A constructive study of the module structure of rings of partial differential operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media

Realization in Generalized State Space form for 2-D Polynomial System Matrices

In this paper, a direct realization procedure is presented that brings a general 2-D polynomial system matrix to generalized state space (GSS) form, such that all the relevant properties including the zero structure of the system matrix are retained. It is shown that the transformation linking the original 2-D polynomial system matrix with its associated GSS form is zero coprime system equivalence. The exact nature of the resulting system matrix in GSS form and the transformation involved are established

Equivalence of wave linear repetitive processes and the singular 2-D Roesser state-space model

This paper develops a direct method for transforming a polynomial system matrix describing a discrete wave linear repetitive process to a 2-D singular state-space Roesser model description where all relevant properties, including the zero coprimeness properties of the system matrix, are retained. It is shown that the transformation is zero coprime system equivalence. The structure of the resulting system matrix in singular form and the transformation are also establishe

Equivalent 2-D nonsingular Roesser models for discrete linear repetitive processes

The elementary operations algorithm is used to establish that a system matrix describing a discrete linear repetitive process can be transformed to that of a 2-D nonsingular Roesser model where all the input–output properties are preserved. Moreover, the connection between these system matrices is shown to be input–output equivalence. The exact forms of the resulting system matrix and the transformation involved are established. Some areas for possible future use/application of the developed results are also briefly discussed

Reduction of wave linear repetitive processes to singular Roesser model form

Using the elementary operations algorithm, it is shown that a system matrix describing a wave discrete linear repetitive process can be reduced to that for a 2D singular Roesser model. The transformation linking the original polynomial system matrix with its associated 2D singular Roesser form is input-output equivalence. The nature of the resulting system matrix in singular form and the transformation involved are established

Reduction of discrete linear repetitive processes to nonsingular Roesser models via elementary operations

A method based on the elementary operations algorithm (EOA) is developed that reduces a system matrix describing a discrete linear repetitive process to a 2-D nonsingular Roesser form such that all the input-output properties, including the transfer-function matrix, are preserved. Some areas for possible future use/application of the developed results will also be briefly discussed

A companion matrix for 2-D polynomials

Consiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal