Normal forms, canonical forms, and invariants of single input nonlinear systems under feedback

We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener (1992) based on analysing, step by step, the action of homogeneous transformations on the homogeneous part of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang (1994, 1995). We also propose a canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms coincide. We give an explicit construction of transformations bringing the system to its normal, dual normal, and canonical form