can carryout fault alarm and diagnosis timely During the monitoringprocess it does not require the dedicated testers and doesnot need professional and technical personnel to make ananalysis and judgment

2 Analysis of Factors Affecting the ContinuousRolling Production

In order to build the monitoring and fault diagnosis modelwe analyze the influencing factors for continuous rollingproduction firstly

As shown in Figure 1 the continuous rolling process canbe divided into three stages

(1) Bite StageAs shown in Figure 1 the head of steel tubemoves from point a1015840 to point b1015840 and the tail movesfrom point A1015840 to point B1015840

(2) Stable Rolling Stage As shown in Figure 1 the head ofsteel tube moves from point b1015840 to point c1015840 and the tailmoves from point B1015840 to point C1015840

(3) Steel Leaving Stage As shown in Figure 1 the head ofsteel tube moves from point c1015840 to point d1015840 and the tailmoves from point C1015840 to point D1015840

Factors affecting seamless steel rolling production processare mainly the following roller rotational speed roller inputcurrent and roller torque

(1) Roller Rotational Speed In continuous rolling process theroller rotational speed is very important If the speed is toofast the surface of shell cannot be completely eliminated

Mathematical Problems in Engineering 3

Batches

Time

Variable1

1

X

K

J

J

2J KJ

I

I

X

Figure 2 Arrangement and decomposition of a three-way array byMPCA

In addition if the speed is too slow the consumption ofelectricity and other energy will increase

(2) Roller Torque The roller torque is one of the controlledvariables It is controlled by the input current The torquedirectly affects the quality of steel tube If torque is too small itcannot completely remove the blank of tube surface If torqueis too large shell will deform under the pressure Reasonablycontrolled torque of the rolling process is an important factorwhich cannot be ignored In order to save energy resourcesduring the wait state the torque will be reduced When thetube is rolled the torque increases When the tube leaves theroll the torque decreases

(3) Roller Input Current Roller current is one of the mostimportant control variables in the process of rolling It isan important factor to control torque and rotational speedBecause the sensitivity of the current is higher when thepierced tube enters the stand mill current increases quicklyThen the roller rotational speed and roller torque increase inorder to roll smoothly Current and torque are proportionalto the relationship The current increases when the steel tubegets into the rollThen it tends to a stable stateWhen the tubeleaves the roll current drops rapidly So correctly controllingthe size and direction of current is an important part of theprocess of rolling and the current is themain control variablein the rolling process

3 Multiway Principal Component Analysis

Batch processes are repetitive production process Theirdata sets have one more dimension than the continuousproduction process data set We can use three-dimensionaldata matrix 119883 = (119868 times 119869 times 119870) instead of the batch processdata collection where the three dimensions 119868 119869 and 119870respectively represent the batch number of samples numberof process variables and the number of measuring points ineach operation [19 20] MPCA will unfold119883 = (119868 times 119869 times119870) in

such a way as to put each of its vertical slices (119868 times 119869) side byside to the right which start with the one corresponding tothe first time interval The resulting two-dimensional matrixhas dimensions119883(119868 times 119869119870) [21 22] (Figure 2)

After three-dimensional data matrix 119883 is expanded intotwo-dimensional data it will be decomposed into a seriesof principal components consisting of score vectors 119905

119895and

loading matrices 119875119895 together with a residual matrix 119864 by the

principles of PCA The MPCA model can be written as

119883 =

119896

sum

119895=1

119905119895otimes 119901119895+ 119864 (1)

where the score vectors 119905119895is related only to batches and the

loading matrices 119875119895is related to variables and their time

variation The noise or residual part 119864 is as small as possiblein a least squares sense

4 Establish a Multiperiod ContinuousRolling Production Process MonitoringModel Based on 119870-Means and MPCA

41 Collecting the Production Data Based on actual produc-tion data characteristics of seamless steel rolling process inthis paper we use ibaAnalyzer software to collect 20 dataunder normal production condition As shown in Table 1 weselect 24 production process indicator variables to establish amonitoring model In this paper the three-dimensional datamatrix is 119883(119868 times 119869 times 119870) The three dimensions respectivelyrepresent the batch number of samples number of processvariables and the number of measuring points in eachoperation (119868 = 20 119869 = 24 119870 = 400) In order to obtainvertical data slice 119883

119896(119868 times 119869) we cut the three-dimensional

matrix along the direction of the third dimension Thusduring a period we can get 400 time slice matrixes By usingPCA for the 400 two-dimensional time slice matrix we got400 load matrixes and got the whole model load matrix bytaking an average

42 Online Monitoring Based on PCA Method The wholePCA model can be defined as follows

119875lowast=1

400

400

sum

119896=1

119875119896 (2)

where 119896 is the number of load matrixThe number of principal components 119860lowast can be calcu-

lated by the cumulative contribution rate In this paper weset 119860lowast = 6

The cumulative contribution rate =sum119860lowast

119895=1120582lowast

119895

trace (119878lowast)ge 90 (3)

where 119878lowast = diag(1205821 1205822 120582

119869) is an eigenvalues diagonal

matrix of the matrix119883The whole PCA load matrix 119875lowast is divided into two parts

the main component space 119875lowast(24 times 6) and the residual

4 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40014

145

15

155

16

165

Time

T2

(a) 1198792 control limits

0 50 100 150 200 250 300 350 40020

40

60

80

100

120

140

Time

SPE

(b) SPE control limits

Figure 3 Seamless tube rolling process 1198792 control limits and SPE control limits

Table 1 Measured variables for seamless steel rolling productionprocess

Symbol Variables Unit1 1st roller speed rs2 1st roller current A3 1st roller torque Nlowastm4 2nd roller speed rs5 2nd roller current A6 2nd roller torque Nlowastm7 3rd roller speed rs8 3rd roller current A9 3rd roller torque Nlowastm10 4th roller speed rs11 4th roller current A12 4th roller torque Nlowastm13 5th roller speed rs14 5th roller current A15 5th roller torque Nlowastm16 6th roller speed rs17 6th roller current A18 6th roller torque Nlowastm19 7th roller speed rs20 7th roller current A21 7th roller torque Nlowastm22 8th roller speed rs23 8th roller current A24 8th roller torque Nlowastm

space 119875lowast

(24 times 18) Similarly eigenvalue diagonal matrix 119878lowast

is correspondingly divided into two parts 119878lowast and 119878lowast

= 119883 (119875lowast

)119879

119883 = (119875

lowast

)119879

119864 = 119883 minus119883

(4)

