Systems over a Principal Ideal Domain. A Polynomial Model Approach

The polynomial model approach to linear dynamical systems over a field was developed principally by P. A. Fuhrmann, starting in 1976 [J. Franklin Inst., 305 (1976), pp. 521–540]. In this paper an analogous approach is proposed for systems over a principal ideal domain.When the concept of extended linear i/o map is introduced, fractional representations of transfer function matrices arise naturally in this theoretical framework. A correspondence between fractional representations of the transfer function matrix of a given i/o map and its reachable or observable realizations is established. The McMillan degree of a linear i/o map is proved to be equal to the degree of the determinant of the matrix appearing as “denominator” in a coprime fractional representation of the associated transfer function matrix.