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Research ArticleResearch on Fault Diagnosis of Flexible Material R2RManufacturing System Based on Quality Control Chart and SoV
Yaohua Deng ,1,2 Na Zhou ,1 Xiali Liu ,3 and Qiwen Lu 3
1School of Electro-Mechanical Engineering, Guangdong University of Technology, Guangzhou, Guangdong, China2Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA3Foshan World Intelligent Technology Co., Ltd., Foshan, Foshan 528000, China
Correspondence should be addressed to Na Zhou; [email protected], Xiali Liu; [email protected],and Qiwen Lu; [email protected]
Received 21 February 2018; Accepted 3 April 2018; Published 21 May 2018
Academic Editor: Guangming Xie
Copyright ยฉ 2018 Yaohua Deng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Stream of Variation (SoV) model and control chart are combined to study the fault diagnosis method of flexible materials R2Rmanufacturing system. Based on the analysis of the correlation between the fault source and product quality in the manufacturingprocess and also the statistical distribution rule of the processing quality characteristic vector ๐ฟ ๐ and the fault source ๐๐, SoVmodelunder controlled or uncontrolled states and themathematical model of the probability distribution of the statistic๐2๐,๐ of the qualitycharacteristic variable ๐ฟ ๐ are deduced. And the calculation equation of the centerline, the upper limit, and the lower limit of thecontrol chart are deduced. The experimental results show that, under controlled or uncontrolled condition, when the programruns to 500 steps, the Average Run Length (ARL) of the performance parameters tends to be stable; and when program reaches1000 steps, the actual ARL value is almost the same as the theoretical value. The fault diagnosis experiment shows that, under thecondition when the fault source is strongly correlated or the fault source correlation coefficient is the same, using the control chartestablished in this paper can simply and quickly determine the fault location in the system.
1. Introduction
R2R manufacturing system is a typical multistation con-tinuous manufacturing system [1]. Since the factors thataffect the quality of R2R are caused by many related processcharacteristics such asmanufacturing system faults ormotionabnormalities, it is difficult for the conventional predictionmethod to determine the fault source when a manufacturingquality problem occurs. The fault diagnosis method basedon the quality control chart classifies the various patternsof control charts from processing quality data, establishesan abnormal pattern set and a fault set, and correlates theabnormal pattern set and the fault set in order to diagnosethe fault source of the manufacturing system [2].The existingquality control chart fault diagnosis methods are univariatecontrol chart [3], multivariate control chart [4], regressionadjustment control chart [5], and so on. However, when
using these methods to monitor multistation systems, thecontrol charts have a high false alarm rate. In particular,when the process data is autocorrelated, it is impossible tomonitor the abnormal faults in the manufacturing process byusing conventional control charts under the assumption ofindependence.
Based on the analysis of the correlation between eachstation of R2Rmanufacturing system, combining the physicalanalysis and data-driven method, this paper establishes therelation equation describing the process deviation of mul-tistation and the final quality of the product, constructs aSoV model under controlled or uncontrolled manufacturingsystems, works out corresponding quality control chart forproduct quality characteristic variables to monitor autocor-relation data, and detects and isolates multiple faults. Hencethis paper lays a theoretical foundation for the subsequentintelligent maintenance of R2R manufacturing system.
HindawiMathematical Problems in EngineeringVolume 2018, Article ID 6350380, 8 pageshttps://doi.org/10.1155/2018/6350380
http://orcid.org/0000-0002-5722-1037http://orcid.org/0000-0002-3879-2221http://orcid.org/0000-0001-7351-0886http://orcid.org/0000-0003-4602-7180https://doi.org/10.1155/2018/6350380
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2 Mathematical Problems in Engineering
Guide rollerGuide roller
Unwinding module Winding module
Feed transfer module
Figure 1: A simplified working principle diagram of manufacturingsystem.
