controlled variable and measurement selection
DESCRIPTION
CONTROLLED VARIABLE AND MEASUREMENT SELECTION. Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) Trondheim, Norway Bratislava, Jan. 2011. Outline. Plantwide control. The controlled variables (CVs) interconnect the layers. - PowerPoint PPT PresentationTRANSCRIPT
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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)