On a Problem of Minimal Realization

Abstract

It’s supposed that for a discrete-time linear time-invariant system § the McMillan
degree ± and a finite sequence of the Markov parameters G1; : : : ;Gm, m > 2±, are known.
The problems of reconstruction a transfer function G(z) of the system, minimal indices
and coprime fractional factorizations of G(z), minimal solutions of the appropriate Bezout
equations, the minimal realization of § from these dates are considered. There are various
algorithms to solve each of these problems. In the work we propose an unified approach
to study the problems. The approach is based on the method of indices and essential
polynomials of a finite sequence of matrices. This method was developed in connection with
the problem of an explicit construction of the Wiener – Hopf factorization for meromorphic
matrix functions. It is shown that we can obtain the solutions of all the above problems
as soon as we find the indices and essential polynomials of the sequence G1; : : : ;Gm. The
calculation of the indices and essential polynomials can be realized by means of linear
algebra. For matrices with entries from the field of rational numbers we have implemented
the algorithm in procedure ExactEssPoly in Maple.