Polynomial systems theory applied to the analysis and design of multidimensional systems

The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p)\in F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators { p1,p2,..., pn-1}, a polynomial ring F[p1,p2,...,pn] acting on X can be extended to F(p1,p2,..., pn-1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of `ordinary' polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.