Process Control: Designing Process and Control
Systems for Dynamic Performance
Chapter 20. Multiloop Control – Relative Gain Analysis
Copyright © Thomas Marlin 2013
The copyright holder provides a royalty-free license for use of this material at non-profit
educational institutions
RELATIVE GAIN MEASURE OF INTERACTION
We have seen that interaction is important. It affects
whether feedback control is possible, and if possible, its
performance.
Do we have a quantitative measure of interaction?
The answer is yes, we have several! Here, we will learn
about the RELATIVE GAIN ARRAY.
Our main challenge is to understand the correct
interpretations of the RGA.
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
1.
DEFINITION OF THE RGA
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
PRELIMINARY CONTROL DESIGN
IMPLICATIONS OF RGA
Let’s start
here to build
understanding
RELATIVE GAIN MEASURE OF INTERACTION
The relative gain
between MVj and
CVi is ij . It given
in the following
equation.
Explain in words.
MV1(s)
G11(s)
+
G21(s)
+
CV1(s)
Gd1(s)
D(s)
G12(s)
MV2(s)
G22(s)
Gd2(s)
+
+
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
CV2(s)
Assumes
that loops
have an
integral
mode
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
1.
DEFINITION OF THE RGA
Now, how do
we determine
the value?
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
PRELIMINARY CONTROL DESIGN
IMPLICATIONS OF RGA
RELATIVE GAIN MEASURE OF INTERACTION
1. The RGA can be
calculated from openloop values.
CV   K MV 
MV   K 1 CV 
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
 CVi

k ij  


MV
j

 MVi

kI ij  


CV
j

The relative gain array is the element-by-element
product of K with K-1
 
  K  K1
T
ij  kij kI ji 
RELATIVE GAIN MEASURE OF INTERACTION
1. The RGA can be
calculated from openloop values.
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
The relative gain array for a 2x2 system is given in the
following equation.
1
11 
K12 K 21
1
K11 K 22
What is true for the RGA to have 1’s on diagonal?
RELATIVE GAIN MEASURE OF INTERACTION
2. The RGA elements
are scale independent.
Original units
 CV1  10 10  MV1 


 MV 
CV
9
10
 2 
 2 
Changing the units of
the CV or the capacity
of the valve does not
change ij .
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
Modified units
 CV1   1 10 MV *1 
CV *   .09 1   MV 
2



