Interpreting the RGA
Andrew Kim
Stephanie Cleto
What is the RGA?
• Relative Gain Array is an analytical tool used to
determine the optimal input-output variable
pairings for a multi-input-multi-output (MIMO)
system.
Significance of the RGA
• Below is a general relative gain array
Significance of the RGA
• Below is a general relative gain array
• Different columns represent the different
manipulated (input) variables
Significance of the RGA
• Below is a general relative gain array
• Different columns represent the different
manipulated (input) variables
• Different rows represent the different controlled
(output) variables
Significance of the RGA
• The values in the array describe the relationship
between the input and output variables
• Negative values indicate an unstable
relationship
• A value of 0 indicates no relationship
• A value of 1 indicates that specific input variable
is the only influence on that output variable
Example 1
• Assume a mixing tank with constant mass and two
inputs as shown below:
• wA and wB are manipulated flowrates entering the tank
• w is the flowrate leaving the tank and xA is the
concentration of A in the tank
Example from
http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html
Example 1 (cont.)
• This process can be
modeled by the
following equations:
w = wA + wB
xA = wA/(wA + wB)
• The RGA can be
solved for this
system:
1 x 
 x
RelativeGain Array  



1

x
x


Example 1
Solution
• What does the RGA tell us?
– If a concentration of xA=0.5 is desired, either
wA or wB can be used
– If a concentration of xA>0.5 is desired, then
the concentration loop should be paired with
wA
– If a concentration of xA<0.5 is desired, then
the concentration loop should be paired with
wB
Example 2
Run
R(kg/min)
S(kg/min)
xD
xB
1
125
22
0.97
0.04
2
150
22
0.93
0.06
3
150
20
0.91
0.08
Adapted from
http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html
Example 2
Solution
• RGA Matrix
2
-1
-1
2
• Pairing of variables: RGA matrix value
should be 1)positive, then 2)close to 1.
• Because each combination has only one
positive value, that is the combination to
be paired (R,xD and P,xB)
Example 3
• Suppose you
calculate the following
RGA matrix. How
should pairing of the
variables occur?
-.25
0
1.25
0.75
0.8
-0.55
0.5
0.2
0.3
Example 3
Solution
• In the first row,
only x3 gives a
positive result, and
then we go with
the closest values
to 1 for the others.
• y1,x3
y2,x2
y3,x1
-.25
0
1.25
0.75
0.8
-0.55
0.5
0.2
0.3