Canonical State Representations and Hilbert Functions of Multidimensional Systems
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Canonical State Representations and Hilbert Functions of Multidimensional Systems Ulrich Oberst
Received: 22 June 2005 / Accepted: 24 July 2006 / Published online: 27 September 2006 © Springer Science + Business Media B.V. 2006
Abstract A 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 differential or difference equations with constant coefficients is isomorphic to a canonical state behavior which is constructed by means of Gröbner bases. In contrast to the onedimensional situation, to J.C. Willems’ multidimensional state space models and and to J.F. Pommaret’s modified Spencer form the canonical state behavior is not necessarily a first order system. Further first order models are due E. Zerz. As a byproduct of the state space construction we derive a new variant of the algorithms for the computation of the Hilbert function of finitely generated polynomial modules or behaviors. J.F. Pommaret, J. Wood and P. Rocha discussed the Hilbert polynomial in the systems theoretic context. 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. A two-page example also gives the complete proof of Kalman’s one-dimensional theorem mentioned above. We believe that for this standard case the algorithms of the present paper compare well with their various competitors from the literature. Key words state · Hilbert function · behavior · multidimensional system · partial differential equation · partial difference equation · polynomial module. Mathematics Subject Classifications (2000) 93B · 93C · 13P. U. Oberst (B) Institut für Mathematik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria e-mail: [email protected]
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Acta Appl Math (2006) 94: 83–135
1 Introduction One of the basic and substantial theorems of systems theory, originally due to Kalman [15], says that any input/output-behavior with proper transfer matrix admits an observable state realization. In more detail, let
B :=
y ∈ D (R ) p+m ; P(d/dt)y = Q(d/dt)u u
(1)
be an IO-behavior where t is the independent variable, usually time, P ∈ C [s] p× p and Q ∈ C [s] p×m are polynomial matrices with non-zero determinant det(P) and proper transfer matrix P−1 Q and where the input u and output y are vector distributions. The case of real instead of complex coefficients is mathematical
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