In current online monitoring and fault diagnosis judgingwhether statistics1198792 and SPE are over the limit is usually usedto determine whether faults happen The control limits of 1198792approximately obey 119865 distribution

119863 sim

119860lowast(1198992minus 1)

119899 (119899 minus 119860lowast)119865119860lowast119899minus119860lowast120572 (5)

where 119870 = 400 and 119899 is the number of samples data formodeling The control limits of 1198792 is shown in Figure 3(a)For residual subspace SPE

119896of the PCAmodel approximately

obey 1205942 distribution [23 24] at time 119896

SPE119896120572= 1198921198961205942

ℎ119896120572

119892119896=

V119896

2119898119896

ℎ119896=2 (119898119896)2

V119896

(6)

where 119892 is a constant ℎ is the freedom degree of the 1205942distribution and V

119896and 119898

119896are respectively the mean and

variance of the square prediction error at time 119896 SPE controllimits are shown in Figure 3(b)

In order to monitor the rolling process first we needto obtain the measured data of new production periodstandardize the new data calculate the main component andthe prediction error of the data by formula (7) and checkwhether the 1198792 and SPE are beyond their own control limitIf the statistic exceeds the control limit it indicates that afault may occur at the time We now should analyze possiblecauses of the failure by variable 1198792 and SPE time-varyingcontribution plots and exclude or isolate the faults Consider

119905 = 119909119875lowast

119890 = 119909 (119868 minus 119875lowast

(119875lowast

)119879

)

1198792= 119905119879(119878lowast

)minus1

119905

SPE = 119890119890119879

(7)

Mathematical Problems in Engineering 5

0 50 100 150 200 250 300 350 40002468

1012141618

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 4 1198792 and SPE plot for MPCA monitoring results of the normal rolling process

As shown in Figure 4 under normal conditions moni-toring plot of SPE has an obvious alarm phenomenon duringthe first 50 sampling times and the last 50 sampling timesWecan explain the reason for alarm in the process of monitoringcombining with the rolling process The first 50 samples arein the bite stage where current speed and torque suddenlychange due to steel tube enter The last 50 samples are inthe steel leaving stage where speed current and torquedecrease sharply because of tube leaving Because of suddenchanges of variables standardized data still remain largedeviations Therefore PCA monitoring model appears alarmphenomenon So it is necessary to establish a multistageMPCA monitoring model

43 Build a Multistage MPCA Monitoring Model accordingto Production Process According to the process rollingproduction process can be divided into three stages bitestage stable rolling stage and steel leaving stage Then thispaper established a multistage MPCA monitoring modelaccording to the three stages

As shown in Figure 5 the situation of the steel leavingstage has improved But alarm phenomenon still exists andthe monitoring effect still needs to be improved So thebite stage and steel leaving stage should be further divided119870-means and Fuzzy 119862-means (FCM) clustering algorithmare commonly used FCM algorithm does not consider anyinformation related to the image space continuity so it ishighly sensitive to noise However 119870-means algorithm issimple and fast Particularly when dealing with the large datasets it has a very high efficiency So this paper chooses 119870-means clustering algorithm as a segmentation method

44 119870-Means and Multistage MPCA Combined to Build aMonitoring Model 119870-means algorithm is to cluster 119899 objectsbased on attributes into 119870(119870 lt 119899) partitions It assigns eachobject to the cluster which has the nearest center The centeris defined as the average of all the objects in the cluster whichstarts from a set of random initial centers The main steps of119870-means clustering algorithm is as follows

(1) Set up the cluster number 119870(2) Directly generate119870 random points as cluster centers(3) Assign each other points to the nearest cluster center(4) Recalculate the new cluster centers after new points

are clustered into the clusters(5) Repeat 3 and 4 until cluster centers do not change

According to process and the 119870-means algorithm forsegmentation number of principal components retained ineach substage PCA model is 119860lowast

119888 which can be obtained by

formula (3) The whole PCA load matrix 119875lowast119888is divided into

two parts the main component space 119875lowast119888and the residual

space 119875lowast

119888 which can be obtained by formula (4) Similarly

eigenvalue diagonalmatrix 119878lowast119888is correspondingly divided into

two parts 119878lowast119888and 119878

lowast

119888 SPE control limits can be obtained by

formula (4) 1198792 control limits is defined as follows

119863119896sim

119860lowast

119888(1198992

stage 119888 minus 1)

119899stage 119888 (119899stage 119888 minus 119860lowast

119888)

119865119860lowast

119888119899stage 119888minus119860

lowast

119888120572 (8)

As shown in Figure 6 when the number of stages is 7(bite stage is divided into three stages no segmentation stablerolling stage and steel leaving stage is divided into threestages) 1198792 and SPE are beyond their own control limits Themodel has a good performance in monitoring

According to the contrast of the foregoing analysis wecan easily conclude that seamless tube continuous rollingproduction process has too many characteristics includ-ing multiperiod strong nonlinearity and quickly changingdynamic characteristics It is hard tomonitor production pro-cess by the traditional MPCA method In this paper we usemultistage MPCA method according to production processand clustering algorithm This method can solve nonlinearproblem of seamless tube rolling production process andimprove the accuracy of online monitoring At the sametime the method has a strong guiding significance for theproduction

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40002468

101214161820

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 5 1198792 and SPE plot for multistage MPCA monitoring results of the normal rolling process

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 6 1198792 and SPE plot for 7 stages monitoring results of the normal rolling process

Through the experiment we find that the number ofsegments is larger and the precision is higher But at thesame time it easily causes misjudgment In this paper afterseveral experiments we finally decided to select 119870 = 7which both can be very good for rolling production processmonitoring and avoid misjudgment Compared with thesituation without segmentation segmentation model canmore accurately judge running state of rolling process whichhas a positive meaning for actual production

5 Fault Diagnosis

By monitoring the test method based on statistic can onlymonitorwhether faults occur and the approximate time of theoccurrence but it could not determine the source of the faultThe method of contribution plot provides possibilities ofdetermining the fault sources It can reflect the contributionto the statistics from variables at each moment

For the main component and residual subspace there aretwo contribution plots that can be used for fault diagnosismdash1198792 contribution plot and SPE contribution plot

The contribution to the 119886th principal component 119905119886from

119895th process variable 119909119895can be defined as follows

119862119905119886119909119895

=

119909119895119901119895119886

119905119886

(119886 = 1 119860 119895 = 1 119898) (9)

