Loss Method: A Static Estimator Applied for Product ...folk.ntnu.no/skoge/publications/2011/ghadrdan-aiche-annual-2011/... · Loss Method: A Static Estimator Applied for Product Composition

Loss Method: A Static Estimator Applied for ProductComposition Estimation From Distillation Column

Temperature Pro�le

M. Ghadrdan1 C. Grimholt1 S. Skogestad1 I.J. Halvorsen2

1Norwegian University of Science & Technology, Department of Chemical Engineering,

7491, Trondheim, Norway

2SINTEF ICT, Applied Cybernetics, 7465 Trondheim, Norway

AIChE Meeting, 2011

M. Ghadrdan, C. Grimholt, S. Skogestad, I.J. Halvorsen (Norwegian University of Science & Technology, Department of Chemical Engineering, 7491, Trondheim, Norway, SINTEF ICT, Applied Cybernetics, 7465 Trondheim, Norway)Loss Method AIChE Meeting, 2011 1 / 27

Motivation

Some process variables can not be measured frequently

Example: Composition measurement using online analyzers (like GasChromatograph)

Large measurement delaysHigh investment/maintenance costsLow reliability

Sensors:

TemperaturePressureFlow rateetc.

An estimator attempts to approximate the unknown parameters using themeasurements


Motivation




Sensors:




Motivation




Sensors:




Outline

1 IntroductionEstimationPartial Least Squares

2 Loss MethodOptimal estimators for di�erent scenariosNecessary data for the task of estimation

3 Examples


Introduction Estimation

Estimators

Di�erent categories: Static / Dynamic, Data-based / Model-based,Open-loop / Close-loop

Static Estimators

Model-based

Example: Brasilow estimator1

Our method is in this category

Data-based

Example: Partial Least Square (PLS)

Dynamic Estimators

Model-based

Example: Kalman �lter

Data-based

Time variant reliability analysis of existing structures using data

1R. Weber, C. Brosilow, The Use of Secondary Measurements to Improve Control, AIChE J., 18, 3,

p. 614-623



Estimators


Static Estimators

Model-based



Data-based


Dynamic Estimators

Model-based


Data-based

Time variant reliability analysis of existing structures using data

1R. Weber, C. Brosilow, The Use of Secondary Measurements to Improve Control, AIChE J., 18, 3,

p. 614-623



Estimators


Static Estimators

Model-based



Data-based


Dynamic Estimators

Model-based


Data-based

Time variant reliability analysis of existing structures using data1R. Weber, C. Brosilow, The Use of Secondary Measurements to Improve Control, AIChE J., 18, 3,

p. 614-623


Introduction Partial Least Squares

Partial Least Squares

PC regression = weights are calculated from the covariance matrix ofthe predictors

PLS = weights re�ect the covariance structure between predictors andresponse � mostly requires a complicated iterative algorithm

Nipals and SIMPLS algorithms probably most common

The goal is to maximize the correlation between the response(s) andcomponent scores

PLS can extends to multiple outcomes and allows for dimensionreduction

No collinearity � Independence of observations not required


Introduction Partial Least Squares

PLS

Y = BX

PLS: is not optimal for any particular problem

Loss method: optimal for certain well-de�ned problems


Loss Method

Loss Method

OBJECTIVE

The main objective is to �nd a linear combination of measurements suchthat keeping these constant indirectly leads to nearly accurate estimationwith a small loss L in spite of unknown disturbances, d, and measurementnoise, nx .

minH‖e‖2 = ‖y− y‖2


Loss Method

Loss Method

"Open-loop" (for thepurpose of Monitoring):

1 No control (u is a freevariable)

2 Primary variables y arecontrolled (u is used tokeep y = ys).

3 Secondary variables zare controlled (u isused to keep z = zs).

"Close-loop" (for thepurpose of Control)

PlantH

PlantH

PlantH

PlantH

d

d

d

d

u

u

u

u

y

y

y

y

x

x

x

x

K

K

K

+

+

+

+

nx

nx

nx

nx

ŷ

ŷ

ŷ

ŷ

xm

xmys

-

z

-zs xm

xmys

-


Loss Method

Loss Method

Assumption: Linear models for the primary variables y, measurements x,and secondary variables z

y = Gyu +Gdy d

Gy =(

∂y∂u

)d, Gd

y =(

∂y∂d

)u

x = Gxu +Gdx d

Gx =(

∂x∂u

)d, Gd

x =(

∂x∂d

)u

z = Gzu +Gdz d

Gz =(

∂z∂u

)d, Gd

z =(

∂z∂d

)u

The actual measurements xm, containing measurement noise nx is

xm = x + nx

The linear estimator is of the form

y =Hxm


Loss Method

Loss Method

Gx

Gdx

Hŷ

d

nx

xxmLinear model from inputs

to measurements

Linear model from disturbances to measurements

Gy

Gdy

Linear model from inputs to primary

variables

Linear model from disturbances to primary

variables

u

y

+-

prediction error

Gx

Gdx

Hŷ

d

nx


to measurements


Gy

Gdy

Linear model from inputs to primary variables


variables

u

y

+-

prediction error


Loss Method

Loss Method

Gx

Gdx

Hŷ

d

nx


to measurements


Gy

Gdy

Linear model from inputs to primary

variables


variables

u

y

+-

prediction error

Gx

Gdx

Hŷ

d

nx


to measurements


Gy

Gdy

Linear model from inputs to primary variables


variables

u

y

+-

prediction error


Loss Method Optimal estimators for di�erent scenarios

Optimal estimators for di�erent scenarios (Loss Method)

"Open-loop" 1

H1 = Y1X†1

Y1 =[GyWu GdyWd 0

]X1 =

[GxWu GdxWd Wnx

]2

"Open-loop" 3

H3 = Y3X†3

Y3 =[Gcly Wzs F′yWd 0

]X3 =

[Gclx Wzs F′xWd Wnx

]

"Open-loop" 2

H2 = Y2X†2

Y2 =[Wys 0 0

]X2 =

[Gclx Wys FWd Wnx

]4

"Closed-loop"

minH

∥∥H[ FWd Wnx]∥∥

F

s.t.HGx = Gy

* All subject to the constraint of independent variables values



Optimal "open-loop" estimator, when y=ys (Loss Method)Plant

B

PlantB

PlantB

PlantB

d

d

d

d

u

u

u

u

y

y

y

y

x

x

x

x

K

K

K

+

+

+

+

nx

nx

nx

nx

ŷ

ŷ

ŷ

ŷ

xm

xmys

-

z

-

zs xm

xmys

-

H2 = Y2X†2

Y2 =[Wys 0 0

]X2=

[Gclx Wys FWd Wnx

]InitialEquations

y = Gyu+Gdy dx = Gxu+Gdx dxm = x+nx

y =Hxm

u =G−1y ys −G−1y Gdy d

y =H[GxG

−1y ys +

(Gdx −GxG

−1y G

dy

)d+nx

]

e =[ (I−HGcl

x

)Wys −HFWd −HWnx

]︸︷︷︸M

ol(H)

y’sd’nx′

‖e(H)‖2 =1

2‖Mol (H)‖2F

minH

‖[Wys 0 0

]−H

[Gclx Wys FWd Wnx

]‖=min

H

‖Y2−HX2‖M. Ghadrdan, C. Grimholt, S. Skogestad, I.J. Halvorsen (Norwegian University of Science & Technology, Department of Chemical Engineering, 7491, Trondheim, Norway, SINTEF ICT, Applied Cybernetics, 7465 Trondheim, Norway)Loss Method AIChE Meeting, 2011 12 / 27



B

PlantB

PlantB

PlantB

d

d

d

d

u

u

u

u

y

y

y

y

x

x

x

x

K

K

K

+

+

+

+

nx

nx

nx

nx

ŷ

ŷ

ŷ

ŷ

xm

xmys

-

z

-

zs xm

xmys

-

H2 = Y2X†2

Y2 =[Wys 0 0

]X2=

[Gclx Wys FWd Wnx

]InitialEquations


y =Hxm

u =G−1y ys −G−1y Gdy d

y =H[GxG

−1y ys +

(Gdx −GxG

−1y G

dy

)d+nx

]

e =[ (I−HGcl

x

)Wys −HFWd −HWnx

]︸︷︷︸M

ol(H)

y’sd’nx′

‖e(H)‖2 =1

2‖Mol (H)‖2F

minH

‖[Wys 0 0

]−H

[Gclx Wys FWd Wnx

]‖=min

H

‖Y2−HX2‖M. Ghadrdan, C. Grimholt, S. Skogestad, I.J. Halvorsen (Norwegian University of Science & Technology, Department of Chemical Engineering, 7491, Trondheim, Norway, SINTEF ICT, Applied Cybernetics, 7465 Trondheim, Norway)Loss Method AIChE Meeting, 2011 12 / 27



B

PlantB

PlantB

PlantB

d

d

d

d

u

u

u

u

y

y

y

y

x

x

x

x

K

K

K

+

+

+

+

nx

nx

nx

nx

ŷ

ŷ

ŷ

ŷ

xm

xmys

-

z

-

zs xm

xmys

-

H2 = Y2X†2

Y2 =[Wys 0 0

]X2=

[Gclx Wys FWd Wnx

]a

InitialEquations


y =Hxm

d =

∥∥∥∥∥∥ u′

d′

nx′

∥∥∥∥∥∥∼N(0, Inu+n

d+nx

)

‖e(H)‖2,exp =1

2E[tr(MddTMT

)]=

1

2tr(M

TME

[ddT

])E[ddT

]= Cov

(d, d)+µµ

T

minH‖[Wys 0 0

]−H

[Gclx Wys FWd Wnx

]‖=min

H‖Y2−HX2‖



Optimal "close-loop" estimator (Loss Method)

PlantB

PlantB

PlantB

PlantB

d

d

d

d

u

u

u

u

y

y

y

y

x

x

x

x

K

K

K

+

+

+

+

nx

nx

nx

nx

ŷ

ŷ

ŷ

ŷ

xm

xmys

-

z

-

zs xm

xmys

-

minH


F

s.t.HGx = Gy

InitialEquations


y =Hxm

u =−(HGx )−1H

(Gdx d+nx

)+(HGx )

−1 ys

y =−Gy (HGx )−1H

Gdx −GxG−1y G

dy︸︷︷︸

F

d+nx

+Gy (HGx )−1 ys

e = y− y = y−ys =−Gy (HGx )−1H(Fd+nx )+[Gy (HGx )

−1− I]

ys



Optimal "close-loop" estimator (Loss Method)

PlantB

PlantB

PlantB

PlantB

d

d

d

d

u

u

u

u

y

y

y

y

x

x

x

x

K

K

K

+

+

+

+

nx

nx

nx

nx

ŷ

ŷ

ŷ

ŷ

xm

xmys

-

z

-

zs xm

xmys

-

minH


F

s.t.HGx = Gy

InitialEquations


y =Hxm

u =−(HGx )−1H

(Gdx d+nx

)+(HGx )

−1 ys

y =−Gy (HGx )−1H

Gdx −GxG−1y G

dy︸︷︷︸

F

d+nx

+Gy (HGx )−1 ys

e = y− y = y−ys =−Gy (HGx )−1H(Fd+nx )+[Gy (HGx )

−1− I]

ys



Optimal "close-loop" estimator (contd.)

The prediction error e

e = y− y = y−ys =−Gy (HGx)−1H(Fd + nx) +[Gy (HGx)−1− I

]ys

Introducing the normalized variables:

e =−Gy (HGx)−1H[FWd Wnx

][ d′

nx′

]︸︷︷︸

e1

+[Gy (HGx)−1− I

]ys︸︷︷︸

e2

Degree of Freedom

e1 (H) = e1 (DH)



Optimal "close-loop" estimator (contd.)

If F=[FWd Wnx

]is full rank, which is always the case if we include

independent measurement noise, then 2

H=D

((XoptX

Topt

)−1Gx

)T

where

D= Gy

(GTx

(XoptX

Topt

)−1Gx

)−1

2Alstad et al. (2009), Optimal measurement combinations as controlled variables, J. Proc. Control,

19 (1), 138-148


Loss Method Necessary data for the task of estimation

Necessary data for the task of estimation (Model-based)

Model-Based Estimation

Yall =

[Y

X

]=

[Gy 0Gx Xopt

]where Xopt =

[FWd Wnx

]aa

Y =[Ynon−opt 0

]X =

[Xnon−opt Xopt

]



Necessary data for the task of estimation (Data-based)

Theorem

Closed Loop Regressor (CLR) a. The data matrices can be transformed to the�optimal � non-optimal� structure by

1 Performing a singular value decomposition on the data matrix Y

2 Multiplying the data matrices X and Y with the unitary matrix V

YV=[Ynon−opt 0

]XV=

[Xnon−opt Xopt

]aSkogestad et al (2011). Selected Topics on Constrained and Nonlinear Control Workbook

Proof.

Since V is unitary, so ‖YV−HXV‖F = ‖Y−HX‖FWriting the unitary matrix U in block form as U=

[U1 U2

], we will have

YV=US=[U1 U2

][ S10

]=[U1S1 0

]



Necessary data for the task of estimation (Data-based)

Theorem

Closed Loop Regressor (CLR) a. The data matrices can be transformed to the�optimal � non-optimal� structure by

1 Performing a singular value decomposition on the data matrix Y

2 Multiplying the data matrices X and Y with the unitary matrix V

YV=[Ynon−opt 0

]XV=

[Xnon−opt Xopt

]aSkogestad et al (2011). Selected Topics on Constrained and Nonlinear Control Workbook

Proof.

Since V is unitary, so ‖YV−HXV‖F = ‖Y−HX‖FWriting the unitary matrix U in block form as U=

[U1 U2

], we will have

YV=US=[U1 U2

][ S10

]=[U1S1 0

]M. Ghadrdan, C. Grimholt, S. Skogestad, I.J. Halvorsen (Norwegian University of Science & Technology, Department of Chemical Engineering, 7491, Trondheim, Norway, SINTEF ICT, Applied Cybernetics, 7465 Trondheim, Norway)Loss Method AIChE Meeting, 2011 18 / 27

Examples

Example 1: Results

Binary Distillation (Col. A), 41 trays, 8 measurements

Secondary variables: Re�ux, temperature in 25th tray

The mean prediction error of the model-based estimators applied to fouroperation scenarios

Validation Data

CaliberationData S1 S2 S3 S4S1 0.0085 0.2749 0.0215 0.0506S2 0.0591 0.0093 0.0104 0.0104S3 0.0599 0.0166 0.0098 0.0132S4 0.0099 0.0099 0.0099 0.0099


Examples

Example 1: Results

0 5 10 15 20 25 30 35 404

5

6

7

8

9

10

11

12

13

14

x 10−3

Number of X variables (Temperature measurements)

Med

ian

of 1

50 p

redi

ctio

n er

ror

norm

s

H2

H4

Bls

Bclr

Bpcr

Bpls

B†clr

Median prediction error for 150 data set with 200 samples


Examples

Example 2: Multi-component distillation

ABCD

ABCD

ABCD

1234...

36

Estimator

PID

PID

u = y =[xC3 inD xC2 inB

]Gy = I

Gdx = F

Gdy = 0


Examples

Example 2: Results

H=

0.0004 0.00140.0081 −0.0045−0.005 0.0074−0.0047 0.00060.0062 −0.0104−0.003 0.0126−0.0013 0.00510.0024 −0.0162−0.0028 0.0042

a

B=

0.0002 0.00130.0087 −0.0041−0.006 0.0068−0.0051 0.00030.0077 −0.0096−0.0034 0.0124−0.0016 0.00490.0026 −0.016−0.0031 0.004


Examples

Example 2: Results

0 100 200 300 400 500 600 700 8009.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9x 10

−3

time(min)

xC3

in D

istil

late

Feed disturbance: +5%

y

D, Loss

yD

true value

yD

, PLS

0 100 200 300 400 500 600 700 8000.0375

0.038

0.0385

0.039

0.0395

0.04

0.0405

0.041

0.0415

0.042

time(min)

xC2

in B

ot

Feed disturbance: +5%

x

B, Loss

xB true value

xB, PLS

(a) +5% disturbance in feed �ow

0 100 200 300 400 500 600 700 800

8.5

9

9.5

10x 10

−3

time(min)

xC3

in D

istil

late

zF1 disturbance: −1%

y

D, Loss

yD

true value

yD

, PLS

0 100 200 300 400 500 600 700 8000.037

0.0375

0.038

0.0385

0.039

0.0395

0.04

0.0405

time(min)

xC2

in B

ot

zF1 disturbance: −1%

x

B, Loss

xB

true value

xB

, PLS

(b) -1% disturbance in Feed composition z1,F

Figure: Estimated and model Composition values for the case with twotemperature controls and with the consideration of 8 measurements


Examples

Example 3: Kaibel distillation column

Figure 2. Schematic of a 4-product dividing wall column

The model used for this study is simulated in UNISIM. The feed stream is an equimolal mixture of Methanol, Ethanol, 1-Propanol, 1-butanol and saturated liquid. All the optimal operating points for different sets of the disturbances are found by applying an optimisation solver in MATLAB with the full non-linear model in UNISIM. The right figure in Figure 2 shows the composition profiles in different sections of the dividing-wall column. As it is obvious, the most difficult separation is taking place in the prefractionator and the other sections are performing close to binary separation with small light or heavy impurity. Because of this, the focus of our study is on the prefractionator part. Partial Least Square (PLS) Method In chemometrics, partial least squares (PLS) regression has become an established tool for modelling linear relations between multivariate measurements (Martens and Næs 1989). This biased regression method is used to compress the predictor data matrix 1 2, ,..., pX x x x = , that contains the values of p predictors for n samples, into a set of A latent variable or factor scores 1 2[ , ,..., ]AT t t t= , where A p≤ . The factors at , 1, 2,...,a A= , are determined sequentially using the nonlinear iterative partial least squares (NIPALS) algorithm [2]. The orthogonal factor scores are used to fit a set of n observations to m dependent variables

1 2[ , ,..., ]mY y y y= . The main attraction of the method is that it finds a parsimonious model even when the predictors are highly collinear or linearly dependent. So, the final fitting

D

S1

S2

B

Feed

DoF

u =[RL RV L V S1 S2

]aExtra Degrees of Freedom:

Vapor Split (RV )

Liquid Split (RL)

sDisturbances:

Feed �owrate, composition andquality

Column Pressure

Setpoints for splitsM. Ghadrdan, C. Grimholt, S. Skogestad, I.J. Halvorsen (Norwegian University of Science & Technology, Department of Chemical Engineering, 7491, Trondheim, Norway, SINTEF ICT, Applied Cybernetics, 7465 Trondheim, Norway)Loss Method AIChE Meeting, 2011 24 / 27

Examples

Example 3: Results

0 20 40 60 80 100-8

-7

-6

-5

-4

-3

-2

-1

0

1

2 x 10-3

trays

Hop

t

H1H2H3H4

a

Possible Improvement for Loss method: Structured H3

3Yelchuru et al., MIQP formulation for Controlled Variable Selection in Self Optimizing Control


Examples

Conclusion

Loss method is more systematic method to design soft-sensorcompared to PLS

For the example we showed, PLS and Loss method show almost thesame result although two di�erent approaches are used


Examples

Comment on PLS

Shrinkage properties4

MSE = E (b−β )′S (b−β ) = Σiλi (Eai −αi )2︸︷︷︸

Bias term

+ ΣiλiVar (ai )

ai = f (λi )a0i

f (λi ) = 0or 1 for OLS, PCR, RidgeButler et al.: PLS is not a shrinkage method. PLSR nearly always can beimproved

4Butler et al, The peculier shrinkage properties of partial least squares regression, J. R. Stat. Soc.,

B 62 (2000) 585-593


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Loss Method: A Static Estimator Applied for Product ...folk.ntnu.no/skoge/publications/2011/ghadrdan-aiche-annual-2011/... · Loss Method: A Static Estimator Applied for Product Composition