A Reachability Test for Systems over Polynomial Rings using Gröbner Bases

Conditions for the reachability of a system over a polynomial ring are well known in the literature. However, verification of these conditions is in general a difficult problem. In this paper, the reachability problem is expressed in terms of polynomial ideals, and Gröbner Bases are used to carry out the test explicitly. Four different methods are proposed, and their effectiveness is compared. With one of these methods, it is also possible to obtain a right-or left-inverse of a general non-square polynomial matrix, often required in the design of feedback compensators.