The contribution to the statistic SPE from the 119895th processvariables is

119862SPE119909119895

=

(119909119895minus 119909119895)2

SPE

(10)

In order to verify the performance of the multistagemonitoring model this paper introduces two typical faultdata for monitoring and fault diagnosis

Fault 1 1st roller speed fault from75th to 125th sampling timethe roller speed is 0

Fault 2 1st roller current fault from 70th to 130th samplingtime the roller current is 0

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Batches

Time

Variable1

1

X

K

J

J

2J KJ

I

I

X

Figure 2 Arrangement and decomposition of a three-way array byMPCA

In addition if the speed is too slow the consumption ofelectricity and other energy will increase

(2) Roller Torque The roller torque is one of the controlledvariables It is controlled by the input current The torquedirectly affects the quality of steel tube If torque is too small itcannot completely remove the blank of tube surface If torqueis too large shell will deform under the pressure Reasonablycontrolled torque of the rolling process is an important factorwhich cannot be ignored In order to save energy resourcesduring the wait state the torque will be reduced When thetube is rolled the torque increases When the tube leaves theroll the torque decreases

(3) Roller Input Current Roller current is one of the mostimportant control variables in the process of rolling It isan important factor to control torque and rotational speedBecause the sensitivity of the current is higher when thepierced tube enters the stand mill current increases quicklyThen the roller rotational speed and roller torque increase inorder to roll smoothly Current and torque are proportionalto the relationship The current increases when the steel tubegets into the rollThen it tends to a stable stateWhen the tubeleaves the roll current drops rapidly So correctly controllingthe size and direction of current is an important part of theprocess of rolling and the current is themain control variablein the rolling process

3 Multiway Principal Component Analysis

Batch processes are repetitive production process Theirdata sets have one more dimension than the continuousproduction process data set We can use three-dimensionaldata matrix 119883 = (119868 times 119869 times 119870) instead of the batch processdata collection where the three dimensions 119868 119869 and 119870respectively represent the batch number of samples numberof process variables and the number of measuring points ineach operation [19 20] MPCA will unfold119883 = (119868 times 119869 times119870) in

such a way as to put each of its vertical slices (119868 times 119869) side byside to the right which start with the one corresponding tothe first time interval The resulting two-dimensional matrixhas dimensions119883(119868 times 119869119870) [21 22] (Figure 2)

After three-dimensional data matrix 119883 is expanded intotwo-dimensional data it will be decomposed into a seriesof principal components consisting of score vectors 119905

119895and

loading matrices 119875119895 together with a residual matrix 119864 by the

principles of PCA The MPCA model can be written as

119883 =

119896

sum

119895=1

119905119895otimes 119901119895+ 119864 (1)

where the score vectors 119905119895is related only to batches and the

loading matrices 119875119895is related to variables and their time

variation The noise or residual part 119864 is as small as possiblein a least squares sense

4 Establish a Multiperiod ContinuousRolling Production Process MonitoringModel Based on 119870-Means and MPCA

41 Collecting the Production Data Based on actual produc-tion data characteristics of seamless steel rolling process inthis paper we use ibaAnalyzer software to collect 20 dataunder normal production condition As shown in Table 1 weselect 24 production process indicator variables to establish amonitoring model In this paper the three-dimensional datamatrix is 119883(119868 times 119869 times 119870) The three dimensions respectivelyrepresent the batch number of samples number of processvariables and the number of measuring points in eachoperation (119868 = 20 119869 = 24 119870 = 400) In order to obtainvertical data slice 119883

119896(119868 times 119869) we cut the three-dimensional

matrix along the direction of the third dimension Thusduring a period we can get 400 time slice matrixes By usingPCA for the 400 two-dimensional time slice matrix we got400 load matrixes and got the whole model load matrix bytaking an average

42 Online Monitoring Based on PCA Method The wholePCA model can be defined as follows

119875lowast=1

400

400

sum

119896=1

119875119896 (2)

where 119896 is the number of load matrixThe number of principal components 119860lowast can be calcu-

lated by the cumulative contribution rate In this paper weset 119860lowast = 6

The cumulative contribution rate =sum119860lowast

119895=1120582lowast

119895

trace (119878lowast)ge 90 (3)

where 119878lowast = diag(1205821 1205822 120582

119869) is an eigenvalues diagonal

matrix of the matrix119883The whole PCA load matrix 119875lowast is divided into two parts

the main component space 119875lowast(24 times 6) and the residual

4 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40014

145

15

155

16

165

Time

T2

(a) 1198792 control limits

0 50 100 150 200 250 300 350 40020

40

60

80

100

120

140

Time

SPE

(b) SPE control limits

Figure 3 Seamless tube rolling process 1198792 control limits and SPE control limits

Table 1 Measured variables for seamless steel rolling productionprocess

Symbol Variables Unit1 1st roller speed rs2 1st roller current A3 1st roller torque Nlowastm4 2nd roller speed rs5 2nd roller current A6 2nd roller torque Nlowastm7 3rd roller speed rs8 3rd roller current A9 3rd roller torque Nlowastm10 4th roller speed rs11 4th roller current A12 4th roller torque Nlowastm13 5th roller speed rs14 5th roller current A15 5th roller torque Nlowastm16 6th roller speed rs17 6th roller current A18 6th roller torque Nlowastm19 7th roller speed rs20 7th roller current A21 7th roller torque Nlowastm22 8th roller speed rs23 8th roller current A24 8th roller torque Nlowastm

space 119875lowast

(24 times 18) Similarly eigenvalue diagonal matrix 119878lowast

is correspondingly divided into two parts 119878lowast and 119878lowast

= 119883 (119875lowast

)119879

119883 = (119875

lowast

)119879

119864 = 119883 minus119883

(4)

In current online monitoring and fault diagnosis judgingwhether statistics1198792 and SPE are over the limit is usually usedto determine whether faults happen The control limits of 1198792approximately obey 119865 distribution

119863 sim

119860lowast(1198992minus 1)

119899 (119899 minus 119860lowast)119865119860lowast119899minus119860lowast120572 (5)

where 119870 = 400 and 119899 is the number of samples data formodeling The control limits of 1198792 is shown in Figure 3(a)For residual subspace SPE

