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CONTROLLED VARIABLE AND MEASUREMENT SELECTION

Sigurd Skogestad

Department of Chemical EngineeringNorwegian University of Science and Technology (NTNU)Trondheim, Norway

Bratislava, Jan. 2011

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Outline

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cs = y1s

MPC

PID

y2s

RTO

u (valves)

Switch+Follow path (+ look after other variables)

CV=y1 (+ u); MV=y2s

Stabilize + avoid drift CV=y2; MV=u

Min J (economics); MV=y1s

OBJECTIVE

Plantwide control

Process control The controlled variables (CVs)interconnect the layers

MV = manipulated variableCV = controlled variable

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Need to find new “self-optimizing” CVs (c=Hy) in each region of active constraints

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3 unconstrained degrees of freedom -> Find 3 CVs

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2

1

1

Control 3 active constraints

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• Operational objective: Minimize cost function J(u,d)• The ideal “self-optimizing” variable is the gradient (first-order

optimality condition) (Halvorsen and Skogestad, 1997; Bonvin et al., 2001; Cao, 2003):

• Optimal setpoint = 0

• BUT: Gradient can not be measured in practice• Possible approach: Estimate gradient Ju based on measurements y

• Approach here: Look directly for c as a function of measurements y (c=Hy) without going via gradient

Ideal “Self-optimizing” variables

Unconstrained degrees of freedom:

I.J. Halvorsen and S. Skogestad, ``Optimizing control of Petlyuk distillation: Understanding the steady-state behavior'', Comp. Chem. Engng., 21, S249-S254 (1997)Ivar J. Halvorsen and Sigurd Skogestad, ``Indirect on-line optimization through setpoint control'', AIChE Annual Meeting, 16-21 Nov. 1997, Los Angeles, Paper 194h. .

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H

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H

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Optimal measurement combination

H

•Candidate measurements (y): Include also inputs u

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H

Loss method. Problem definition

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Nullspace method

No measurement noise (ny=0)

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Amazingly simple!

Sigurd is told how easy it is to find H

Proof nullspace methodBasis: Want optimal value of c to be independent of disturbances

• Find optimal solution as a function of d: uopt(d), yopt(d)

• Linearize this relationship: yopt = F d

• Want:

• To achieve this for all values of d:

• To find a F that satisfies HF=0 we must require

• Optimal when we disregard implementation error (n)

V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', Ind.Eng.Chem.Res, 46 (3), 846-853 (2007).

No measurement noise (ny=0)

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( , ) ( , )opt optL J u d J u d

Ref: Halvorsen et al. I&ECR, 2003

Kariwala et al. I&ECR, 2008

Generalization (with measurement noise)

21/2 1( )yavg uu F

L J HG HYLoss with c=Hym=0 due to(i) Disturbances d(ii) Measurement noise ny

31( , ) ( , ) ( ) ( ) ( )

2T

opt u opt opt uu optJ u d J u d J u u u u J u u

1

[ ],d n

y yuu ud d

Y FW W

F G J J G

u

J

( )opt ou d

Loss

'd

Controlled variables,c yH

ydG

cs = constant +

+

+

+

+

- K

H

yG y

'yn

c

u

dW nW

d

optu

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1/2 1min ( )yuu FH

J HG HY Non-convex optimization problem

(Halvorsen et al., 2003)

Hmin HY F

st 1/ 2yuuHG J

Improvement 1 (Alstad et al. 2009)

st yHG Q

Improvement 2 (Yelchuru et al., 2010)

Hmin HY F

-1 -1 -1 1 -11 y 1 y y y (H G ) H = (DHG ) DH = (HG ) D DH = (HG ) H

1H DH D : any non-singular matrix

Have extra degrees of freedom

[ ]d nY FW W

Convex optimization

problem

Global solution

- Full H (not selection): Do not need Juu

- Q can be used as degrees of freedom for faster solution- Analytical solution when YYT has full rank (w/ meas. noise):

1/2 1 1 1( ( ) ) ( )yT T y yT TuuH J G Y Y G G Y Y

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Special cases

• No noise (ny=0, Wny=0):

Optimal is HF=0 (Nullspace method)• But: If many measurement then solution is not unique

• No disturbances (d=0; Wd=0) + same noise for all measurements (Wny=I):

Optimal is HT=Gy (“control sensitive measurements”)• Proof: Use analytic expression

'd

ydG

cs = constant +

+

+

+

+

- K

H

yG y

'yn

c

u

dW nW

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'd

ydG

cs = constant +

+

+

+

+

- K

H

yG y

'yn

c

u

dW nW

“Minimize” in Maximum gain rule( maximize S1 G Juu

-1/2 , G=HGy )

“Scaling” S1-1 for max.gain rule

“=0” in nullspace method (no noise)

Relationship to maximum gain rule

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Special case: Indirect control = Estimation using “Soft sensor”

• Indirect control: Control measurements c= Hy such that primary output y1 is constant

– Optimal sensitivity F = dyopt/dd = (dy/dd)y1

– Distillation: y=temperature measurements, y1= product composition

• Estimation: Estimate primary variables, y1=c=Hy

– y1 = G1 u, y = Gy u

– Same as indirect control if we use extra degrees of freedom (H1=DH) such that (H1Gy)=G1

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Current research:“Loss approach” can also be used for Y = data

• Alternative to “least squares” and extensions such as PCR and PLS (Chemiometrics)

• Why is least squares not optimal?– Fits data by minimizing ||Y1-HY||

– Does not consider how estimate y1=Hy is going to be used in the future.

• Why is the loss approach better?– Use the data to obtain Y, Gy and G1 (step 1).

– Step 2: obtain estimate y1=Hy that works best for the future expected disturbances and measurement noise (as indirectly given by the data in Y)

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Loss approach as a Data Regression Method

• Y1 – output to be estimated

• X – measurements (called y before)

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Test 1. Gluten data

• 1 y1 (gluten), 100 x (NIR), 100 data sets (50+50).

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but....

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Test 2. Wheat data

• 2 y1, 701 x , 100 data sets (50+50).

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Test 3. Our own

• 2 y1, 7 x (generated by 2u + 2d),

• 8 or 32 data sets + noisefree as validation

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Conclusion ”Loss regression method”

• Works well but not always the winner– Not surprising, PLS and PCR are well-proven methods...

• Needs sufficent data to get estimate of noise

• If we have model: Very promising as ”soft sensor” (estimate y1)

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The end in Bratislava....

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Self-optimizing control vs. NCO-tracking

1. Find controlled variables c=Hy (inputs u are generated by PI-feedback to keep c=0)

2. c=Hy may be viewed as estimate of gradient Ju

3. Use model offline to find H• HF=0

4. Optimal for modelled (expected) disturbances

5. Fast time scale

1. Find input values , Δu=-β Juu-1 Ju

2. Measure gradient Ju (if possible) or estimate by perturbing u

3. No explicit model needed

4. Optimal also for “new” disturbances

5. Slow time scale

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Self-optimizing control and NCO-tracking

• Similar idea: Use measurements to drive gradient to zero

• Not competing but complementary

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Toy Example

Reference: I. J. Halvorsen, S. Skogestad, J. Morud and V. Alstad, “Optimal selection of controlled variables”, Industrial & Engineering Chemistry Research, 42 (14), 3273-3284 (2003).

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Toy Example: Single measurements

Want loss < 0.1: Consider variable combinations

Constant input, c = y4 = u

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Toy Example

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Conclusion

• Systematic approach for finding CVs, c=Hy

• Can also be used for estimation

S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), 219-234 (2004).

V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', Ind.Eng.Chem.Res, 46 (3), 846-853 (2007).

V. Alstad, S. Skogestad and E.S. Hori, ``Optimal measurement combinations as controlled variables'', Journal of Process Control, 19, 138-148 (2009)

Ramprasad Yelchuru, Sigurd Skogestad, Henrik Manum , MIQP formulation for Controlled Variable Selection in Self Optimizing Control IFAC symposium DYCOPS-9, Leuven, Belgium, 5-7 July 2010

Download papers: Google ”Skogestad”

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Other issues

• C=Hy. Want to use as few measurements y as possible (RAMPRASAD)

• The c’s need to be controlled by some MV. Would like to use local measurements (RAMPRASAD)

• Simplify switching: Would like to use same H in several regions? (RAMPRASAD)

• Nonlinearity (JOHANNES)