2. Fault Diagnosis Based on QualityControl Chart
The flexible material R2R manufacturing system is a con-tinuous multistation manufacturing system as previouslydescribed. Figure 1 is a simplified working principle diagramof manufacturing system, the output of the previous stationis the input of the next station duringmanufacturing process,due to incidental factors and system factors, each stationoutput product quality characteristic produces a certaindeviation, this deviation then enters the next station, andthus the final quality characteristic deviation of product isthe result of gradual accumulation of quality characteristicdeviation of all previous stations [6]. Therefore, not only arethe systemโs input and output variables considered, but alsothe systemโs real-time statuses are followed in the control ofproduct quality. In order to better analyze the main influenc-ing factors of manufacturing process quality deviation, thispaper divides the state of R2R manufacturing process intocontrolled state and uncontrolled state. Under the controlledstate, it is inevitable that there is a random deviation inmanufacturing process, the quality characteristic value showsa certain regularity, and the product quality fluctuation issmall. It is difficult to eliminate this deviation, and it is notnecessary to eliminate it. Under the uncontrolled state, thereare abnormal factors besides the effect of random error in theproduction process; the values of the quality characteristicsfluctuate greatly and have a great impact on the quality.To ensure that the quality maintains the original regularity,the deviation must be eliminated to make production runsmoothly [7].
Based on the working principle, as shown in Figure 1, itis possible to judge whether the manufacturing system hasabnormal factors by the state of quality control chart of theprocessing object; Figure 2 shows the basic form of controlchart, where the abscissa denotes the number of samples,the ordinate denotes the sample quality characteristic value,UCL denotes upper control limit, CL denotes centerline, LCLdenotes lower control limit, quality characteristic values aresampled in chronological order, which are described in theform of a scatter plot in the coordinate system sequentially,
Qua
lity
char
acte
ristic
val
ue
1510 20 302550The number of samples
UCL
CL
LCL
Figure 2: The basic form of quality control chart.
and finally the control chart is obtained through the con-nection to reflect the quality fluctuation in the productionprocess. Under the controlled state, all sample points arebetween the upper and lower control limits. If the samplepoints are beyond this limit, then there is an exception or faultduring the production process, which means at this momentthe system is under an uncontrolled state, and when the datapoint exceeds the control limit area, an alarm occurs on thecontrol chart.
R2R manufacturing system usually includes hundredsof operations to complete the product processing, manyfailures of the system may occur at the same time, andthe conventional Statistical Process Control (SPC) takes theentire multistation system as a whole; thus it is not ableto identify abnormal station. The SoV model based on thephysical model of tension transfer can remedy this problem;this method uses the SoV model to obtain the quality data ofdifferent workstations to establish the fault diagnosis controlchart; after the process change is detected, the fault signalis extracted by means of estimation or pattern matching todetermine the root cause so that fault detection and isolationcan be achieved.
3. Construction of the Relationshipbetween Fault Source and ProductQuality Model Based on SoV
If ๐ denotes the ๐th station, ๐ฟ ๐ is the quality characteristicsvector of ๐ ร 1-dimensional output product of the ๐thstation (this paper takes the length of the printed image asthe quality characteristics), ๐บ denotes a constant coefficientmatrix determined by a production system, ๐๐ is the ๐ ร 1-dimensional fault source vector of the ๐th station, ๐๐ is themeasurement noise of ๐ ร 1 dimension in the ๐th stationand the noise vector which is not included in the model, ๐ isconstant, and then the SoV basic model that reflects therelationship between fault source and product quality isshown in ๐ฟ ๐ = ฮ๐๐ + ๐๐. (1)
From (1), it is obvious that ๐ฟ ๐, the product qualitycharacteristic, becomes abnormal immediatelywhen the faultsource is affected by abnormal factors.
Therefore, the definition in this paper is as follows: ifthe fault source ๐๐ is not affected by abnormal factors, then
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Mathematical Problems in Engineering 3
the manufacturing process is under controlled state, and thedistribution of the fault source vector obeys๐๐ โผ ๐(0, ฮฃ๐) (ฮฃ๐is the covariance matrix of ๐๐). The manufacturing processis considered under uncontrolled state if the expectationof fault source๐๐ or the covariance matrix is shifting or ifexpectation of fault source ๐๐ is shifting simultaneously withthe covariance matrix. Due to the length limitation of thisarticle, only the anomalies caused by the expected offset of๐๐ are discussed in this paper; at this time, the distributionof fault source vectors obeys ๐๐ โผ ๐(๐๐, ฮฃ๐) (๐ โฅ ๐),where ๐๐ (๐๐ ฬธ= 0) is the mean vector of ๐๐ after anomaliesoccur.
