On a Problem of Minimal Realization

It's supposed that for a discrete-time linear time-invariant system $\Sigma$ the McMillan degree $\delta$ and a finite sequence of the Markov parameters $G_1,\ldots,G_m$, $m\geqslant 2\delta$, 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 $\Sigma$ 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 $G_1,\ldots,G_m$. 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.