Generalised resultants, dynamic polynomial combinants and the minimal design problem

Abstract

The theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems (DAP), which provides the unifying description of the dynamic, as well as static pole and zero dynamic assignment problems in linear systems. The assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original DAP. This paper demonstrates the origin of dynamic polynomial combinants from linear systems, examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree, and examines their 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 of 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 parameterisation 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, which is referred to as the dynamic combinant minimal design (DCMD) problem. An algorithmic approach based on rank tests of Sylvester matrices is given, which produces the minimal order and degree solution in a finite number of steps. Such solutions provide low bounds for the respective dynamic assignment control problems.

Keywords:

• linear systems
• spectrum assignment
• generalised resultants
• polynomial combinants
• minimal design