3.1. SoVModel under Controlled State. Because the R2Rman-ufacturing process is affected by the internal system factors,the process output of each station is not independent; thatis, the quality data of the previous station affect the qualitydata of the next station, so that there is autocorrelation amongthe data.
If ฮฆ represents the unit diagonal matrix, V๐ representsa random number matrix that is independent and normaldistribution, and the fault source of the current station islinearly related to the fault source of the previous station, then๐๐ = ฮฆ๐๐โ1 + V๐. Thus assuming V๐ and ๐๐ are independentof each other and obey normal distributions V๐ โผ ๐(0, ฮฃV),๐๐ โผ ๐(0, ฮฃ๐) the distribution of ๐๐ obeys ๐๐ โผ ๐(0, ฮฃ๐), thedistribution of ๐ฟ ๐ obeys ๐ฟ ๐ โผ ๐(0, ฮฃ๐ฟ), and the SoV modelunder the controlled state of R2R manufacturing system isobtained as follows:๐ฟ ๐ = ฮ๐๐ + ๐๐๐๐ = ฮฆ๐๐โ1 + V๐ฮฃ๐ฟ = ฮฮฆ๐ฮฃ๐ (ฮฮฆ๐) + ฮฮฆ๐โ1ฮฃV (ฮฮฆ๐โ1) + โ โ โ + ฮฮฆฮฃV (ฮฮฆ) + ฮฮฃV (ฮ) + ฮฃ๐.
(2)
Further derivation is as follows:๐ฟ ๐ = ฮ๐๐ + ๐๐= ฮ (ฮฆ๐๐โ1 + V๐) + ๐๐= ฮฮฆ๐๐โ1 + ฮV๐ + ๐๐= ฮฮฆ (ฮฆ๐๐โ2 + V๐โ1) + ฮV๐ + ๐๐= ฮฮฆ2๐๐โ2 + ฮฮฆV๐โ1 + ฮV๐ + ๐๐...= ฮฮฆ๐๐0 + ฮฮฆ๐โ1๐1 + ฮฮฆ๐โ2๐2 + โ โ โ + ฮฮฆV๐โ1 + ฮV๐+ ๐๐.
(3)
In the equation above, the diagonalmatrixฮฆ indicates therelevant level of the fault source. In general, ฮฆ = ๐๐ผ (โ1
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4 Mathematical Problems in Engineering
4.1. The Distribution Mathematical Expression Derivation ofStatistic ๐2๐,๐. When the mean value of the fault source ๐๐shifts, themean value of the corresponding ๐ฟ ๐ shifts, and eachoffset direction of ๐๐ corresponds to an offset direction of ๐ฟ ๐.It is assumed that a total of ๐ก stations have failure, each stationcorresponds to an offset direction ๐ฟ, the system has a total of๐ก offset directions ๐๐ (๐ = 1, 2, . . . , ๐ก, |๐๐| = 1), and thenaverage offset of ๐๐ is ๐๐,๐ = ๐ฟ๐๐.
When ๐ = ๐, a station failure of R2R manufacturingsystem occurs, the mean of๐๐ shifts, and the change directionof๐ฟ ๐ can be calculated by themean change direction of๐๐withreference to (4); that is, the updated quality characteristicvector ๐ฟ ๐,๐ is obtained:๐ฟ ๐,๐ = ฮ (๐๐ + ๐๐,๐) + ๐๐= ฮฮฆ๐โ๐๐๐ + ฮฮฆ๐โ(๐+1)V๐+1 + ฮฮฆ๐โ(๐+2)V๐+2 + โ โ โ + ฮฮฆV๐โ1 + ฮV๐ + ๐๐ + ฮฮฆ๐โ๐๐๐,๐+ ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ2๐๐,๐ + ฮฮฆ๐๐,๐+ ฮ๐๐,๐.
