bayesian filtering of "smearing effect" -- fault isolation in chemical process monitoring
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Bayesian Filtering of "Smearing Effect" --Fault Isolation in
Chemical Process Monitoring
Jia Lin LiuCenter for Energy and Environmental ResearchNational Tsing Hua University, Hsinchu, Taiwan
David Shan Hill WongDepartment of Chemical Engineering
National Tsing Hua University, Hsinchu, Taiwan
Shujie LiuDepartment of Control Science and Engineering
Hua Zhong University of Science and Technology, Wuhan, China
NCTS-IS Workshop26th October, 2012
CONTENT
Fault Isolation Relative Contribution and Smearing Effect Reconstructive Based Contribution Bayesian Decision Applications Conclusions
NCTS-IS WORKSHOP2012.10.26
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MONITORING, DIAGNOSIS AND ISOLATION
The possibility of using multivariate statistical analysis to monitor manufacturing processes have been extensively researched.
For example, PCA have been widely practiced to project sensor data in high dimension to a latent structure. Hotelling T2 or Q statistics can be used to monitor whether the process is in control.
If a faulty signal appears, it is desirable to diagnosis the root cause of the fault.
Efficient diagnosis is facilitated by isolation of major contributing variables NCTS-IS WORKSHOP
2012.10.263
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
X2
X1
99% Confidence Limit
PC1PC2
SUPERVISED APPROACH (1/2)
Fault Signatures1 Projecting each known event data onto the PC and residual subspaces, the fault
signatures of the two subspaces can be obtained. The detected faults are decomposed into two subspaces and inner product with each fault signature are calculated.
Fuzzy Logic Knowledge-based Expert Systems2
Generating fuzzy rules from different operational-mode data, the new data were classified into the known groups according the fuzzy rules.
Modified Fault Tree Analysis (FTA)3
Match the trend patterns of the candidates with the standard fault propagation trends to identify the root causes.
Possibilistic c-means4
Separate the known event data into groups, and to classify new data into groups according to the membership values.
SUPERVISED APPROACH (2/2)
Bayesian Classification5 Cluster data into the denser regions, and faults were identified according to the
posterior probabilities. Support Vector Machine (SVM)6
Building decision boundaries between two groups of data from different operating modes, the new fault was tested for each SVM.
Pattern-matching Approach7, 8
Several PCA models were built using known event datasets. The statistical distances and angles of the new data were measured with each group.
Fault Subspace Extraction9
Each fault subspace was extracted from each known event dataset. The detected fault can be identified by minimizing the reconstructed statistics.
An event set must be available. A new fault may lead to incorrect diagnosis.
UNSUPERVISED APPROACH -- DISCRIMINATION BASED Pairwise Fisher Discriminant Analysis10
The pairwise FDA was then applied to the normal data and each class of faulty data to find fault directions that were used to generate contribution plots for isolating faulty variables.
Dissimilarities Between Normal and Abnormal Groups11
The dissimilarities between normal and abnormal cluster centers and covariances are measured. The faulty variables of new faults can be isolated using the maximal values of the dissimilarities.
This type of approach is based on a restrictive assumption that the faulty data can be formed into groups.
Sufficient data of the faulty group must be accumulated to correctly isolate the faulty variables.
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SampleNCTS-IS WORKSHOP
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PROPAGATION OF SIGNALS DUE TO CONTROL ACTION
CAF
LT
FTFC
TTTC
QFTF
QCTCF
T
h
QCTC
LC
FC FT
CAQ
CAF
LT
FTFC
TTTC
QFTF
QCTCF
T
h
QCTC
LC
FC FT
CAQ
LC
FC FT
CAQ
0 2 4 6 8 10 12 14 16 18 20 22 24400
401
402
403
404
405
406
T (
K)
Hour
0 2 4 6 8 10 12 14 16 18 20 22 2414
15
16
17
QC (
L/m
in)
Hour
0 2 4 6 8 10 12 14 16 18 20 22 24342
343
344
345
346
347
TC (
K)
Hour
0 2 4 6 8 10 12 14 16 18 20 22 2436
38
40
42
CA (
mol
/m3)
Hour
1. Adding a bias of 1 K to the measurement of the reactor temperature after the eighth hour.
2. The coolant flow rate was increasing for compensating this abnormality.
4. The actual reactor temperature was lower than its set point; therefore, the reactant concentration would be higher than the normal operating data due to the lower reaction rate.
