Minimal bases of polynomial modules, structural indices and Brunovsky-transformations

Abstract

Structural indices and state feedback of finite dimensional linear systems are analysed with the help of elementary module theory. The basic modules are polynomial and have a natural graduation. With respect to this graduation the concepts of minimal bases and degree preserving isomorphisms are introduced. Thus it becomes possible to give a module theoretic interpretation of the different definitions of structural indices. They are reduced to invariants of a special submodule of Km[s] which is also crucial for the understanding of Brunovsky-transformations. It is shown that the Brunovsky-transformations induce families of degree-preserving module isomorphisms. A detailed study of this fact leads to results concerning pole assignment via state feedback, classification of orbits of the Brunovsky-group and canonical forms.