1. Andrew Kim Stephanie Cleto
Interpreting the RGA
Andrew Kim
Stephanie Cleto

2. 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.

3. Significance of the RGA
Below is a general relative gain array

4. Significance of the RGA
Below is a general relative gain array
Different columns represent the different manipulated (input) variables

5. 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

6. 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

7. 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

8. 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:
No calculation step through b/c focusing more on the pairing for inputs and outputs

9. What does the RGA tell us?
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

10. Example 2 Run R(kg/min) S(kg/min) xD xB 1 125 22 0.97 0.04 2 150 0.93
0.06 3 20
0.91
0.08
Adapted from

11. Example 2 Solution RGA Matrix
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) 2 -1

12. Example 3
Suppose you calculate the following RGA matrix. How should pairing of the variables occur?
-.25
1.25
0.75
0.8
-0.55
0.5
0.2
0.3

13. 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
1.25
0.75
0.8
-0.55
0.5
0.2
0.3