The purpose of this paper is to present a significant generalization to the direct-construction realization approach proposed recently by the authors, with an emphasis on the coefficient-dependent nature of the realization problem of multidimensional (n-D) systems in Roesser model. The central idea in the approach is to reduce the n-D realization problem to the construction of an admissible n-D polynomial matrix where careful treating of the structure-dependent and coefficient-dependent properties is required. It is shown that though the direct construction of an admissible n-D polynomial matrix is very difficult, it is possible to construct first an admissible n-D monomial matrix, say Ψ, which can be constructively obtained based only on the structure-dependent property, and then to build the polynomial one by combining some monomial entries into polynomial ones when the corresponding coefficients satisfy certain proportional conditions. Specifically, the relationship among the monomial entries of Ψ is first investigated thoroughly and some new concepts and notations necessary for the technique development are introduced. Then, some fundamental facts on combinability conditions and combination techniques are clarified, and algorithms for the generation of an admissible n-D polynomial matrix and the corresponding realization are established. Furthermore, a preprocessing strategy on entry re-assignment is explored, which can lead to possible further significant reduction in the realization order. Several symbolic and numerical examples are presented to illustrate the basic ideas as well as the effectiveness of the proposed approach.
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