119896of the PCAmodel approximately

obey 1205942 distribution [23 24] at time 119896

SPE119896120572= 1198921198961205942

ℎ119896120572

119892119896=

V119896

2119898119896

ℎ119896=2 (119898119896)2

V119896

(6)

where 119892 is a constant ℎ is the freedom degree of the 1205942distribution and V

119896and 119898

119896are respectively the mean and

variance of the square prediction error at time 119896 SPE controllimits are shown in Figure 3(b)

In order to monitor the rolling process first we needto obtain the measured data of new production periodstandardize the new data calculate the main component andthe prediction error of the data by formula (7) and checkwhether the 1198792 and SPE are beyond their own control limitIf the statistic exceeds the control limit it indicates that afault may occur at the time We now should analyze possiblecauses of the failure by variable 1198792 and SPE time-varyingcontribution plots and exclude or isolate the faults Consider

119905 = 119909119875lowast

119890 = 119909 (119868 minus 119875lowast

(119875lowast

)119879

)

1198792= 119905119879(119878lowast

)minus1

119905

SPE = 119890119890119879

(7)

Mathematical Problems in Engineering 5

0 50 100 150 200 250 300 350 40002468

1012141618

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 4 1198792 and SPE plot for MPCA monitoring results of the normal rolling process

As shown in Figure 4 under normal conditions moni-toring plot of SPE has an obvious alarm phenomenon duringthe first 50 sampling times and the last 50 sampling timesWecan explain the reason for alarm in the process of monitoringcombining with the rolling process The first 50 samples arein the bite stage where current speed and torque suddenlychange due to steel tube enter The last 50 samples are inthe steel leaving stage where speed current and torquedecrease sharply because of tube leaving Because of suddenchanges of variables standardized data still remain largedeviations Therefore PCA monitoring model appears alarmphenomenon So it is necessary to establish a multistageMPCA monitoring model

43 Build a Multistage MPCA Monitoring Model accordingto Production Process According to the process rollingproduction process can be divided into three stages bitestage stable rolling stage and steel leaving stage Then thispaper established a multistage MPCA monitoring modelaccording to the three stages

As shown in Figure 5 the situation of the steel leavingstage has improved But alarm phenomenon still exists andthe monitoring effect still needs to be improved So thebite stage and steel leaving stage should be further divided119870-means and Fuzzy 119862-means (FCM) clustering algorithmare commonly used FCM algorithm does not consider anyinformation related to the image space continuity so it ishighly sensitive to noise However 119870-means algorithm issimple and fast Particularly when dealing with the large datasets it has a very high efficiency So this paper chooses 119870-means clustering algorithm as a segmentation method

44 119870-Means and Multistage MPCA Combined to Build aMonitoring Model 119870-means algorithm is to cluster 119899 objectsbased on attributes into 119870(119870 lt 119899) partitions It assigns eachobject to the cluster which has the nearest center The centeris defined as the average of all the objects in the cluster whichstarts from a set of random initial centers The main steps of119870-means clustering algorithm is as follows

(1) Set up the cluster number 119870(2) Directly generate119870 random points as cluster centers(3) Assign each other points to the nearest cluster center(4) Recalculate the new cluster centers after new points

are clustered into the clusters(5) Repeat 3 and 4 until cluster centers do not change

According to process and the 119870-means algorithm forsegmentation number of principal components retained ineach substage PCA model is 119860lowast

119888 which can be obtained by

formula (3) The whole PCA load matrix 119875lowast119888is divided into

two parts the main component space 119875lowast119888and the residual

space 119875lowast

119888 which can be obtained by formula (4) Similarly

eigenvalue diagonalmatrix 119878lowast119888is correspondingly divided into

two parts 119878lowast119888and 119878

lowast

119888 SPE control limits can be obtained by

formula (4) 1198792 control limits is defined as follows

119863119896sim

119860lowast

119888(1198992

stage 119888 minus 1)

119899stage 119888 (119899stage 119888 minus 119860lowast

119888)

119865119860lowast

119888119899stage 119888minus119860

lowast

119888120572 (8)

As shown in Figure 6 when the number of stages is 7(bite stage is divided into three stages no segmentation stablerolling stage and steel leaving stage is divided into threestages) 1198792 and SPE are beyond their own control limits Themodel has a good performance in monitoring

According to the contrast of the foregoing analysis wecan easily conclude that seamless tube continuous rollingproduction process has too many characteristics includ-ing multiperiod strong nonlinearity and quickly changingdynamic characteristics It is hard tomonitor production pro-cess by the traditional MPCA method In this paper we usemultistage MPCA method according to production processand clustering algorithm This method can solve nonlinearproblem of seamless tube rolling production process andimprove the accuracy of online monitoring At the sametime the method has a strong guiding significance for theproduction

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40002468

101214161820

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 5 1198792 and SPE plot for multistage MPCA monitoring results of the normal rolling process

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 6 1198792 and SPE plot for 7 stages monitoring results of the normal rolling process

Through the experiment we find that the number ofsegments is larger and the precision is higher But at thesame time it easily causes misjudgment In this paper afterseveral experiments we finally decided to select 119870 = 7which both can be very good for rolling production processmonitoring and avoid misjudgment Compared with thesituation without segmentation segmentation model canmore accurately judge running state of rolling process whichhas a positive meaning for actual production

5 Fault Diagnosis

By monitoring the test method based on statistic can onlymonitorwhether faults occur and the approximate time of theoccurrence but it could not determine the source of the faultThe method of contribution plot provides possibilities ofdetermining the fault sources It can reflect the contributionto the statistics from variables at each moment

For the main component and residual subspace there aretwo contribution plots that can be used for fault diagnosismdash1198792 contribution plot and SPE contribution plot

The contribution to the 119886th principal component 119905119886from

119895th process variable 119909119895can be defined as follows

119862119905119886119909119895

=

119909119895119901119895119886

119905119886

(119886 = 1 119860 119895 = 1 119898) (9)

The contribution to the statistic SPE from the 119895th processvariables is

119862SPE119909119895

=

(119909119895minus 119909119895)2

SPE

(10)

In order to verify the performance of the multistagemonitoring model this paper introduces two typical faultdata for monitoring and fault diagnosis

Fault 1 1st roller speed fault from75th to 125th sampling timethe roller speed is 0

Fault 2 1st roller current fault from 70th to 130th samplingtime the roller current is 0

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40014

145

15

155

16

165

Time

T2

(a) 1198792 control limits

0 50 100 150 200 250 300 350 40020

40

60

80

100

120

140

Time

SPE

(b) SPE control limits

Figure 3 Seamless tube rolling process 1198792 control limits and SPE control limits

Table 1 Measured variables for seamless steel rolling productionprocess

Symbol Variables Unit1 1st roller speed rs2 1st roller current A3 1st roller torque Nlowastm4 2nd roller speed rs5 2nd roller current A6 2nd roller torque Nlowastm7 3rd roller speed rs8 3rd roller current A9 3rd roller torque Nlowastm10 4th roller speed rs11 4th roller current A12 4th roller torque Nlowastm13 5th roller speed rs14 5th roller current A15 5th roller torque Nlowastm16 6th roller speed rs17 6th roller current A18 6th roller torque Nlowastm19 7th roller speed rs20 7th roller current A21 7th roller torque Nlowastm22 8th roller speed rs23 8th roller current A24 8th roller torque Nlowastm

space 119875lowast

(24 times 18) Similarly eigenvalue diagonal matrix 119878lowast

is correspondingly divided into two parts 119878lowast and 119878lowast

= 119883 (119875lowast

)119879

119883 = (119875

lowast

)119879

119864 = 119883 minus119883

(4)

In current online monitoring and fault diagnosis judgingwhether statistics1198792 and SPE are over the limit is usually usedto determine whether faults happen The control limits of 1198792approximately obey 119865 distribution

119863 sim

119860lowast(1198992minus 1)

119899 (119899 minus 119860lowast)119865119860lowast119899minus119860lowast120572 (5)

where 119870 = 400 and 119899 is the number of samples data formodeling The control limits of 1198792 is shown in Figure 3(a)For residual subspace SPE

119896of the PCAmodel approximately

obey 1205942 distribution [23 24] at time 119896

SPE119896120572= 1198921198961205942

ℎ119896120572

119892119896=

V119896

2119898119896

ℎ119896=2 (119898119896)2

V119896

(6)

where 119892 is a constant ℎ is the freedom degree of the 1205942distribution and V

119896and 119898

119896are respectively the mean and

variance of the square prediction error at time 119896 SPE controllimits are shown in Figure 3(b)

In order to monitor the rolling process first we needto obtain the measured data of new production periodstandardize the new data calculate the main component andthe prediction error of the data by formula (7) and checkwhether the 1198792 and SPE are beyond their own control limitIf the statistic exceeds the control limit it indicates that afault may occur at the time We now should analyze possiblecauses of the failure by variable 1198792 and SPE time-varyingcontribution plots and exclude or isolate the faults Consider

119905 = 119909119875lowast

119890 = 119909 (119868 minus 119875lowast

(119875lowast

)119879

)

1198792= 119905119879(119878lowast

)minus1

119905

SPE = 119890119890119879

(7)

Mathematical Problems in Engineering 5

0 50 100 150 200 250 300 350 40002468

1012141618

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 4 1198792 and SPE plot for MPCA monitoring results of the normal rolling process

As shown in Figure 4 under normal conditions moni-toring plot of SPE has an obvious alarm phenomenon duringthe first 50 sampling times and the last 50 sampling timesWecan explain the reason for alarm in the process of monitoringcombining with the rolling process The first 50 samples arein the bite stage where current speed and torque suddenlychange due to steel tube enter The last 50 samples are inthe steel leaving stage where speed current and torquedecrease sharply because of tube leaving Because of suddenchanges of variables standardized data still remain largedeviations Therefore PCA monitoring model appears alarmphenomenon So it is necessary to establish a multistageMPCA monitoring model

43 Build a Multistage MPCA Monitoring Model accordingto Production Process According to the process rollingproduction process can be divided into three stages bitestage stable rolling stage and steel leaving stage Then thispaper established a multistage MPCA monitoring modelaccording to the three stages

As shown in Figure 5 the situation of the steel leavingstage has improved But alarm phenomenon still exists andthe monitoring effect still needs to be improved So thebite stage and steel leaving stage should be further divided119870-means and Fuzzy 119862-means (FCM) clustering algorithmare commonly used FCM algorithm does not consider anyinformation related to the image space continuity so it ishighly sensitive to noise However 119870-means algorithm issimple and fast Particularly when dealing with the large datasets it has a very high efficiency So this paper chooses 119870-means clustering algorithm as a segmentation method

44 119870-Means and Multistage MPCA Combined to Build aMonitoring Model 119870-means algorithm is to cluster 119899 objectsbased on attributes into 119870(119870 lt 119899) partitions It assigns eachobject to the cluster which has the nearest center The centeris defined as the average of all the objects in the cluster whichstarts from a set of random initial centers The main steps of119870-means clustering algorithm is as follows

(1) Set up the cluster number 119870(2) Directly generate119870 random points as cluster centers(3) Assign each other points to the nearest cluster center(4) Recalculate the new cluster centers after new points

are clustered into the clusters(5) Repeat 3 and 4 until cluster centers do not change

According to process and the 119870-means algorithm forsegmentation number of principal components retained ineach substage PCA model is 119860lowast

119888 which can be obtained by

formula (3) The whole PCA load matrix 119875lowast119888is divided into

two parts the main component space 119875lowast119888and the residual

space 119875lowast

119888 which can be obtained by formula (4) Similarly

eigenvalue diagonalmatrix 119878lowast119888is correspondingly divided into

two parts 119878lowast119888and 119878

lowast

119888 SPE control limits can be obtained by

formula (4) 1198792 control limits is defined as follows

119863119896sim

119860lowast

119888(1198992

stage 119888 minus 1)

119899stage 119888 (119899stage 119888 minus 119860lowast

119888)

119865119860lowast

119888119899stage 119888minus119860

lowast

119888120572 (8)

As shown in Figure 6 when the number of stages is 7(bite stage is divided into three stages no segmentation stablerolling stage and steel leaving stage is divided into threestages) 1198792 and SPE are beyond their own control limits Themodel has a good performance in monitoring

According to the contrast of the foregoing analysis wecan easily conclude that seamless tube continuous rollingproduction process has too many characteristics includ-ing multiperiod strong nonlinearity and quickly changingdynamic characteristics It is hard tomonitor production pro-cess by the traditional MPCA method In this paper we usemultistage MPCA method according to production processand clustering algorithm This method can solve nonlinearproblem of seamless tube rolling production process andimprove the accuracy of online monitoring At the sametime the method has a strong guiding significance for theproduction

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40002468

101214161820

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 5 1198792 and SPE plot for multistage MPCA monitoring results of the normal rolling process

