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E. S. Hori, Maximum Gain Rule
Maximum Gain Rule for Selecting Controlled Variables
Eduardo Shigueo Hori, Sigurd SkogestadNorwegian University of Science and Technology – NTNUN-7491 Trondheim, Norway
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E. S. Hori, Maximum Gain Rule
Outline
1. Introduction: What should we control?
2. Self-optimizing Control
3. Maximum Gain Rule
4. Application: Indirect control of Distillation Column
5. Combination of Measurements
6. Conclusions
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E. S. Hori, Maximum Gain Rule
Optimal operation of Sprinter (100m)
• Objective function J=T• What should we control ?
– Active constraint control:• Maximum speed (”no thinking required”)
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E. S. Hori, Maximum Gain Rule
Optimal operation of Marathon runner
• Objective function J=T
• Unconstrained optimum• What should we control?
– Any ”self-optimizing” variable c (to control at constant setpoint)?
• c1 = distance to leader of race
• c2 = speed
• c3 = heart rate
• c4 = ”pain” (lactate in muscles)
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E. S. Hori, Maximum Gain Rule
2. What is a good variable c to control?
• Self-optimizing control… is when acceptable operation can be achieved using constant
set points (cs)
for the controlled variables c (without the need
for re-optimizing when disturbances occur).
• Desirable properties for a ”self-optimizing” CV (c) :- Small optimal variation (”obvious”)
- Large sensitivity (large gain from u to c) (ref. Moore, 1992)
- Small implementation error (”obvious”)
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E. S. Hori, Maximum Gain Rule
How do we find ”self-optimizing” variables in a systematic manner?
• Assume cost J determined by steady-state behavior• Effective tool for screening: MAXIMUM GAIN RULE
c – candidate controlled variable (CV)u – independent variable (MV)G – steady-state gain matrix (c = G u)
G’ = S1 G S2 - scaled gain matrix
S1 – output scaling
S2 = Juu-1/2 – input ”scaling”
• Maximum gain rule: Maximize• This presentation: Importance of input scaling, S2 = Juu
-1/2
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E. S. Hori, Maximum Gain Rule
u
cost J
uopt
c = G u
Halvorsen, I.J., S. Skogestad, J. Morud and V. Alstad (2003). ”Optimal selection of controlled variables”. Ind. Eng. Chem. Res. 42(14), 3273–3284.
3. Maximum Gain Rule: Derivation
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E. S. Hori, Maximum Gain Rule
3. Maximum Gain Rule: Derivation (2)
Maximum Gain Rule Simplified Maximum Gain Rule
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E. S. Hori, Maximum Gain Rule
3. Maximum Gain Rule: Output Scaling S1
• The outputs are scaled with respect to their ”span”:
( ){ }1 1 iS diag span c=
( )= + = +opt. var. (d) implem. error opt ci i i
span c c n
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E. S. Hori, Maximum Gain Rule
3. Maximum Gain Rule: Input Scaling S2
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E. S. Hori, Maximum Gain Rule
4. Application: indirect control
Selection/Combination of measurements
Primary variables
Disturbances
Measurements
Noise
Inputs
Constant setpoints
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E. S. Hori, Maximum Gain Rule
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E. S. Hori, Maximum Gain Rule
Column Data
• Column A:
- Binary mixture
- 50% light component
- AB = 1.5
- 41 stages (total condenser)
- 1% heavy in top
- 1% light in bottom
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E. S. Hori, Maximum Gain Rule
Application to distillation
Selection/Combination of measurements,e.g. select two temperatures
Primary variables: xHtop, xL
btm
Disturbances: F, zF, qF
Measurements:All T’s + inputs (flows)
Noise (meas. Error)0.5C on T
Inputs: L, V
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E. S. Hori, Maximum Gain Rule
Distillation Column: Output Scaling S1
( ){ }1 1 iS diag span c=
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E. S. Hori, Maximum Gain Rule
Distillation Column: Input Scaling S2=Juu-1/2
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E. S. Hori, Maximum Gain Rule
Distillation Column: Maximum Gain rule
• Select two temperatures (symmetrically located)
• This case: Input scaling (Juu-
1/2) does not change order….
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E. S. Hori, Maximum Gain Rule
Distillation Column: Maximum Gain rule and effect of Input Scaling
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E. S. Hori, Maximum Gain Rule
5. Linear combination of Measurements
• Consider temperatures only (41):
Nullspace method: Possible to achive no disturbance loss : – Need as many measurements as u’s + d’s: need 4 T’s
• Two-step approach (”nullspace method”):
1. Select measurements (4 T’s): Maximize min. singular value of
2. Calculate H-matrix that gives no disturbance loss:
1y y y y
dS G G G Gé ù¢ ¢ ¢= =ê úë û% %
†1H GG= %%
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E. S. Hori, Maximum Gain Rule
5. Combination of Measurements
,40%
,75%1
,85%2
,40%
b
b
t
t
T
TcH
Tc
T
é ùê úê úé ùê úê ú= ê úê úë û ê úê úë û
0.0152 0.0010 0.0020 0.0014
0.0009 0.0008 0.0016 0.0148H
é ù- -ê ú=ê ú- -ë û
2. Same 4 T’s, but minimize for both d and n: J=0.58
1. Nullspace method:
Composition deviation: J=0.82 (caused by meas. error n )
Alternative approaches:
3. Optimal combination of any 4 T’s: J=0.44 (branch & bound; Kariwala/Cao)
4. Optimal combination of all 41 T’s: J=0.23
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E. S. Hori, Maximum Gain Rule
6. Conclusions
• Identify candidate CVs
• Simplified Maximum Gain Rule,
- easy to apply – Juu not needed
- usually good assumption
• Maximum Gain Rule:
- results very close to exact local method (but not exact)- better for ill-conditioned plants