(6)
Obviously, the mean change of ๐ฟ ๐,๐ is as follows:โ๐ฟ,๐ = ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ2๐๐,๐+ ฮฮฆ๐๐,๐ + ฮ๐๐,๐. (7)Further, the resulting direction vector ๐๐ฟ,๐ of โ๐ฟ,๐ is๐๐ฟ,๐ = โ๐ฟ,๐โ๐ฟ,๐= ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ2๐๐,๐ + ฮฮฆ๐๐,๐ + ฮ๐๐,๐ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ2๐๐,๐ + ฮฮฆ๐๐,๐ + ฮ๐๐,๐= ฮ (๐๐โ๐ + ๐๐โ(๐+1) + โ โ โ + ๐2 + ๐ + 1) ๐ผ๐๐,๐ฮ (๐๐โ๐ + ๐๐โ(๐+1) + โ โ โ + ๐2 + ๐ + 1) ๐ผ๐๐,๐ = ฮ๐๐ฮ๐๐ .
(8)
From (8), ๐๐ฟ,๐ does not relate to ๐ ๐or ๐; it relates to ฮ and๐๐,๐; when designing control charts with statistic, accordingto Hawkinsโs proof [8], the statistic shown in (9) is the mosteffective: ๐2๐,๐ = ๐๐ฟ,๐ โ1โ
๐ฟ
๐ฟ ๐. (9)Because ๐ฟ ๐ obeys distribution ๐ฟ ๐ โผ ๐(๐๐ฟ, ฮฃ๐ฟ), the mean
vector and the covariance matrix are calculated separately,and the distribution of the statistic ๐2๐,๐ is obtained as follows:๐2๐,๐ โผ ๐(๐๐ฟ,๐ โ1โ
๐ฟ
๐๐ฟ, ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ,๐) , ๐ = 1, . . . , ๐ก. (10)Based on the distribution mathematical expression of
statistic ๐2๐,๐, the equation of control centerline, upper limit,and lower limit under controlled or uncontrolled state incontrol chart is going to be derived.
4.2. The Equation Derivation of Centerline, the Upper ControlLimit, and the Lower Control Limit in Control Chart. (1)When ๐ < ๐ system is under controlled state, ๐๐ and ๐ฟ ๐ obeythe distributions ๐๐ โผ ๐(0, ฮฃ๐), ๐ฟ ๐ โผ ๐(0, ฮฃ๐ฟ), and then thedistribution of statistic ๐2๐,๐ is
๐2๐,๐ โผ ๐(0, ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ,๐) . (11)Since ๐๐ has ๐ก possible directions of change, so ๐ก pieces
of control chart are needed to monitor the changes of ๐๐; ifthere is an alarm in a control chart, it indicates that the man-ufacturing process is under uncontrolled state. In order toestablish Shewhart control charts [9, 10] for ๐2๐,๐, the cen-terline, upper control limit, and lower control limit of eachcontrol chart are shown in
CL๐ = ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ = 0UCL๐ = ๐๐ฟ,๐ โ1โ
๐ฟ
๐๐ฟ + ๐ด๐๐2๐,๐
= ๐ด๐๐2๐,๐
LCL๐ = ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ โ ๐ด๐๐2๐,๐
= โ๐ด๐๐2๐,๐,
(12)
where๐ = 1, 2, . . . , ๐ก, ๐ด is control limit factor, and ๐๐2๐,๐is the
standard deviation of statistic ๐2๐,๐.(2) When ๐ = ๐ the fault has just started and ๐๐ obeys the
distribution ๐๐ โผ ๐(๐๐, ฮฃ๐), then the distribution of statistic๐2๐,๐ is shown in๐2๐,๐ โผ ๐(๐๐ฟ,๐ โ1โ
๐ฟ
(ฮ๐๐) , ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ,๐) . (13)The centerline, the upper control limit, and the lower
control limit of each control chart are shown in
CL๐ = ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ = ๐๐ฟ,๐ โ1โ๐ฟ
(ฮ๐๐)UCL๐ = ๐๐ฟ,๐ โ1โ
๐ฟ
๐๐ฟ + ๐ด๐๐2๐,๐
= ๐๐ฟ,๐ โ1โ๐ฟ
(ฮ๐๐) + ๐ด๐๐2๐,๐
LCL๐ = ๐๐ฟ,๐ โ1โ๐ฟ
๐๐ฟ โ ๐ด๐๐2๐,๐
= ๐๐ฟ,๐ โ1โ๐ฟ
(ฮ๐๐) โ ๐ด๐๐2๐,๐.
