1 E. S. Hori, Maximum Gain Rule Maximum Gain Rule for Selecting Controlled Variables Eduardo Shigueo Hori, Sigurd Skogestad Norwegian University of Science and Technology – NTNU N-7491 Trondheim, Norway

2 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

3 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”)

4 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)? c 1 = distance to leader of race c 2 = speed c 3 = heart rate c 4 = ”pain” (lactate in muscles)

5 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 (c s ) 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”)

6 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’ = S 1 G S 2 - scaled gain matrix S 1 – output scaling S 2 = J uu -1/2 – input ”scaling” Maximum gain rule: Maximize This presentation: Importance of input scaling, S 2 = J uu -1/2

7 E. S. Hori, Maximum Gain Rule u cost J u opt 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– Maximum Gain Rule: Derivation

8 E. S. Hori, Maximum Gain Rule 3. Maximum Gain Rule: Derivation (2) Maximum Gain Rule Simplified Maximum Gain Rule

9 E. S. Hori, Maximum Gain Rule 3. Maximum Gain Rule: Output Scaling S 1 The outputs are scaled with respect to their ”span”:

10 E. S. Hori, Maximum Gain Rule 3. Maximum Gain Rule: Input Scaling S 2

11 E. S. Hori, Maximum Gain Rule 4. Application: indirect control Selection/Combination of measurements Primary variables Disturbances Measurements Noise Inputs Constant setpoints

12 E. S. Hori, Maximum Gain Rule

13 E. S. Hori, Maximum Gain Rule Column Data Column A: - Binary mixture - 50% light component -  AB = stages (total condenser) - 1% heavy in top - 1% light in bottom

14 E. S. Hori, Maximum Gain Rule Application to distillation Selection/Combination of measurements, e.g. select two temperatures Primary variables: x H top, x L btm Disturbances: F, z F, q F Measurements: All T’s + inputs (flows) Noise (meas. Error) 0.5C on T Inputs: L, V

15 E. S. Hori, Maximum Gain Rule Distillation Column: Output Scaling S 1

16 E. S. Hori, Maximum Gain Rule Distillation Column: Input Scaling S 2 =J uu -1/2

17 E. S. Hori, Maximum Gain Rule Distillation Column: Maximum Gain rule Select two temperatures (symmetrically located) This case: Input scaling (J uu -1/2 ) does not change order….

18 E. S. Hori, Maximum Gain Rule Distillation Column: Maximum Gain rule and effect of Input Scaling

19 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:

20 E. S. Hori, Maximum Gain Rule 5. Combination of Measurements 2. Same 4 T’s, but minimize for both d and n: J= 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

21 E. S. Hori, Maximum Gain Rule 6. Conclusions Identify candidate CVs Simplified Maximum Gain Rule, - easy to apply – J uu not needed - usually good assumption Maximum Gain Rule: - results very close to exact local method (but not exact) - better for ill-conditioned plants