# Sensitivity of variable pairing in multivariable process control to model uncertainties

This project proposes a method for obtaining the variations in the Relative Gain Array (RGA) due to uncertainties in models of 2×2 multivariable systems. The uncertainties are represented as element by element multiplicative uncertainty using the Schur multiplier.

An introduction and some references about the background of the project are given, the topics are: uncertainties, decentralized control, decoupling, and interaction measures, with a deeper overview of the RGA.

The conclusions obtained about uncertainties and RGA are applied to some 2×2 examples which illustrate the following cases:

A diagonally dominant system, with values of the RGA close to 1 for the diagonal elements. The RGA of these systems is the most robust to uncertainties; it’s robustness is shown, and an example about this reliability is given.

A system with large values of the RGA. Large values of the RGA are usually advised against pairing purposes because they are related to ill-conditioned plants; nevertheless, it is not clear how large can these values be and still be used for pairing purposes. These systems are the most sensible to uncertainties, and the larger the values of the RGA are, the higher sensitivity of the RGA to uncertainties. An analysis of the sensitivity of these systems is made in order to suggest upper bounds for the values of the nominal RGA. Values of the RGA above these ones will be considered highly sensitive to uncertainties, therefore, small perturbations can yield to large values of the RGA, and they should be discarded for pairing purposes.

A system with positive values of the RGA moderately lower than 1 for the diagonal elements. Take as an example a system with value 0.8 for the element lambda_{11} of the nominal RGA; this value is usually taken as valid for pairing purposes, but, can perturbations modify the value of the RGA in such a way that the results would not be valid for pairing purposes?. A system with these characteristics is introduced, and a method for analyzing the bounds of the RGA due to uncertainties is proposed.

Author: Castaño Arranz, Miguel

Source: Luleå University of Technology