3. The excess of the coolant flow rate induced the coolant exit temperature to be lower than its normal operating values.
1
2
3
4
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UNSUPERVISED APPROACH, CONTRIBUTION BASED
Contribution plots The most popular tool for identifying which variables are pushing the statistics out of
their control limits. It is well known that this approach suffers from the smearing effect. Reconstruction-based Contribution (RBC)12
The RBC differs from traditional contributions by a scaling factor that also appears in the corresponding control limits. Therefore, The RBC approach still suffers the smearing effect when implementing the control limits on the RBC.
Branch and Bound (BAB) method13
The time-consuming task of continuously optimizing the nonlinear integer programming problem for every sampling data is needed.
Contribution of the Reduced Statistics14
Repeatedly insert a variable with the maximal reduction of the statistics into the faulty variable set until the reconstructed statistics under the control limits. Since the selected faulty variables do not equally contribute to the faults, contribution plots of the reduced statistics are used to find the faulty variables with the most contributions.
ISOLATION BY CONTRIBUTION
Isolation is the procedure of identifying the variables contributing to a fault signal detected by multivariate analysis
This is normally by relative contribution in engineering literature
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T T T1
1m
S X X PΛP PΛP
×N MRX
T T ˆ X XPP XPP X E
TT TQ x x x x xPP x
2 1 T T 1 TT xPΛ P x tΛ t
2 2Q Q T T
2 TQ Qi i i
i
c , ,Q c xCξ C PP
0 52 1 T 2.T Ti i i
i
c , ,T c xDξ D PΛ P
2 2i i i
i
c , Q T , c xΦξ Φ C D
is a column vector in which the ith element is one and the others are zero.
iξ
CONTROL LIMITS OF CONTRIBUTIONSBOX 1954
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2T T TQi i i ic x Cξ x Cξ ξ Cx
2Q Q Q,i i ig h
22 TT
2T T1
i ii iQ Qi i
i i i i
trtrg ,h
tr tr
SCξ ξ CSCξ ξ C
SCξ ξ C SCξ ξ C
2T 0 5 T 0 5 T 0 5T . . .i i i ic x D ξ x D ξ ξ D x
2T T T,i i ig h
20 5 T 0 5
T 0 5 0 5
0 5 T 0 5
. .i iT . .
i i i. .i i
trg
tr
SD ξ ξ Dξ D SD ξ
SD ξ ξ D
1Tih
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T2
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cT 2
INCONSISTENT DIAGNOSIS (1/2)
Signaling of the overall process may be caused by signaling of individual variables Signaling of overall process may not induce signals of individual variables Signaling of individual variables do not guarantee signaling of the overall limit
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00000000000
INCONSISTENT DIAGNOSIS (2/2)
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cT 1
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cT 2
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cT 2
SMEARING EFFECT (1/2)
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VARIABLE CONTRIBUTIONS OF CSTR EXAMPLE
CAF
LT
FTFC
TTTC
QFTF
QCTCF
T
h
QCTC
LC
FC FT
CAQ
CAF
LT
FTFC
TTTC
QFTF
QCTCF
T
h
QCTC
LC
FC FT
CAQ
LC
FC FT
CAQ
0 2 4 6 8 10 12 14 16 18 20 22 24
1
2
3
4
5
6
7
8
9
Hour
Var
iabl
e 5%
10%
20%
40%
80%
T
CA
TC
QC
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RECONSTRUCTION BASED CONTRIBUTION ALCALA AND QIN AUTIOMATICA 2009
