Canonical State Representations and Hilbert Functions of Multidimensional Systems

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 one-dimensional 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 by-product 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.     

Electrodeposition and cycling of polypyrrole

Polypyrrole has been electrodeposited at a constant current density of 4 mA cm⁻² onto Pt from 0.1 M pyrrole, 0.1 M NBu₄ BF₄ in dry CH₃CN. The thickness of the polypyrrole has been varied over a wide range of 0.02 to 50 μm. Elemental analysis reveals an excess of hydrogen in the polymer. The electrochemical equivalent corresponds to an insertion of 25–35 mole % BF₄-anions and a current efficiency of 80–100% for the electrodeposition process. SEM technique shows a highly textured material at a thickness larger than 1 μm. Reversible water vapour adsorption of this material has been detected. The potential/time curve during galvanostatic electrodeposition is without special features. The start potential, U_SSCE = 0.9 V, is relatively negative. A slight decay of potential in the course of the electrodeposition of thick layers is explained in terms of increasing surface roughness. Cyclic voltammetry has been used for a systematic investigation into the role of the positive and negative endpotentials. The importance of a total primary discharge after electrodeposition has been clarified. At a film thickness d exceeding 1 μm, the cyclovoltammetric curve degenerates, and the anodic peak flattens and shifts to more positive potentials. Active mass utilization decreases with increasing d and with increasing voltage scan rate v_s. The plot of anodic peak current density vs. v_s is linear for layers below 1 μm thickness. For thick layers and at high v_s, deviations occur, indicating a transport limitation in the film. As the film is not homogeneous, a quantitative evaluation is not possible.