The 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, which 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 via its input and output variables, this does not 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 transfer function relating to a Sakai fractional distillation system. Specifically, for this transfer function we assess whether or not the choice of unit, which in this case relates to temperature, affects the choice of loop pairings implied 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. Index Terms— Control systems, Moore-Penrose pseudoinverse, Process control, Relative Gain Array (RGA), UC inverse, Unit consistency.