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 6 1198792 and SPE plot for 7 stages monitoring results of the normal rolling process

Through the experiment we find that the number ofsegments is larger and the precision is higher But at thesame time it easily causes misjudgment In this paper afterseveral experiments we finally decided to select 119870 = 7which both can be very good for rolling production processmonitoring and avoid misjudgment Compared with thesituation without segmentation segmentation model canmore accurately judge running state of rolling process whichhas a positive meaning for actual production

5 Fault Diagnosis

By monitoring the test method based on statistic can onlymonitorwhether faults occur and the approximate time of theoccurrence but it could not determine the source of the faultThe method of contribution plot provides possibilities ofdetermining the fault sources It can reflect the contributionto the statistics from variables at each moment

For the main component and residual subspace there aretwo contribution plots that can be used for fault diagnosismdash1198792 contribution plot and SPE contribution plot

The contribution to the 119886th principal component 119905119886from

119895th process variable 119909119895can be defined as follows

119862119905119886119909119895

=

119909119895119901119895119886

119905119886

(119886 = 1 119860 119895 = 1 119898) (9)

The contribution to the statistic SPE from the 119895th processvariables is

119862SPE119909119895

=

(119909119895minus 119909119895)2

SPE

(10)

In order to verify the performance of the multistagemonitoring model this paper introduces two typical faultdata for monitoring and fault diagnosis

Fault 1 1st roller speed fault from75th to 125th sampling timethe roller speed is 0

Fault 2 1st roller current fault from 70th to 130th samplingtime the roller current is 0

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 50 100 150 200 250 300 350 40002468

1012141618

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 4 1198792 and SPE plot for MPCA monitoring results of the normal rolling process

As shown in Figure 4 under normal conditions moni-toring plot of SPE has an obvious alarm phenomenon duringthe first 50 sampling times and the last 50 sampling timesWecan explain the reason for alarm in the process of monitoringcombining with the rolling process The first 50 samples arein the bite stage where current speed and torque suddenlychange due to steel tube enter The last 50 samples are inthe steel leaving stage where speed current and torquedecrease sharply because of tube leaving Because of suddenchanges of variables standardized data still remain largedeviations Therefore PCA monitoring model appears alarmphenomenon So it is necessary to establish a multistageMPCA monitoring model

43 Build a Multistage MPCA Monitoring Model accordingto Production Process According to the process rollingproduction process can be divided into three stages bitestage stable rolling stage and steel leaving stage Then thispaper established a multistage MPCA monitoring modelaccording to the three stages

As shown in Figure 5 the situation of the steel leavingstage has improved But alarm phenomenon still exists andthe monitoring effect still needs to be improved So thebite stage and steel leaving stage should be further divided119870-means and Fuzzy 119862-means (FCM) clustering algorithmare commonly used FCM algorithm does not consider anyinformation related to the image space continuity so it ishighly sensitive to noise However 119870-means algorithm issimple and fast Particularly when dealing with the large datasets it has a very high efficiency So this paper chooses 119870-means clustering algorithm as a segmentation method

44 119870-Means and Multistage MPCA Combined to Build aMonitoring Model 119870-means algorithm is to cluster 119899 objectsbased on attributes into 119870(119870 lt 119899) partitions It assigns eachobject to the cluster which has the nearest center The centeris defined as the average of all the objects in the cluster whichstarts from a set of random initial centers The main steps of119870-means clustering algorithm is as follows

(1) Set up the cluster number 119870(2) Directly generate119870 random points as cluster centers(3) Assign each other points to the nearest cluster center(4) Recalculate the new cluster centers after new points

are clustered into the clusters(5) Repeat 3 and 4 until cluster centers do not change

According to process and the 119870-means algorithm forsegmentation number of principal components retained ineach substage PCA model is 119860lowast

119888 which can be obtained by

formula (3) The whole PCA load matrix 119875lowast119888is divided into

two parts the main component space 119875lowast119888and the residual

space 119875lowast

119888 which can be obtained by formula (4) Similarly

eigenvalue diagonalmatrix 119878lowast119888is correspondingly divided into

two parts 119878lowast119888and 119878

lowast

119888 SPE control limits can be obtained by

formula (4) 1198792 control limits is defined as follows

119863119896sim

119860lowast

119888(1198992

stage 119888 minus 1)

119899stage 119888 (119899stage 119888 minus 119860lowast

119888)

119865119860lowast

119888119899stage 119888minus119860

lowast

119888120572 (8)

As shown in Figure 6 when the number of stages is 7(bite stage is divided into three stages no segmentation stablerolling stage and steel leaving stage is divided into threestages) 1198792 and SPE are beyond their own control limits Themodel has a good performance in monitoring

According to the contrast of the foregoing analysis wecan easily conclude that seamless tube continuous rollingproduction process has too many characteristics includ-ing multiperiod strong nonlinearity and quickly changingdynamic characteristics It is hard tomonitor production pro-cess by the traditional MPCA method In this paper we usemultistage MPCA method according to production processand clustering algorithm This method can solve nonlinearproblem of seamless tube rolling production process andimprove the accuracy of online monitoring At the sametime the method has a strong guiding significance for theproduction

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40002468

101214161820

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 5 1198792 and SPE plot for multistage MPCA monitoring results of the normal rolling process

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 6 1198792 and SPE plot for 7 stages monitoring results of the normal rolling process

Through the experiment we find that the number ofsegments is larger and the precision is higher But at thesame time it easily causes misjudgment In this paper afterseveral experiments we finally decided to select 119870 = 7which both can be very good for rolling production processmonitoring and avoid misjudgment Compared with thesituation without segmentation segmentation model canmore accurately judge running state of rolling process whichhas a positive meaning for actual production

5 Fault Diagnosis

By monitoring the test method based on statistic can onlymonitorwhether faults occur and the approximate time of theoccurrence but it could not determine the source of the faultThe method of contribution plot provides possibilities ofdetermining the fault sources It can reflect the contributionto the statistics from variables at each moment

For the main component and residual subspace there aretwo contribution plots that can be used for fault diagnosismdash1198792 contribution plot and SPE contribution plot

The contribution to the 119886th principal component 119905119886from

119895th process variable 119909119895can be defined as follows

119862119905119886119909119895

=

119909119895119901119895119886

119905119886

(119886 = 1 119860 119895 = 1 119898) (9)

