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.
[1]
Β. L. HO,et al.
Editorial: Effective construction of linear state-variable models from input/output functions
,
1966
.
[2]
J. Massey,et al.
Invertibility of linear time-invariant dynamical systems
,
1969
.
[3]
L. Silverman.
Inversion of multivariable linear systems
,
1969
.
[4]
Elwyn R. Berlekamp,et al.
Algebraic coding theory
,
1984,
McGraw-Hill series in systems science.
[5]
B. McMillan.
Introduction to formal realizability theory — II
,
1952
.
[6]
G. David Forney,et al.
Convolutional codes I: Algebraic structure
,
1970,
IEEE Trans. Inf. Theory.
[7]
Yen-Long Kuo,et al.
On the irreducible Jordan form realization and the degree of a rational matrix
,
1970
.