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, ,1
mi 1 i 1 i 1 2
xx =argmin m x , x y x x m=Q,T,
‘
, , , ,mi 1 i 1 i i 1 2 1 i 1 i i 1 2RBC =m x , x x x x m x , x x x x
m=Q,T,
mm ii m
ii
cRBC =
Cm 2 2mj i ijm m m 2 mi
i j ij ii i i ii jm mii jj jj
c x Ccx x C C RBC = =x C RBC = =
C C C
2 m mii jj ij i jC C C RBC RBC
CONTROL LIMITS OF RELATIVE CONTRIBUTIONS
NCTS-IS WORKSHOP2012.10.26
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2 1T T T T1Qi i i i i i
i ,i
RBCc
x Cξ x Cξ ξ Cξ ξ Cx
2
21T T
1T T
21T T
21T T
1
1
RQ RQ RQ,i i i
i i i iRQ Qi i
i ,ii i i i
i i i iRQi
i i i i
g h
trg g
ctr
trh
tr
SCξ ξ Cξ ξ C
SCξ ξ Cξ ξ C
SCξ ξ Cξ ξ C
SCξ ξ Cξ ξ C
Q Qi iQ RQ
,i ,i
c RBC
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2 1T T T T1Ti i i i i i
i ,i
RBCd
x Dξ x Dξ ξ Dξ ξ Dx
2
21T T
T T 1 T
1T T
21T T
21T T
1 1
1
RT RT RT,i i i
i i i iRTi i i i i
i ,i i ,ii i i i
i i i iRTi
i i i i
g h
trg
d dtr
trh
tr
SDξ ξ Dξ ξ Dξ DSDξ ξ PΛ P ξ
SDξ ξ Dξ ξ D
SDξ ξ Dξ ξ D
SDξ ξ Dξ ξ D
2T 1 T
0 52 T 1 T
1
1
Ti i
.RT,i i i
RBC
x PΛ P ξ
ξ PΛ P ξ
T Ti iT RT
,i ,i
c RBC
BAYESIAN DECISION
1
1 11
j jj
The posterior probability of class w after an observation x
P Pw wxP x =w
w wP Px
,1 2Let w w be a set of decisions classes
11P is the prior probability of class w w
11P x is the conditional probability of observation x given class w w
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BAYESIAN FILTERING APPLIED TO FAULT ISOLATION
NCTS-IS WORKSHOP2012.10.26
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,
, , .
m mi iLet N F be two classes of decisions for variable i,normal, fault
according respective indices m T Q
m mi t i t
t
P F and P N be the posterior probabilities
of the two classes after observation was recieved.
x x
x
m mi i
t
P F and P N be the prior probabilities
of the two classes before observation was recieved.
x
m mt ti i
tht
P and P are the conditional probabilitiesF N
of observing given the i variable is
faulty or normal respectively
x x
x
CONDITIONAL PROBABILITY BASED ON RBC
For the ith variable there are two classes, in fault (Fi) or normal (Ni).
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min max min mi
mt i z
1P =P + P P F
1 e
x
max
.mi
i m
RBC0 5z s
RBC
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Con
ditio
nal P
roba
bilit
ies
Ci / C
max
max
mi
m
RBC
RBC
mt i
P Fx
EVOLUTION OF POSTERIOR
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REVISIT THE CSTR EXAMPLE
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Hour
Var
iab
le 51%
60%
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90%
Bay
esia
n I
nfe
ren
ce
Plo
t
T
CA
TC
QC
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FAULT 1 OF TE PROCESS
Fault 1: A/C feed ratio changes and B composition remains constant (Stream 4)
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iab
le 51%
60%
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90%
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COMPOSITION A CONTROLLER IN REACTOR FEED
The scenario of Fault 1 was that the composition of A of Stream 4 was changed from 48.5 mol% to 45.5 mol%; meanwhile, the composition of C was changed from 51 mol% to 54 mol%.
Stream 1 flow rate (x1) was increasing through opening the valve (x44) and trying to maintain the composition A in the reactor feed flow
.
