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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.
Normalized Coprime Factorizations in Continuous and Discrete Time - A Joint State-Space Approach
(1995)
Based on state-space formulas for coprime factorizations over ... and an algebraic characterization of J-inner functions, normalized doubly-coprime factorizations for different classes of continuous- and discrete-time transfer functions are derived by using a single general construction method. The parametrization of the factors is in terms of the stabilizing solutions of general degenerate continuous- respectively discrete-time Riccati equations, which are obtained by examining state-space representations of J-normalized factor matrices.
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.