(14)
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Mathematical Problems in Engineering 5
(3) When ๐ > ๐ the fault continues to occur and ๐๐ obeysthe distribution ๐๐ โผ ๐(๐๐, ฮฃ๐), then the distribution ofstatistic ๐2๐,๐ is shown in
๐2๐,๐ โผ ๐(๐๐ฟ,๐ โ1โ๐ฟ
(ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ๐๐,๐ + ฮ๐๐,๐) , ๐๐ฟ,๐ โ1โ
๐ฟ
๐๐ฟ,๐) . (15)Hence the centerline, the upper control limit, and the
lower control limit of each control chart are shown in
CL๐ = ๐๐ฟ,๐ โ1โ๐ฟ
(ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ๐๐,๐ + ฮ๐๐,๐)UCL๐ = ๐๐ฟ,๐ โ1โ
๐ฟ
(ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ๐๐,๐ + ฮ๐๐,๐) + ๐ด๐LCL๐ = ๐๐ฟ,๐ โ1โ
๐ฟ
(ฮฮฆ๐โ๐๐๐,๐ + ฮฮฆ๐โ(๐+1)๐๐,๐ + โ โ โ + ฮฮฆ๐๐,๐ + ฮ๐๐,๐) โ ๐ด๐.
(16)
The coordinate system is established on the centerline,the upper control limit, and the lower control limit cal-culated according to (12), (14), and (16), respectively, andthe manufacturing process data are collected and tracedin the coordinate system; then the control chart design iscompleted.
5. Verification Experiment
5.1. Control Chart Performance Verification Experiment. ARLis used as a measure of control chart performance [11,12]. ARL of control chart under controlled state ARLin =โโ๐=1 ๐(1โ๐1)๐โ1๐1 = 1/๐1, where๐1 is alarmprobability of the๐กth control chart; ARL of control chart under uncontrolledstate ARLout = 1 ร ๐2 + (1 โ ๐2) โโ๐=2 ๐(1 โ ๐3)๐โ2๐3 =(1 โ ๐2 + ๐3)/๐3, where ๐2 indicates the alarm probabilityof each control chart at the beginning of the fault when ๐ = ๐,๐3 indicates probability of alarm of each control chart when๐ > ๐, the fault continues to occur, and the mean value of thefault source ๐ changes while the covariance matrix does notchange.
There are 4 directions of fault source (๐ = 4); using theMento Carlo method to simulate the autocorrelation data ofmultistation manufacturing process [13], the validity of theproposed control chart is verified.
A Taking control coefficient ๐ด = 3 and according to3 ๐ principle, the average length of the operation undercontrolled state is calculated as ARLin = 1/๐1 = 370.40, ๐, ฮ,๐ are randomly generated by computer numerical simulation
10008006004002000steps
150
200
250
300
350
400
450
ARL
Figure 3: Distribution of ARLin under controlled state.
010008006004002000
80
60
40
20
ARL
steps
Figure 4: Distribution of ARLin under uncontrolled state.
software, wherein ๐ โ (0, 1), ฮ is ๐ก ร ๐ก dimension, ๐ is ๐ก ร 1dimension, and the result data are as follows: ๐ = 0.4, ๐ =(0.0856 โ0.2793 0.8457 0.3587)๐:
ฮ = ( 0.7256 โ1.5284 1.2148 โ0.83691.5243 1.3692 โ0.7648 0.80520.2968 1.2658 โ0.6839 1.1583โ0.9254 โ0.4835 1.0258 0.4236 ) . (17)The distribution of ARLin in controlled state shown in
Figure 3 is further obtained.As analysis from Figure 3, when the program runs to
1000 steps, ARLin = 369.254, which is basically the same asits theoretical value. Although the previous period fluctuatesgreatly, the program becomes stable after 500 steps.
B Using the above randomly generated ๐, ฮ, ๐, if theexpected length of change in size ๐ฟ = 2, then ๐๐ =2๐. At this moment, multistation manufacturing process isunder uncontrolled state; that is, a fault has occurred; aftercalculating ARLout = (1 โ ๐2 + ๐3)/๐3 = 56.87, the ARLoutdistribution chart under uncontrolled state shown in Figure 4is obtained.
