Andrew Kim Stephanie Cleto 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: No calculation step through b/c focusing more on the pairing for inputs and outputs

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

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 http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html

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

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

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