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TitleCanonical State Representations and Hilbert Functions of Multidimensional Systems
Author(s) Ulrich Oberst
TextAMS subject classification: 93B,93C,13P
TypeTechnical Report, Misc
AbstractA basic and substantial theorem of one-dimensional systems theory,
due to R. Kalman, says that an arbitrary input/output behavior with
proper transfer matrix admits an observable state representation which,
in particular, is a realization of the transfer matrix. The state equations
have the characteristic property that any local, better temporal, state at
time zero and any input give rise to a unique global state or trajectory of
the system or, in other terms, that the global state is the unique solution
of a suitable Cauchy problem. With an adaption of this state property
to the multidimensional situation or rather its algebraic counter-part we
prove that any behavior governed by a linear system of partial differen-
tial or difference equations with constant coe±cients is isomorphic to a
canonical state behavior which is constructed by means of Groebner bases.
In contrast to the one-dimensional situation and to J.C. Willems' multi-
dimensional state space models the canonical state behavior is not nec-
essarily a first order system. Further Kalman representations and first
order models are due to J.-F- Pommaret and E. Zerz. As a by-product of
the state space construction we derive a new algorithm for the computa-
tion of the Hilbert function of any finitely generated polynomial module
or behavior. J. Wood, P. Rocha et al. recognized the systems theoretic
significance of this Hilbert function in context with complexity and struc-
ture indices. The theorems of this paper are constructive and have been
implemented in MAPLE in the two-dimensional case and demonstrated
in a simple, but instructive example. For the standard one-dimensional
systems the algorithms of this paper compare well with those from the
Keywordsstate, Hilbert function, behavior, multidimensional system, partial differential equation, partial difference equation, polynomial module
Translation No
Refereed No
Organization University of Innsbruck
Institution Institut für Mathematik (Universität Innsbruck)