On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems

Abstract

A recently-derived alternative method for computing the relative gain array (RGA) for singular and/or non-square systems has been proposed that provably guarantees unit invariance. This property is not offered by the conventional method that uses the Moore-Penrose (MP) pseudoinverse. In this paper we note that the absence of the scale-invariance property by the conventional MP-RGA does not necessarily imply a practical disadvantage in real-world applications. In other words, while it is true that performance of a controller should not depend on the choice of units on its input and output variables, this does not {\em necessarily} imply that the resulting MP-RGA measures of component interaction lead to different controller-design input-output pairings. In this paper we consider the application of the MP-RGA to a realistic system (a Sakai fractional distillation system) to assess whether or not the choice of unit, which in this case relates to temperature, affects the choice of input-output pairings determined by the resulting RGA matrix. Our results show that it does, thus confirming that unit-sensitivity of the MP-RGA undermines its rigorous use for MIMO controller design.