On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion

A novel method is presented to determine the SmithMacmillan form of a rational
m \times n
matrix
R(p)
from Laurent expansions in its poles and zeros. Based on that method, a numerically stable algorithm is deduced, which uses only a minimal number of terms of the Laurent expansion, hence providing a shortcut with respect to cumbersome and unstable procedures based on elementary transformations with unimodular matrices. The method can be viewed as a generalization of Kublanovkaya's algorithm for the complete solution of the eigenstructre problem for
\lambda I - A
. From a system's point of view it provides a handy and numerically stable way to determine the degree of a zero of a transfer function and unifies a number of results from multivariable realization and invertibility theory. The paper presents a systematic treatment of the relation between the eigen-information of a transfer function and the information contained in partial fraction or Laurent expansions. Although a number of results are known, they are presented in a systematic way which considerably simplifies the total picture and introduces in a natural way a number of novel techniques.