Process modelling and optimization aid FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering

Process modelling and optimization aid

FONTEIX ChristianProfessor of Chemical Engineering

Polytechnical National Institute of Lorraine

Chemical Engineering Sciences Laboratory

Process modelling and optimization aid

Parametric identificationfrom experimental data

FONTEIX ChristianProfessor of Chemical Engineering

Polytechnical National Institute of Lorraine

Chemical Engineering Sciences Laboratory

Parametric identificationLikelyhood Maximum Method

• Number of measurement i={1, 2, …, nj}

• Number of component j={1, 2, …, m}example : concentration, temperature, molecular weight…

• Operating conditions (ith measurement of jth component) :example for continuous experiments : time of measurement

• Measured value (ith measurement of jth component) :• Corresponding predicted value :• Corresponding unknown true value :• Parameters unknown true values : *• Measurement error :

ˆ y j x ij ,

ˆ y j x ij ,

ij y ij ˆ y j x ij ,

x ij

y ij


• Measurement error modelling (replications) : independant gaussian errors with average =0

multiplicative errors :

additive errors : • Probability density :

• Likelyhood maximum :

• Identified parameters values :

V j jˆ y j x ij ,

2 i

V j constan t j i

1

2V j

exp ij

2

2V j

i1

n j

j1

m

Ln

0 Ln V j

0 j

ˆ


• Likelyhood function :

• Estimation of the measurement errors :

• Parameters estimation :

•

ˆ V j 1

n j

y ij ˆ y j x ij ,ˆ 2

i1

n j

Ln

01ˆ V jj1

m

y ij ˆ y j x ij , 2

i1

n j

ˆ

Ln1

2

1

V j

y ij ˆ y j x ij, 2

i1

n j

j1

m

1

2n jLn 2V j

j1

m


• Parameters estimation by minimization of :

• Total number of freedom degree :total number of measurements - parameters number - variances number + 1

• Unbiased estimation of measurement error variances :

n jLn y ij ˆ y j x ij , 2i1

n j

j1

m

n j

j1

m

n m 1nm n m 1

ˆ V UBj nm

nm n m 1ˆ V j


• Example 1 : unknown average and variance of n gaussian hazards yi

• Parametric identification :

• Likelyhood maximum method gives biased variance :

A y i m A y i m 2 V A y i m y j m T 0j i i

Ln1

2Vy i m 2 n

2i1

n

Ln 2V

LnV

01

2 ˆ V 2y i ˆ m 2

i1

n

n

2 ˆ V ˆ V

1

ny i ˆ m 2

i1

n

Lnm

01ˆ V

y i ˆ m ˆ m 1

ni1

n

y i

i1

n

A ˆ m m A ˆ V n 1

nV ˆ V UB

1

n 1y i ˆ m 2

i1

n

numberof freedomdegreen 1"m" 1"V"1n 1


• Example 2 : unknown C (and T), measurement of P and T

• Parametric identification :

PV nRT P CRT unknown var iancesVP andVT

2Ln Pm1 CRT1 2 Pm2 CRT2 2 2Ln Tm1 T1 2 Tm2 T2 2

C04R

ˆ T 1 ˆ C R ˆ T 1 Pm1 ˆ T 2 ˆ C R ˆ T 2 Pm 2 Pm1 ˆ C R ˆ T 1 2 Pm 2 ˆ C R ˆ T 2 2

ˆ C ˆ T 1Pm1 ˆ T 2Pm2

R ˆ T 12 ˆ T 2

2 T1

04ˆ C R ˆ C R ˆ T 1 Pm1

Pm1 ˆ C R ˆ T 1 2 Pm 2 ˆ C R ˆ T 2 2 4

ˆ T 1 Tm1

Tm1 ˆ T 1 2 Tm2 ˆ T 2 2

T2

04ˆ C R ˆ C R ˆ T 2 Pm 2

Pm1 ˆ C R ˆ T 1 2 Pm 2 ˆ C R ˆ T 2 2 4

ˆ T 2 Tm1

Tm1 ˆ T 1 2 Tm2 ˆ T 2 2


• Example 3 : terpolymerization in tubular reactors • (69 parameters) styrene/a-methylstyrene/acrylic acid

• 1rst step : simultaneously indentification of 23 parameters (3 times)

• 2nd step : simultaneously indentification of the 69 parameters

Parametric identificationParameters confidence domain

• Vectors of nm elements (independant for e) :

• Confidence domain calculation :projection of e on a tangential plane to the modelb is the projection of e and h the distance between experiments and modele2 = b2 + h2

b and h are independant (orthogonal)

ey ij ˆ y j x ij ,

V j

numberof freedom deg ree nm

ˆ e y ij ˆ y j x ij ,

ˆ V j

numberof freedomdeg ree nm n m 1

e2 ij2

i1

n j

j1

m

2 nm

h2 n j

ˆ V jV jj1

m

2 nm n m 1

b2 e2 h2 2 n m 1

b2

n m 1h2

nm n m 1

F n m 1,nm n m 1


Measurement 1

Measurement 2Experimental point

ˆ

ˆ h

e

b

h Model

ˆ e ˆ h

ˆ b 0

h ˆ h


• Definitions and properties :

