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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
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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)