2 
MV or1
10MV1
CV1
10
CV2 or CV2 / 10
9
MV2
9
10
RELATIVE GAIN MEASURE OF INTERACTION
3. The rows and
columns of the RGA
sum to 1.0.
CV1
CV2
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
MV1
10
9
MV2
9
10
For a 2x2 system, how many elements are independent?
RELATIVE GAIN MEASURE OF INTERACTION
4. In some cases, the
RGA is very sensitive
to small errors in the
gains, Kij.
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
1
11 
K12 K 21
1
K11 K 22
When is this equation very sensitive to errors in the
individual gains?
RELATIVE GAIN MEASURE OF INTERACTION
5. We can evaluate the
RGA of a system with
integrating processes,
such as levels.
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
Redefine the output as the derivative of the level; then,
calculate as normal.
m1
m2
 = density
L
A
D = density
dL
A
 A  m1  m2  Fout
dt
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
1.
DEFINITION OF THE RGA
How do we
use values to
evaluate
behavior?
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
PRELIMINARY CONTROL DESIGN
IMPLICATIONS OF RGA
RELATIVE GAIN MEASURE OF INTERACTION
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
MVj  CVi
ij < 0 In this case, the steady-state gains have different
signs depending on the status (auto/manual) of
other loops
A
Solvent
A
CA
A
CA0
A B
Discuss
interaction in
this system.
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
ij < 0 In this case, the steady-state gains have different
signs depending on the status (auto/manual) of
other loops
We can achieve stable multiloop feedback by using the
sign of the controller gain that stabilizes the multiloop
system.
Discuss what happens when the other interacting loop is
placed in manual!
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
ij < 0 the steady-state gains have different signs
For ij < 0 , one of three BAD situations occurs
1. Multiloop is unstable with all in automatic.
2. Single-loop ij is unstable when others are in manual.
3. Multiloop is unstable when loop ij is manual and other
loops are in automatic
Example of pairing on a negative RGA (-5.09). XB
controller has a Kc with opposite sign from single-loop
control! The system goes unstable when a constraint is
encountered.
FR  XB
IAE = 0.58326 ISE = 0.0041497
XB, Bottoms Lt Key
0.985
0.98
0.975
0
100
200
300
400
0.025
0.02
0.015
0.01
0.005
13.8
9
13.7
8.9
Reflux Flow
XD, Distillate Lt Key
0.03
0.99
Reboiled Vapor
FRB  XD
IAE = 0.3338 ISE = 0.0012881
0.995
13.6
13.5
13.4
13.3
0
100
200
300
400
8.8
8.7
8.6
0
100
200
300
Time
400
500
8.5
0
100
200
300
Time
400
500
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
ij = 0 In this case, the steady-state gain is zero when all
other loops are open, in manual.
T
L
Could this control
system work?
What would happen if
one controller were in
manual?
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
0<ij<1 In this case, the steady-state (ML) gain is larger
than the SL gain.
What would be the effect on tuning of opening/closing the
other loop?
Discuss the case of a 2x2 system paired on ij = 0.1
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
ij= 1 In this case, the steady-state gains are identical in
both the ML and the SL conditions.
MV1(s)
What is generally
true when ij= 1 ?
G11(s)
+
G21(s)
+
CV1(s)
Gd1(s)
D(s)
Does ij= 1 indicate
no interaction?
G12(s)
MV2(s)
G22(s)
Gd2(s)
+
+
CV2(s)
RELATIVE GAIN MEASURE OF INTERACTION
ij= 1 In this case, the
steady-state gains are
identical in both the ML
and the SL conditions.
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
Solvent
Zero heat
of reaction
Reactant
FS >> FR
AC
TC
Calculate the
relative gain.
Discuss
interaction in
this system.
RELATIVE GAIN MEASURE OF INTERACTION
ij= 1 In this case, the
steady-state gains are
identical in both the ML
and the SL conditions.
Diagonal gain
matrix
k 11

k 22

K

 0


0 


..

.. 
..
Lower
diagonal gain
matrix
k 11

k 21 k 22
K   ..

 ..
k
..
 n1

0 


..

.. 
.. .. ..
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
Both give an RGA
that is diagonal!
1

0


 1

I
RG A  
1


1
 0


1
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
1<ij In this case, the steady-state (ML) gain is larger
than the SL gain.
What would be the effect on tuning of opening/closing
the other loop?
Discuss a the case of a 2x2 system paired on ij = 10.
RELATIVE GAIN MEASURE OF INTERACTION
MVj  CVi
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
ij=  In this case, the gain in the ML situation is zero.
We conclude that ML control is not possible.
How can we improve the situation?
RELATIVE GAIN MEASURE OF INTERACTION
OUTLINE OF THE PRESENTATION
1.
DEFINITION OF THE RGA
2.
EVALUATION OF THE RGA
3.
INTERPRETATION OF THE RGA
4.
PRELIMINARY CONTROL DESIGN
IMPLICATIONS OF RGA
Let’s evaluate
some design
guidelines based
on RGA
RELATIVE GAIN MEASURE OF INTERACTION
Proposed Guideline #1
Select pairings that do
not have any ij<0
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
• Review the interpretation, i.e., the effect on behavior.
• What would be the effect if the rule were violated?
• Do you agree with the Proposed Guideline?
RELATIVE GAIN MEASURE OF INTERACTION
Proposed Guideline #2
Select pairings that do
not have any ij=0
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
• Review the interpretation, i.e., the effect on behavior.
• What would be the effect if the rule were violated?
• Do you agree with the Proposed Guideline?
RELATIVE GAIN MEASURE OF INTERACTION
RGA and INTEGRITY
•
We conclude that the RGA provides excellent insight
into the INTEGRITY of a multiloop control system.
•
INTEGRITY: A multiloop control system has good
integrity when after one loop is turned off, the
remainder of the control system remains stable.
•
“Turning off” can occur when (1) a loop is placed in
manual, (2) a valve saturates, or (3) a lower level
cascade controller no lower changes the valve (in
manual or reached set point limit).
•
Pairings with negative or zero RGA’s have poor
integrity
RELATIVE GAIN MEASURE OF INTERACTION
Proposed Guideline #3
Select a pairing that has
RGA elements as close as
possible to ij=1
 CVi 
 CVi 




