Degree Two Normal Forms of Control Systems and the Generalized Legendre Clebsch Condition

Abstract

The Brunovsky (or Controller) form is a useful normal form of linear systems under the group of linear state coordinate changes and linear state feedbacks. We discuss normal forms of quadratic systems under quadratic change of coordinates and quadratic feedback modulo cubic and higher terms. We discuss the relationship of these normal forms to the Generalized Legendre-Clebsch of Singular Optimal Control.