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Based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions, the Ober balanced canonical forms for the classes of bounded real and positive real are derived. This uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials.
Given a proper antistable rational transfer function g, a balanced realization of g is contructed as a matrix representation of the abstract shift realization introduced in Fuhrmann [1976]. The required basis is constructed as a union of sets of polynomials orthogonal with respect to weights given by the square of the absolute values of minimal degree Schmidt vectors of the corresponding Hankel operators. This extends results of Fuhrmann [1991], obtained in the generic case.
The polynomial approach introduced in Fuhrmann [1991] is extended to cover the crucial area of AAK theory, namely the characterization of zero location of the Schmidt vectors of the Hankel operators. This is done using the duality theory developed in that paper but with a twist. First we get the standard, lower bound, estimates on the number of unstable zeroes of the minimal degree Schmidt vectors of the Hankel operator. In the case of the Schmidt vector corresponding to the smallest singular the lower bound is in fact achieved. This leads to a solution of a Bezout equation. We use this Bezout equation to introduce another Hankel operator which have singular values that are the inverse of the singular values of the original Hankel operator.
Diffeomorphisms are given between different subsets of linear systems of fixed McMillan degree. The sets considered are the set of all systems of fixed McMillan degree, the subset of stable systems, the subset of bounded real systems, the subset of positive real systems, the subset of stable systems with Hankel singular values bounded by one. State space techniques are used in the proofs.