The contribution to the statistic SPE from the 119895th processvariables is

119862SPE119909119895

=

(119909119895minus 119909119895)2

SPE

(10)

In order to verify the performance of the multistagemonitoring model this paper introduces two typical faultdata for monitoring and fault diagnosis

Fault 1 1st roller speed fault from75th to 125th sampling timethe roller speed is 0

Fault 2 1st roller current fault from 70th to 130th samplingtime the roller current is 0

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 40002468

101214161820

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 5 1198792 and SPE plot for multistage MPCA monitoring results of the normal rolling process

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(a) 1198792 monitoring plot

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot

Figure 6 1198792 and SPE plot for 7 stages monitoring results of the normal rolling process

Through the experiment we find that the number ofsegments is larger and the precision is higher But at thesame time it easily causes misjudgment In this paper afterseveral experiments we finally decided to select 119870 = 7which both can be very good for rolling production processmonitoring and avoid misjudgment Compared with thesituation without segmentation segmentation model canmore accurately judge running state of rolling process whichhas a positive meaning for actual production

5 Fault Diagnosis

By monitoring the test method based on statistic can onlymonitorwhether faults occur and the approximate time of theoccurrence but it could not determine the source of the faultThe method of contribution plot provides possibilities ofdetermining the fault sources It can reflect the contributionto the statistics from variables at each moment

For the main component and residual subspace there aretwo contribution plots that can be used for fault diagnosismdash1198792 contribution plot and SPE contribution plot

The contribution to the 119886th principal component 119905119886from

119895th process variable 119909119895can be defined as follows

119862119905119886119909119895

=

119909119895119901119895119886

119905119886

(119886 = 1 119860 119895 = 1 119898) (9)

The contribution to the statistic SPE from the 119895th processvariables is

119862SPE119909119895

=

(119909119895minus 119909119895)2

SPE

(10)

In order to verify the performance of the multistagemonitoring model this paper introduces two typical faultdata for monitoring and fault diagnosis

Fault 1 1st roller speed fault from75th to 125th sampling timethe roller speed is 0

Fault 2 1st roller current fault from 70th to 130th samplingtime the roller current is 0

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 7 1198792 monitoring plots of fault 1

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 8 SPE monitoring plots of fault 1

51 Fault Diagnosis for the 1st Roller Speed As shown inFigure 7 for fault 1 monitoring plots of SPE have an obviousalarm phenomenon which 1198792 monitoring plots do nothave For comparison 1198792 contribution plot are still drawntogether with the SPE contribution plot In order to diagnosethe cause of the fault this paper respectively drew maincomponent contribution plots 1198792 contribution plots andSPE contribution plots of 60th 120 and 240th samplingtime As shown in Figure 8(a) MPCA model does not havean obvious alarm phenomenon in the fault time As shownin Figure 8(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 9 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the firstprincipal component from process variables According tocontribution rate from each variable to1198792 and SPE this paper

studied and determined the fault sources They are shown inFigures 10 and 11

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 10does not detect the fault variable As shown in Figure 11 inthe fault time the first variable (1st roller speed) has largercontribution rate to the first principal component From theresults we can see that SPE monitoring plots have an obviousalarm phenomenon According to the SPE contribution ratewe can diagnose the fault So the proposedmethod is correct

52 Fault Diagnosis for the 1st Roller Current As shownin Figure 12 for fault 2 monitoring plots of SPE have anobvious alarm phenomenon which 1198792 monitoring plots donot have For comparison 1198792 contribution plots are stilldrawn together with the SPE contribution plot In order todiagnose the cause of the fault this paper respectively drawsmain component contribution plots 1198792 contribution plotsand SPE contribution plots of 50th 125 and 250th samplingtime As shown in Figure 13(a) MPCA model does not have

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

1 2 3 4 50

00501

01502

02503

03504

04505

Principal components

Con

trib

utio

n

(a) Time = 60

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(b) Time = 120

1 2 3 4 50

0102030405060708

Principal components

Con

trib

utio

n

(c) Time = 240

Figure 9 Principal component contribution plots of fault 1

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 60

0 5 10 15 20 25

001020304

Variables

Con

trib

utio

n

minus04

minus03

minus02

minus01

(b) Time = 120

0 5 10 15 20 25

0

05

1

15

2

Variables

Con

trib

utio

n

minus05

minus1

(c) Time = 240

Figure 10 1198792 contribution plots of fault 1

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 60

0 5 10 15 20 250

002004006008

01012014016018

02

Variables

Con

trib

utio

n

(b) Time = 120

0 5 10 15 20 250

005

01

015

02

025

03

035

Variables

Con

trib

utio

n

(c) Time = 240

Figure 11 SPE contribution plots of fault 1

0 50 100 150 200 250 300 350 40002468

10121416

Time

Control limits

T2

T2

(a) 1198792 monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Time

Control limits

T2

T2

(b) 1198792 monitoring plot for 7 stages MPCA

Figure 12 1198792 monitoring plots of fault 2

an obvious alarm phenomenon in the fault time As shownin Figure 13(b) multistage MPCA model can quickly andaccurately detect the fault

As shown in Figure 14 the contribution rate of eachprincipal component is not the same at different time Nearthe fault time the first principal component contributionrate was larger Away from the fault time the first principalcomponent contribution rate is less than the second principalcomponent This paper analyzes contribution rate to the first

principal component from process variables According tocontribution rate of each variable this paper studies anddetermines the fault sources They are shown in Figures 15and 16

Because 1198792 monitoring plots do not have an obviousalarm phenomenon 1198792 contribution plot shown in Figure 15does not detect the fault variable As shown in Figure 16 in thefault time the second variable (1st roller current) has largercontribution rate to the first principal component From

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Time

SPE

SPEControl limits

(a) SPE monitoring plot for MPCA

0 50 100 150 200 250 300 350 4000

102030405060708090

100

Time

SPE

SPEControl limits

(b) SPE monitoring plot for 7 stages MPCA

Figure 13 SPE monitoring plots of fault 2

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(a) Time = 50

1 2 3 4 50

01020304050607

Principal components

Con

trib

utio

n08

(b) Time = 125

1 2 3 4 50

01

02

03

04

05

06

07

Principal components

Con

trib

utio

n

(c) Time = 250

Figure 14 Principal component contribution plots of fault 2

the results we can see that SPEmonitoring plots have an obvi-ous alarm phenomenon According to the SPE contributionrate we can diagnose the fault So the proposed method iscorrect

