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) ∈ 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.