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6 Mathematical Problems in Engineering
Tachometer
Unwinding roller
Flexible film
Guide roller
Tension sensor Tension sensor Industrial camera
Driving roller
Winding roller
Flexible film
Figure 5: R2Rmanufacturing unwinding-conveyor-winding exper-imental device.
Tachometer Tension sensor
Station 5Station 4Station 3Station 2Station 1
Figure 6: Experimental platform working principle diagram.
As Figure 4, when the program runs to step 1000,ARLout = 57.48 which is relatively close to theoretical value.Similar to the above, the previous period fluctuates greatly,and the program becomes stable after 500 steps.
5.2. R2R Manufacturing System Fault Diagnosis Experi-ment. Theflexiblematerial R2Rmanufacturing experimentalequipment shown in Figure 5 is built for fault predictiontest; the working principle of the experimental equipmentis shown in Figure 6. This equipment contains unwindingroller module, winding roller module, guide roller module,and driving roller module and each module adopts 120Wspeed motor. The driving module adopts the counter-rollermode, the rubber roller is driven by the motor to rotate,the rubber roller is controlled by a hand-held lifting handle,and the upper and lower rollers rotate at the same timeto realize material transmission. The drive module adoptsthe symmetrical roller mode: motor drives lower rubberroller to rotate and upper rubber roller is controlled bya hand-held lifting handle. First, lifting the upper rubberroller, the material is flattened after being placed around theroller, and then after accurate alignment the rubber rolleris put down. Finally, the upper and lower rollers start torotate at the same time to realize material transmission. Themaximum adaptation width of the experimental platform is450mm, which can be used to transfer material thicknessof 0.1mmโ5mm. Blue PET polyester film is selected in thisexperiment; the details of this material are as follows: width is
50mm, thickness is 0.05mm, density is 1450 kg/m3, modulusof elasticity is 3495MPa, and Poissonโs ratio is 0.3.
(1) Unwinding process station number ๐ = 4, theunwinding process affected by faults in four directions isdefined, and a control chart of the autocorrelation datamonitoring during the unwinding process is established.
A Combining the SoV model of R2R unwinding processfor flexible material given in reference [14] and (2), thecoefficient matrix of unwinding process ฮ is calculated:
ฮ = (1.0000 0 0 01.0258 1.0000 0 01.1056 0.9583 1.0000 01.1947 1.1056 1.0258 1.0000) (18)B Considering the four-station situation, the direction
of the fault and offset direction of ๐ฟ ๐ are determined: ๐1 =(1, 0, 0, 0)๐, ๐2 = (0, 1, 0, 0)๐, ๐3 = (0, 0, 1, 0)๐, and ๐4 =(0, 0, 0, 1)๐.C From (8), ๐๐ฟ,๐ = ฮ๐๐/|ฮ๐๐|, ๐๐ฟ,๐ is calculated,
according to the statistic mentioned in (9), CL๐, UCL๐, andLCL๐ of the control chart are determined, and the data areobtained in Table 1.
D According to CL๐, UCL๐, and LCL๐ of the controlchart, the output ๐ฟ ๐ of each station in the unwinding processis monitored. When the data point exceeds the control limitarea, the control chart will create alarm.
E The theoretical and measured values of ARLin andARLout are calculated by the computer numerical simulationsoftware when calculating different values of ๐ and ๐๐. Fromthe previous section, it is known that ARLin = 370.40, andits actual values are shown in Table 2. It is clear that thecharacteristics of the control chart at this moment are basi-cally consistent with the conclusions obtained in the previoussimulation analysis, thereby verifying the effectiveness of thecontrol chart.
F For the unwinding process under uncontrolled state,the unwinding process is supposed to be out of control atthe beginning. The ARL of the control chart is shown inFigure 6 under different correlation coefficient ๐ and faultsource offset ๐ฟ.
