On the factorization of a nonsingular rational matrix

Given a multiport transfer function , the question arises whether it can be realized using feedback loops of shortest possible length. The most elegant way to achieve this is to realize in cascade form, with each subsystem in the cascade of smallest possible degree. Any transfer matrix of finite degree is a matrix whose entries are rational functions of the complex variable , and a cascade synthesis of such a -matrix will bear heavily on the properties of the factorization of rational -matrices into simpler factors. We will call a -matrix of the first degree an elementary factor. It is shown that 1) a nonsingular rational -matrix can be factored into a minimal product of elementary factors provided either all poles or zeros are of the first order but not necessarily of the first degree; 2) that any nonsingular matrix can be factored into a product of elementary factors with the bound on the number of factors being a function of the degree of the matrix; and 3) that there are matrices which cannot be factored out minimally. Factorization procedures in special cases were first deduced by Belevitch and Youla, and later, using an improved criterion for degree reduction, by the first author. These procedures all use unitary or so-called -unitary elementary factors. The theory presented here uses a general type of elementary factor to attack the problem of cascade decomposition of a general nonsingular transfer function. Thus the question whether a general (rational) multiport can be synthesized by means of feedback loops of length one has to be answered negatively although a very large number of systems can, and a criterion is given to distinguish the two cases. Also, an algorithm and an example are presented to exhibit how the factorization (if it exists) can be constructed.