Appendix 13.1: Derivation of the Relative Gain for an n-Input–n-Output System

The relative gain between input j and output i is defined as

Output i is the following steady-state function of all of the inputs:

The relationship between input j and output i with all other inputs constant is

For any desired output vector, y, we can find the corresponding input vector, u, from

Defining (0)=G^-1(0)=K^-1

The relationship between all outputs and input j is

The relationship between output i and input j with all other outputs constant is

and

The relative gain between input j and output i is

Equation 13.23

where element of G^–1(0).

What Equation (13.23) tells us is that the ij element of the RGA is found by multiplying the ij element of the steady-state process gain matrix by the ji element of the inverse of the steady-state process gain matrix.