We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analyzing, step by step, the action of homogeneous transformations on the homogeneous part of the same degree of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms (resp., their dual canonical forms) coincide. We give an explicit construction of transformations bringing the system to its normal, dual normal, canonical, and dual canonical forms. We illustrate our results by simple examples on R3 and R4.
Tall, Issa A. and Respondek, Witold. "Feedback Classification of Nonlinear Single-Input Control Systems with Controllable Linearization: Normal Forms, Canonical Forms, and Invariants." (Jan 2003).