Synthesis of multidimensional control systems by Gröbner basis approach

The purpose of this work is to deal with the synthesis problem of multidimensional (nD) feedback control systems by using Grobner basis approach. Necessary and sufficient solvability conditions are derived for nD general stabilization, regulation and tracking problems under various configurations and/or cases, which show that these problems can be essentially reduced to the solvability problems of coprime and skew prime matrix equations. It is then shown that for the 2D case, these equations, and thus the synthesis problems of various kinds of 2D controllers can be constructively solved by utilizing Grobner basis approach. However, for the general nD (n ges 3) cases, some substantial challenges still remain, which will be summarized as several open problems.