From Figure 7, it is obvious that when the fault sourceoffset is the same, the larger the fault source correlation coeffi-cient is, the larger the ARL of the control graph will be.Whenthe fault source has weak correlation, the performance ofcontrol chart is very good; however, when the fault source hasa strong correlation, although the performance of the controlchart decreases slightly, an alarm will occur quickly. Withthe same correlation coefficient, the larger the fault offsetis, the faster the control chart will create alarm. As a result,which station has failed can be determined and staffmemberscan obtain reference information without checking on thestations one by one.
6. Conclusion
The product quality of the R2R manufacturing process notonly relates to the input and output variables of the system,
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Mathematical Problems in Engineering 7
Table 1: The calculated values of control limits for each fault direction.
Station number ๐๐ ๐๐ฟ,๐ CL๐ UCL๐ LCL๐1 (1, 0, 0, 0)๐ (0.4612, 0.4731, 0.5099, 0.5510)๐ 0 1.059 โ1.0592 (0, 1, 0, 0)๐ (0, 0.5643, 0.5407, 0.6239)๐ 0 1.153 โ1.1533 (0, 0, 1, 0)๐ (0, 0, 0.6980, 0.7161)๐ 0 1.428 โ1.4284 (0, 0, 0, 1)๐ (0, 0, 0, 1)๐ 0 1.736 โ1.736
Table 2: The actual value of ARLin when the program runs to 1000 steps under controlled state and different values of ๐ and ๐๐.๐๐ ๐โ0.7 โ0.5 โ0.3 โ0.1 0.1 0.3 0.5 0.7๐1 365.84 368.84 376.02 371.35 369.07 373.56 365.93 369.24๐2 376.92 368.24 369.02 367.45 368.85 371.75 370.86 374.24๐3 369.57 368.56 368.85 365.47 373.46 370.75 369.34 371.43๐4 368.30 369.43 372.46 371.53 370.29 369.24 369.74 367.87
= 1
= 2
= 3
= 4
0
20
40
๏ผ๏ผ๏ผ๏ผฉ๏ผฏ๏ผฎ
60
80
100
120
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1
Figure 7: ARLout under different values of ๐ and ๐ฟ.but also relates to the real-time status of the system. Thispaper combines the theoretical method of SoV and controlchart and obtains quality data from different stations basedon SoV model to establish the control chart of system faultdiagnosis.
The length ๐ฟ ๐ of printed image is used as a qualitycharacteristic. ๐๐ denotes the fault source vector of a station, ๐denotes the measurement noise and the noise vector which isnot included in themodel, and the SoVbasicmodel reflectingthe relationship between the fault source and the productquality of the manufacturing process is established as ๐ฟ ๐ =ฮ๐๐ + ๐๐. Combining the statistical distribution rules of ๐ฟ ๐and ๐๐ under controlled state and uncontrolled state, thebasic model is expanded, respectively; finally the probabilitydistribution๐2๐,๐ of ๐ฟ ๐โs statistic is obtained.With reference tothis, the calculation equation of the centerline, upper limit,and lower limit of the control chart are deduced.
The ARL is used as a measure of the control chartperformance to conduct the verification experiment. Theresults show that the actual value of the ARL is basically thesame as the theoretical value when the program runs to 1000steps under controlled or uncontrolled state. Although the
fluctuations in the previous period are large, they becomestable after 500 steps. The fault diagnosis experiment of R2Rmanufacturing system shows that when the fault source has aweak correlation, the control chart performance is very good;when the fault source has a strong correlation, althoughthe control chart performance decreases slightly, the alarmoccurs quickly; when the correlation coefficient of the faultsource is the same, the greater the offset of the fault sourceis, the faster the control chart creates alarm. As a result, it issimple and fast to locate the system stationwhich has a failure.
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request.
Conflicts of Interest
The authors declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.
Acknowledgments
The authors disclosed receipt of the following financialsupport for the research, authorship, and/or publicationof this article. This work was supported in part by theNational Natural Science Foundation of China under Grantno. 51675109, Natural Science Foundation of GuangdongProvince, China, under Grant no. 2017A030313308, andProvincial Science and Technology Plan Project of Guang-dong Province, China, under Grant no. 2016B010124002 andin part by the Provincial Science and Technology Plan ofGuangdong Province, China, under Grant no. 2017B010117011and Grant no. 2016B010112003.
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