6 Conclusions

According to strong nonlinearity and dynamic property ofthe seamless tube continuous rolling production process this

paper divides production data into subperiods by 119870-meansclustering algorithm combined with production processThen we establish a continuous rolling production processmonitoring and fault diagnosis model based on multistageMPCAmethodThe results have shown that themodel devel-oped in this paper has better performances in monitoringand fault diagnosis Meanwhile the proposed method can beextended to the other industry processes

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 5 10 15 20 25

0

02

04

06

08

Variables

Con

trib

utio

n

minus06

minus04

minus02

(a) Time = 50

0 5 10 15 20 25

0005

01015

02025

03

Variables

Con

trib

utio

n

minus015

minus01

minus005

(b) Time = 125

0 5 10 15 20 25

0

05

1

Con

trib

utio

n

Variables

minus05

(c) Time = 250

Figure 15 1198792 contribution plots of fault 2

0 5 10 15 20 250

002

004

006

008

01

012

Variables

Con

trib

utio

n

(a) Time = 50

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(b) Time = 125

0 5 10 15 20 250

002004006008

01012014016

Variables

Con

trib

utio

n

(c) Time = 250

Figure 16 SPE contribution plots of fault 2

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (Grant no 61203214) Provincial Science andTechnology Department of Education projects the generalproject (L2013101) and National Natural Science Foundationof China (Grant nos 41371437 61403277 61304121 6147307261203103 and 61374146)

References

[1] D Xiao J C Wang and H X Tian ldquoQuality prediction andcontrol of reducing pipe based on EOS-ELM-RPLS mathemat-ics modeling methodrdquo Journal of Applied Mathematics vol2014 Article ID 298218 13 pages 2014

[2] D Levesque S E Kruger G Lamouche et al ldquoThickness andgrain size monitoring in seamless tube-making process usinglaser ultrasonicsrdquoNDTampE International vol 39 no 8 pp 622ndash626 2006

[3] Y Lv Y R Li and H G Wang ldquoOn-line monitoring and faultdiagnosis for mill drivesrdquoHeavyMachinery vol 2002 no 2 pp56ndash59 2002

[4] M Reggio F McKenty L Gravel J Cortes G Morales andM-A Ladron de Guevara ldquoComputational analysis of theprocess for manufacturing seamless tubesrdquo Applied ThermalEngineering vol 22 no 4 pp 459ndash470 2002

[5] H Jin Fault Diagnosis of Large DC Motor Strip Mill Universityof Science and Technology of China 2000

[6] L Lei and H L Zhang ldquoContinuous rolling-tube unit on-linesupervision system based on virtual instrument technologyrdquoNatural Science Journal of Xiangtan University vol 26 no 1 pp102ndash105 2004

[7] C H Zhao and F R Gao ldquoFault-relevant Principal ComponentAnalysis (FPCA) method for multivariate statistical modelingand process monitoringrdquo Chemometrics and Intelligent Labora-tory Systems vol 133 no 1 pp 1ndash12 2014

[8] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[9] B Zhang C Sconyers C Byington R Patrick M E Orchardand G Vachtsevanos ldquoA probabilistic fault detection approachapplication to bearing fault detectionrdquo IEEE Transactions onIndustrial Electronics vol 58 no 5 pp 2011ndash2018 2011

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S X Ding ldquoData-driven design of monitoring and diagnosissystems for dynamic processes a review of subspace techniquebased schemes and some recent resultsrdquo Journal of ProcessControl vol 24 no 2 pp 431ndash449 2014

[12] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquo

Mechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[13] S Bedoui R FalehH Samet andAKachouri ldquoElectronic nosesystem and principal component analysis technique for gasesidentificationrdquo in Proceedings of the 10th International Multi-Conference on Systems Signals amp Devices (SSD rsquo13) pp 1ndash6IEEE Hammamet Tunisia March 2013

[14] G Georgoulas M O Mustafa I P Tsoumas et al ldquoPrincipalComponent Analysis of the start-up transient and HiddenMarkov Modeling for broken rotor bar fault diagnosis inasynchronous machinesrdquo Expert Systems with Applications vol40 no 17 pp 7024ndash7033 2013

[15] M Evans and J Kennedy ldquoIntegration of adaptive neuro fuzzyinference systems and principal component analysis for thecontrol of tertiary scale formation on tinplate at a hot millrdquoExpert Systems with Applications vol 41 no 15 pp 6662ndash66752014

[16] C M Ringle M Sarstedt R Schlittgen and C R TaylorldquoPLS path modeling and evolutionary segmentationrdquo Journal ofBusiness Research vol 66 no 9 pp 1318ndash1324 2013

[17] E M Abdel-Rahman O Mutanga J Odindi E Adam AOdindo and R Ismail ldquoA comparison of partial least squares(PLS) and sparse PLS regressions for predicting yield of Swisschard grown under different irrigation water sources usinghyperspectral datardquo Computers and Electronics in Agriculturevol 106 pp 11ndash19 2014

[18] B Lu I Castillo L Chiang and T F Edgar ldquoIndustrial PLSmodel variable selection using moving window variable impor-tance in projectionrdquo Chemometrics and Intelligent LaboratorySystems vol 135 no 15 pp 90ndash109 2014

[19] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[20] X-T Doan R Srinivasan P M Bapat and P P WangikarldquoDetection of phase shifts in batch fermentation via statisti-cal analysis of the online measurements a case study withrifamycin B fermentationrdquo Journal of Biotechnology vol 132 no2 pp 156ndash166 2007

[21] M Emparan R Simpson S Almonacid A Teixeira and AUrtubia ldquoEarly recognition of problematic wine fermentationsthrough multivariate data analysesrdquo Food Control vol 27 no 1pp 248ndash253 2012

[22] W Dong Y Yao and F Gao ldquoPhase analysis and identificationmethod formultiphase batch processes with partitioningmulti-way principal component analysis (MPCA) modelrdquo ChineseJournal of Chemical Engineering vol 20 no 6 pp 1121ndash11272012

[23] PNomikos and J FMacGregor ldquoMulti-way partial least squaresin monitoring batch processesrdquo Chemometrics and IntelligentLaboratory Systems vol 30 no 1 pp 97ndash108 1995

[24] Y S Qi P Wang X J Gao and Y J Gong ldquoBatch processmonitoring and fault diagnosis based on improved multi-wayprincipal component analysisrdquo CIESC Journal vol 85 no 1 pp82ndash93 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of