The Minimal Design Problem on Dynamic Polynomial Combinants

The theory of dynamic polynomial combinants is linked to the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the pole and zero dynamic assignment problems in Linear Systems. The fundamentals of the theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Central to this study is the link of dynamic combinants to the theory of ”Generalised Resultants”, which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterization of combinants and respective Generalised Resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, referred to as the ”Dynamic Combinant Minimal Design” (DCMD) problem. Such solutions provide low bounds for the respective Dynamic Assignment control problems.