Ed Bristol (a control engineer with Foxboro) developed a heuristic technique to predict possible interactive effects between control loops when multiple SISO loops are used. He was motivated to develop a dimensionless measure of interaction by some boiler control problems he was working on. His simple measure of interaction is known as the relative gain, or the Bristol relative gain array (RGA). Since then, many papers have been written to provide a more rigorous theoretical basis for the technique.

The relationship between u_{1} and y_{1} at steady state (s = 0), with no change in u_{2}, is

Equation 13.18

This is referred to as the gain between u_{1} and y_{1} with all other loops open (that is, u

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