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Sample
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FAULT 7 OF TE PROCESS
Fault 7: C header pressure loss – reduced availability (Stream 4)
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Hour
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le 0
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Sample
Var
iabl
e 51%
60%
70%
80%
90%
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ROOT CAUSES OF FAULT 7
Since the C header pressure loss could be compensated by increasing the open position of the feed flow valve of Stream 4 (x45), the process would be gradually settled down by the controllers.
Comparing the symptoms of Fault 1 and Fault 7, since the scenario of Fault 7 did not change any compositions in the streams, the diagnosis of Fault 7 was relatively easier than that of Fault 1.
0 200 400 600 80055
60
65
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90
x 45
Sample
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FAULT EVOLUTION OF FAULT 7
Since the condenser cooling water flow rate (x52) almost kept in a constant range, the temperature of the reactor outlet passed through the condenser would be inversely proportional to the flow rate of the output, i.e., the reactor pressure. Therefore, the separator temperature (x11) inversely varied with the reactor pressure. The function of the separator was to separate produces G and H from the reactor outlet. In addition, the vapor pressure of component G was higher than that of component H; therefore, composition G in the purge (x35) was more sensitive with the separator temperature.
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0 40 80 120 160 200 240 280 320 36075
78
81
84
x 11
Sample
0 40 80 120 160 200 240 280 320 3604
5
6
x 35
Sample
0 40 80 120 160 200 240 280 320 36020
22
24
26
x 52
Sample
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Composition E Controllers in Product Flow
160 200 240 280 320 360
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50
Sample
Va
ria
ble 51%
60%
70%
80%
90%
The variation of Stream 4 flow rate (x4) resulted in the variation of composition C in the reactor; therefore, composition E in the Product flow (x38) would be varied as well. The controllers of x38 was trying to maintain the set point that induced the variations of x50, x19 and x18.
29
INDUSTRIAL APPLICATION C
ooli
ng T
ower
Tc, 1
1st Stage 2nd Stage 3rd Stage 4th Stage
1st Intercooler 2nd Intercooler 3rd Intercooler
Cooling Water
Air
Compressed Air
Fa
Pin, 1
Tin, 1
Pout, 1
Tout, 1
Pin, 2
Tin, 2
Pout, 2
Tout, 2
Pout, 3
Tout, 3
Pout, 4
Tout, 4
Pin, 3
Tin, 3
Pin, 4
Tin, 4
Fc ,Tc
Tc, 2 Tc, 3
The compression process used a four-stage centrifugal compressor that was equipped with an intercooler between stages to cool down the compressed air.
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FAULT DETECTION AND ISOLATION
31
0 1 2 3 4 50
2
4
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10
Com
bine
d In
dex
Day
0 1 2 3 4 5
2
4
6
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22
Day
Var
iabl
e 0
1
2
4
Bay
esia
n I
nfe
ren
ce
Plo
tR
BC
Im
ple
men
tin
g C
on
tro
l L
imit
1st AE
2nd AE
3rd AE
0 1 2 3 4 5
2
4
6
8
10
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14
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22
Day
Var
iabl
e 51%
60%
70%
80%
90%
FIRST ABNORMAL EVENT
0 1 2 3 4 5
2
4
6
8
10
12
14
16
18
20
22
Day
Va
ria
ble 51%
60%
70%
80%
90%
The measurements of Tout,1 were compared with the averages of the training and test data, from which the sensor drift can be observed. Calibration of the sensor was requested by the field operators after they were informed about this abnormality.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Test Data
Measurements Averages
Pin
, 1
Day
Training Data
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SECOND ABNORMAL EVENT
0 1 2 3 4 5
2
4
6
8
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Day
Va
ria
ble 51%
60%
70%
80%
90%
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Measurements Averages
To
ut,
2
Day
Training Data Test Data
0 1 2 3 4 5
0.8
0.9
1.0 1st Stage 2nd Stage 3rd Stage 4th Stage
Com
pres
sion
Effi
cien
cy
Day
The left figure shows that it was difficult to convince the operators that the measurements of the sensors were questionable, since the two temperature averages were close. However, they were convinced after the comparison of the compression efficiencies for all stages was displayed, as the right figure shows. The second stage efficiency dramatically surged around day 1.5, and then highly fluctuated; meanwhile the other stages’ efficiencies were maintained in stable ranges. NCTS-IS WORKSHOP
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THIRD ABNORMAL EVENT
0 1 2 3 4 5
2
4
6
8
10
12
14
16
18
20
22
Day
Va
ria
ble 51%
60%
70%
80%
90%
The cooling water flow rate suddenly dropped at day 3.4, bounced back to a lower flow rate, and then dropped again around day 3.5. Since the intercoolers were arranged in a series, the decreases of the cooling water flow rate apparently did not affect the function of the first intercooler; contrarily, the performances of the second and third intercoolers deteriorated due to the insufficient cooling water.