• Fisher Snedecor test for confidence domain :

n jLn n j˜ V j

j1

m

with ˜ V j 1

n j

y ij ˆ y j x ij , 2i1

n j

n jLn n j˜ V j V j V j

j1

m

n jLn n jV j j1

m

n jLn 1˜ V j V j

V j

j1

m

n jLn n jV j j1

m

n j

˜ V j V j

V jj1

m

n jLn n jV j j1

m

nm n j

˜ V j V jj1

m

n j˜ V j

V j

n j

ˆ V jV jj1

m

j1

m

n j

ˆ V jV jj1

m

n m 1

nm n m 1F n m 1,nm n m 1


• Determination of parameters confidence domain :

• 1rst to identify the estimated parameters by optimization

• 2nd to determine the confidence domain of parameters by optimization of the same function than in identification

ˆ nm n m 1 nm n m 1

F n m 1,nm n m 1

n jLn y ij ˆ y j x ij , 2

i1

n j

j1

m

n jLn y ij ˆ y j x ij ,ˆ 2

i1

n j

j1

m

nm n m 1 nm n m 1

F n m 1,nm n m 1


• Example 1 : speed identification of a bullet

1rst measurement : length of shot2nd measurement : time to reach this length

• Y1 and Y2 are the coordinates of the projection of the measurements on the tangent plane to the model

ˆ y 1 2sin cos v 2

gˆ y 2 t

2v sing

v Ln y1 sin2 v 2

g

2

Ln y2 2sin v

g

2

ˆ v gy2

6sin gy2

6sin

2

gy1

3sin2


• Example 1 :

Y2 2v sin

g

Y1 sin2 v 2

g

1

2v cos

Y2 y2

Y1 y1

2v cos

h2 Y1 y1 2 Y2 y2 2

b2 Y1 sin2 v 2

g

2

Y2 2v sin

g

2


• Example 2 : application to a simple enzymatic reaction• An enzyme E with a substrate S transitorily gives a

specific complex enzyme-substrate C before the researched product P (Michaelis-Menten kinetics)

d[S]

dt k1[E][S] k2[C]

d[C]

dtk1[E][S] k2[C] k3[C]

[E][C][E]0 [C]0


• Example 2 : application to a simple enzymatic reaction

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

k2

k1

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

k2

k3

Parametric identificationIdentification quality

• Parameter estimation by evolutionary algorithm (or genetic) = set of solutions (defined number)

• Confidence domain determination by evolutionary algorithm = set of solutions (defined number) with end test by corresponding Fischer Snedecor test

• Set of parameters vector (confidence domain representation) = possibility to calculate correlations between parameters

• Detection of high correlations


• To vizualize the confidence domain = projection of the solutions set on 2 parameters space

• Non elliptical confidence domain = non linear model

• Estimated parameters not at the confidence domain center = non linear model

• Correlations between parameters can be reduced by new experiments


Confidence interval

Parameter 1

Parameter 2

ReducedConfidenceintervalEstimated

parameter

Confidence domain

Inclined confidence domain = correlationbetweenthe 2parameters


• Confidence interval = overall range of the corresponding parameter

• Reduced confidence interval = range of the parameter when the others take their estimated value

• If the reduced confidence interval contains the 0 value, the corresponding parameter is not significantly different to 0

• When a parameter is not significantly different to 0, a model reduction is possible


• Comparison between experimental data with corresponding simulations from model and estimated parameters

• Confidence interval of model prediction from Student test :

• If an experimental data is not include in the corresponding confidence interval the measurement is maybe deviating

ˆ y j x ij ,ˆ ˆ V UBj St

n j

nm

nm n m 1

y ij ˆ y j x ij ,

ˆ ˆ V UBj Stn j

nm

nm n m 1


Estimatedvalue

Measured value

Confidence intervalof model predictions

Experimental dataversus prediction

Deviating data


Example : Modelling of polymer blend Young modulusCorrelations between parameters

1

-0.3218 1

-0.4498 -0.1414 1

-0.5395 0.0884 0.2565 1

ˆ -0.0013 -0.0296 -0.0831 -0.0520 1

0.3759 -0.1188 -0.1042 -0.15

-0

94 -0.053 1

0.1398 0.1321 -0.0.9134 4

r a

73 -0.168 -0.1239 1

0.3834 0.0869 -0.3047 0.051 0.1186 -0.0716 1

0.3843 -0.1316 -0.2107 0.074 -0.1471 0.1609 0.2541

-0.9602

-0.9345 1

2

3

4

5

9

12

13

14

2 3 4 5 9 12 13 14

o

o

a

a

a

a

a

a

a

a

a

a a a a a a a a a


Example : Modelling of polymer blend Young modulusComparison between experimental and calculated young modulus values

300

350

400

450

500

550

600

650

300 350 400 450 500 550 600Predicted (MPa)

Me

sure

d (

MP

a)

Confidence interval

Idenfication experiments

Validation experiments

E

Documents

Process modelling and optimization aid FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering