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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

Outline1. Introduction: What should we control?2. Self-optimizing Control3. Maximum Gain Rule4. Application: Indirect control of Distillation Column5. Combination of Measurements6. 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 matrixS1 – output scalingS2 = 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

TTc

HTcT

é ùê úê úé ù ê úê ú= ê úê úë û ê úê úë û

0.0152 0.0010 0.0020 0.00140.0009 0.0008 0.0016 0.0148

Hé 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