3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.03.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
Day
Flo
w R
ate
Cooling Water
3rd Stage 4th Stage
Out
let
Tem
pera
ture
2nd Intercooler 3rd Intercooler
Out
let
Tem
pera
ture
Day
3rd Stage 4th Stage
Inle
t T
empe
ratu
re
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CONCLUSIONS
Bayesian inference-based fault isolation is derived
Smearing effect of traditional contributions and RBC is eliminated.
Predefined known event datasets are not necessary.
Fault propagation due to the process controllers can be traced.
Multiple sensor faults can be identified.
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REFERENCES
S. Yoon, J.F. MacGregor, Fault diagnosis with multivariate statistical models part I: using steady state fault signatures, J Proc. Cont. 11 (2001) 387-400.
E. Musulin, I. Yélamos, L. Puigjaner, Integration of principal component analysis and fuzzy logic systems for comprehensive process fault detection and diagnosis, Ind. Eng. Chem. Res. 45 (2006) 1739-1750.
Y.S. Oh, K.J. Mo, E.S. Yoon Fault diagnosis based on weighted symptom tree and pattern matching, Ind. Eng. Chem. Res. 36 (1997) 2672-2678.
K.P. Detroja, R.D. Gudi, S.C. Patwardhan, A possibilistic clustering approach to novel fault detection and isolation, J. Proc. Cont. 16 (2006) 1055-1073.
J. Liu, Process monitoring using Bayesian classification on PCA subspace, Ind. Eng. Chem. Res. 43 (2004) 7815-7825.
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REFERENCES, CONT.
6. Y.H. Chu, S.J. Qin, C. Han, Fault detection and operation mode identification based on pattern classification with variable selection, Ind. Eng. Chem. Res. 43 (2004) 1701-1710.
7. A. Raich, A. Çinar, Statistical process monitoring and disturbance diagnosis in multivariable continuous processes, AIChE J. 42 (1996) 995-1009.
8. M.C. Johannesmeyer, A. Singhal, D.E. Seborg, Pattern matching in historical data, AIChE J. 48 (2002) 2022-2038.
9. R. Dunia, S.J. Qin, Subspace approach to multidimensional fault identification and reconstruction, AIChE J. 44 (1998) 1813-1831.
10. Q.P. He, S.J. Qin, J. Wang, A new fault diagnosis method using fault directions in Fisher discriminant analysis, AIChE J. 51 (2005) 555-571.
11. J. Liu, D.S. Chen, Fault detection and identification using modified Bayesian classification on PCA subspace, Ind. Eng. Chem. Res. 48 (2009) 3059-3077.
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REFERENCES, CONT.
12. C.F. Alcala, S.J. Qin, Reconstruction-based contribution for process monitoring, Automatica 45 (2009) 1593-1600.
13. V. Kariwala, P.E. Odiowei, Y. Cao, T. Chen, A branch and bound method for isolation of faulty variables through missing variable analysis, J. Proc. Cont. 20 (2010) 1198-1206.
14. J. Liu, Fault diagnosis using contribution plots without smearing effect on non-faulty variables, J. Proc. Cont. (2012) http://dx.doi.org/10.1016/j.jprocont.2012.06.016
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Thank you for your attentions !Questions?
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