Relative Gain Array


The Relative Gain Array is a classical widely-used method for determining the best input-output pairings for multivariable process control systems. It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.

Definition

Given a linear time-invariant system represented by a nonsingular matrix, the relative gain array is defined as
where is the elementwise Hadamard product of the two matrices, and the transpose operator is necessary even for complex. Each element gives a scale invariant measure of the dependence of output on input.

Properties

The following are some of the linear-algebra properties of the RGA:
  1. Each row and column of sums to 1.
  2. For nonsingular diagonal matrices and,.
  3. For permutation matrices and,.
  4. Lastly,.
The second property says that the RGA is invariant with respect to nonzero scalings of the rows and columns of, which is why the RGA is invariant with respect to the choice of units on different input and output variables. The third property says that the RGA is consistent with respect to permutations of the rows or columns of.

Generalizations

The RGA is often generalized in practice to be used when is singular, e.g., non-square, by replacing the inverse of with its Moore-Penrose inverse. However, it has been shown that the Moore-Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA and that the unit-consistent generalized inverse must therefore be used.