 MV j 
 MV j 

 MVk constant

other loops open
ij 

 CVi 
 CVi 




 MV j 
 MV j 

CVk constant 
other loops closed
• Review the interpretation, i.e., the effect on behavior.
• What would be the effect if the rule were violated?
• Do you agree with the Proposed Guideline?
For set point response, RGA closer to 1.0 is better
FR  XD
FD  XD
FRB  XB
FRB  XB
RGA = 6.09
RGA = 0.39
IAE = 0.25454 ISE = 0.0004554
IAE = 0.045707 ISE = 8.4564e-005
IAE = 0.059056 ISE = 0.00017124
0.986
0.023
0.986
0.022
0.982
0.98
0.022
50
100
150
0.02
200
0.98
0
SAM = 0.31512 SSM = 0.011905
100
150
200
8.6
50
100
150
200
50
100
Time
150
200
13.8
13.7
8.5
8.46
0
50
100
Time
150
200
50
100
150
200
14
8.48
13.5
0
13.9
13.8
13.7
13.6
13.6
0
0.019
SAM = 0.55128 SSM = 0.017408
8.52
Reflux flow
Reboiled vapor
8.7
0
8.54
13.9
8.8
0.02
SAM = 0.10303 SSM = 0.0093095
14
8.9
8.5
50
SAM = 0.28826 SSM = 0.00064734
9
0.021
0.982
0.021
0
0.984
Reboiled vapor
0.984
XB, light key
0.023
XD, light key
0.988
XB, light key
0.024
Reflux flow
XD, light key
IAE = 0.26687 ISE = 0.00052456
0.988
0
50
100
Time
150
200
13.5
0
50
100
Time
150
200
For set point response, RGA farther from 1.0 is better
FR  XD
FD  XD
FRB  XB
FRB  XB
RGA = 6.09
RGA = 0.39
IAE = 0.14463 ISE = 0.00051677
IAE = 0.45265 ISE = 0.0070806
IAE = 0.32334 ISE = 0.0038309
50
100
150
0.01
0
0
100
150
200
8.6
8.55
50
100
Time
150
200
0.01
0
50
100
150
0
200
14
13.5
8.5
13.5
13.4
13.3
8.4
8.3
8.2
8.1
8
0
50
100
Time
150
200
50
100
150
200
SAM = 4.0285 SSM = 0.6871
8.6
13.1
0
SAM = 0.51504 SSM = 0.011985
13.6
13.2
0
0.95
Reflux flow
Reboiled vapor
Reflux flow
50
SAM = 0.38988 SSM = 0.0085339
8.65
0.02
0.015
0.005
SAM = 0.21116 SSM = 0.0020517
8.7
8.5
0.97
0.96
0.005
200
XB, light key
0.015
0.98
Reboiled vapor
0
XD, light key
XB, light key
XD, light key
Smaller
scale
0.975
0.03
0.025
0.02
0.98
IAE = 0.31352 ISE = 0.0027774
0.99
0.025
13
12.5
12
11.5
0
50
100
Time
150
200
11
0
50
100
Time
150
200
RELATIVE GAIN MEASURE OF INTERACTION
The RGA gives useful conclusions from S-S information
• Tells us about the integrity of multiloop
systems and something about the
differences in tuning as well.
• Uses only gains from feedback process!
• Does not use following information
- Control objectives
- Dynamics
- Disturbances
• Lower diagonal gain matrix can have
strong interaction but gives RGAs = 1
Powerful results
from limited
information!
Can we design
controls without
this information?
“Interaction?”