1. Multivariable systems Relative Gain Array (RGA)
Distillation Control Course
December 4-8, 2004
Multivariable systems Relative Gain Array (RGA)

2. 2. MIMO (multivariable case)
Direct Generalization:
“Increasing L from 1.0 to 1.1 changes yD from 0.95 to 0.97, and xB from 0.02 to 0.03”
“Increasing V from 1.5 to 1.6 changes yD from 0.95 to 0.94, and xB from 0.02 to 0.01”
Steady-State Gain Matrix
(Time constant 50 min for yD)
(time constant 40 min for xB)
April 4-8, 2004
KFUPM-Distillation Control Course

3. Effect of Feedback Controlys
C(s) u
G(s) y
G(s): process (distillation column)
C(s): controller (multivariable or single-loop PI’s)
y: output , ys: setpoint for output u: input d: effect of disturbance on output
April 4-8, 2004
KFUPM-Distillation Control Course

4. KFUPM-Distillation Control Course
Negative feedback
With feedback control Eliminate u(s) in (1) and (2), ys=0
S(S)=“sensitivity function” = supression disturbances
April 4-8, 2004
KFUPM-Distillation Control Course

5. KFUPM-Distillation Control Course
In practice: Cannot do anything with fast changes, that is, S(jw)= I at high frequency
S is often used as performance measure
One Direction
Log- scale 1
0.1
Small resonance peak (small peak = large GM and PM)
Bandwidth (Feedback no help)
Slow disturbance d1: 90% effect on y2 removed
w (log)
w B τ
April 4-8, 2004
KFUPM-Distillation Control Course

6. KFUPM-Distillation Control Course
“Summing up of Channels”
“maximum singular value” (worst direction)
0.1 1
w (log) wB
“minimum singular value” (best direction)
 (s)
 (S)
April 4-8, 2004
KFUPM-Distillation Control Course

7. Analysis of Multivariable processes (Distillation Columns)
What is different with MIMO processes to SISO:
The concept of “directions” (components in u and y have different magnitude”
Interaction between loops when single-loop control is used
INTERACTIONS
Process Model y1 u1
g11
g12
g21 y2 G u2
g22
April 4-8, 2004
KFUPM-Distillation Control Course

8. Consider Effect of u1 on y1
“Open-loop” (C2 = 0): y1 = g11(s)·u1
Closed-loop” (close loop 2, C2≠0)
Change caused by “interactions”
April 4-8, 2004
KFUPM-Distillation Control Course

9. KFUPM-Distillation Control Course
Limiting Case C2→∞ (perfect control of y2)
How much has “gain” from u1 to y1 changed by closing loop 2 with perfect control
The relative Gain Array (RGA) is the matrix formed by considering all the relative gains
Example from before
April 4-8, 2004
KFUPM-Distillation Control Course

10. KFUPM-Distillation Control Course
Property of RGA:
Columns and rows always sum to 1
RGA independent of scaling (units) for u and y.
Note: RGA as a function of frequency is the most important for control!
April 4-8, 2004
KFUPM-Distillation Control Course

11. KFUPM-Distillation Control Course
Use of RGA:
Interactions
From derivation: Interactions are small if relative gains are close to 1
Choose pairings corresponding to RGA elements close to 1
Traditional: Consider Steady-state
Better: Consider frequency corresponding to closed-loop time constant
But: Avoid pairing on negative steady-state relative gain – otherwise you get instability if one of the loops
become inactive (e.g. because of saturation)
April 4-8, 2004
KFUPM-Distillation Control Course

12. KFUPM-Distillation Control Course
April 4-8, 2004
KFUPM-Distillation Control Course

13. (2) Sensitivity measure
But RGA is not only an interaction measure:
Large RGA-elements signifies a process that is very sensitive to small changes (errors) and therefore fundamentally difficult to control
Large (BAD!)
April 4-8, 2004
KFUPM-Distillation Control Course

14. KFUPM-Distillation Control Course
Singular Matrix: Cannot take inverse, that is, decoupler hopeless.
Control difficult
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KFUPM-Distillation Control Course

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