Beyond the Existence of Diagonal Controllers: from the Relative Gain Array to the Multivariable Structure Function

A study related to coupling measurement and the existence of diagonal controllers for multivariable linear systems is presented. It is shown that the relative gain array (RGA) can be expressed in terms of the multivariable structure function (MSF). Moreover, the appropriate interpretation and handling of the MSF allows: (a) to establish the dynamical structure of the multivariable control system, (b) to prove the existence of stabilising diagonal controllers, (c) to determine whether design specifications can be met, and (d) the general coupling measurement among channels. In order to show the application of the MSF to the analysis and design of diagonal controllers, a family of plants is studied, ranging from non coupled minimum-phase to highly coupled and non-minimum phase plants. The actual multivariable control designs are included for completeness. The examples here presented may be a good benchmark for further analysis and comparisons.