The effect of feedback on linear multivariable systems

Abstract

This paper describes effective algebraic procedures for performing certain operations on matrix transfer functions, the most important being the calculation of the effect of feedback. Such operations are required, for example, in designing, sequentially, controllers for linear multivariable systems. Previous papers have described algorithms for performing these operations numerically at specific frequencies to obtain, for example, the closed-loop matrix frequency response, in numerical form. However, if algebraic solutions, which are matrix transfer functions whose elements are rational functions, are required, naive use of standard formulae must be avoided, since they result in rational functions of needlessly high degree. This paper shows how this needless increase in the degree of the rational functions may be avoided, thus yielding effective algebraic procedures for the operations considered. Although the results are of interest in their own right, a brief resume of a specific procedure, the sequential return difference method, for designing linear multivariable control systems which utilises these results, is given.