Miniversal deformations of linear systems under the full group action

We consider the equivalence relation in the space of time-invariant linear dynamical systems of the form

under the full group action of state feedback/output injection transformations. As in the case of the reduction of a matrix to its Jordan canonical form, the reduction of a quadruple (A,B,C,D) defining a system as above to its canonical reduced form is an unstable operation. Following V.I. Arnold’s techniques, the starting point for the study of local perturbations is to obtain a miniversal deformation of a differentiable family of quadruples. In this paper, a “real” miniversal deformation and some applications are shown.