On the Relative Gain Array (RGA) with Singular and Rectangular Matrices

TL;DR

This paper critically examines the use of the Moore-Penrose pseudoinverse in the Relative Gain Array (RGA) for singular matrices, proposing an alternative inverse to preserve essential properties.

Contribution

It identifies a key deficiency in applying RGA with singular matrices and introduces a rigorous alternative inverse to address this issue.

Findings

Moore-Penrose pseudoinverse is inappropriate for singular matrices in RGA.

An alternative generalized inverse better preserves RGA properties.

Highlights the importance of proper inverse selection in control system analysis.

Abstract

In this paper we identify a significant deficiency in the literature on the application of the Relative Gain Array (RGA) formalism in the case of singular matrices. Specifically, we show that the conventional use of the Moore-Penrose pseudoinverse is inappropriate because it fails to preserve critical properties that can be assumed in the nonsingular case. We then discuss how such properties can be rigorously preserved using an alternative generalized matrix inverse.