Canonical form associated with the problem of perturbation of one column of a controllable pair

Let (A,B) be a completely controllable matrix pair. When we consider the problem of characterizing the controllability indices of all the matrix pairs obtained by small perturbations on one column of B, a new equivalence relation arises in a natural way.This equivalence relation is a kind of partial feedback equivalence. As a consequence the controllability indices are invariant, but they do not form a complete system of invariants.We have found new invariants for this equivalence relation and we have obtained a canonical form in terms of these invariants.