13.4 The Relative Gain Array

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.

Two-Inputs and Two-Outputs

The relationship between u₁ and y₁ at steady state (s = 0), with no change in u₂, is

Equation 13.18

This is referred to as the gain between u₁ and y₁ with all other loops open (that is, u₂ is constant for this 2 x 2 example).

The steady-state relationship between u₁ and y₁ at steady state, with y₂ maintained constant at its setpoint (using a controller with integral action), is

Equation 13.19

Notice that Equation (13.19) is the gain between input 1 and output 1, assuming output 2 is constant.

Definition of the Relative Gain

The relative gain (l_ij) between input j and output i is defined in the following fashion:

Equation 13.20

By convention, the symbol used for the relative gain is l_ij. Please do not confuse it with the l that was used for the IMC filter or for the l that is commonly used to represent the eigenvalues of a matrix. The relative gain will normally be shown with two subscripts, distinguishing it from eigenvalues or the IMC filter factor.

Relative Gain Between Input 1 and Output 1 for a Two Input–Two Output System

The relative gain between input 1 and output 1 is then [see Equation (13.23)]

We see that the relative gain between u₁ and y₁ for a 2 x 2 system is

Equation 13.21

Remember that Equation (13.21) is only true for the two input–two output case.

The RGA

The RGA is simply the matrix that contains the individual relative gain as elements, that is L = {l_ij} . For a 2 x 2 system, the RGA is

Equation 13.22

Question: What value do we desire for the relative gain?

Think of it this way. If we want to use a u₁-y₁ pairing for a SISO controller, we do not want it to matter whether the other loops in the system are closed or not. This tells us that we desire a relative gain close to 1.0. That is,

Answer: We desire l₁₁ 1.0 if we wish to pair output 1 with input 1.

The l₁₁ calculation for a 2 x 2 system yielded . The more general result is (see Appendix 13.1 for a derivation)

Equation 13.23

where element of G^–1(0). Notice that this is not the same as matrix multiplication. Here, the individual elements of the two matrices are being multiplied together.

We check the general results derived above by applying them to a 2 x 2 (two output, two input) system,

and recalling our rules for inverting a 2 x 2 matrix,

Multiplying the individual elements of G(0) and (0), as in Equation (13.23), we find

Equation 13.24