Overlapping block-balanced canonical forms for various classes of linear systems

Abstract

Through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension. A possible problem of these parametrizations is that they are not overlapping. This could be a drawback for the application of balanced parametrizations in such areas as system identification, model reduction and optimization. It is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations. We first introduce input-normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix. These canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered. Then we introduce so-called block-balanced input normal forms which use the previously constructed input normal forms as building blocks. The classes of systems for which such parametrizations are given are the stable minimal systems, positive-real minimal systems, bounded-real minimal systems and the class of all minimal systems of given McMillan degree. The results include both the single-input single-output and the multivariable case. In the single-input single-output case, however, the issue of choosing nice selections of columns does not occur. Therefore in this case the derivation and presentation of the results is considerably simplified.