Stratification of full rank polynomial matrices

Abstract

We show that perturbations of polynomial matrices of full normal-rank can be analyzed via the study of perturbations of companion form linearizations of such polynomial matrices. It is proved that a full normal-rank polynomial matrix has the same structural elements as its right (or left) linearization. Furthermore, the linearized pencil has a special structure that can be taken into account when studying its stratification. This yields constraints on the set of achievable eigenstructures. We explicitly show which these constraints are. These results allow us to derive necessary and sufficient conditions for cover relations between two orbits or bundles of the linearization of full normal-rank polynomial matrices. The stratification rules are applied to and illustrated on two artificial polynomial matrices and a half-car passive suspension system with four degrees of freedom.