Approximate feedback linearization: homotopy operator approach

In this paper, we present an approach for finding feedback linearizable systems that approximate a given single-input nonlinear system on a given compact region of the state space. We show that homotopy operators can be used to decompose a given one-form annihilating the characteristic distribution of the system into an exact and antiexact part. The exact part is used to define a change of coordinates to a normal form that looks like a linearizable part plus nonlinear perturbation terms. The nonlinear terms in this normal form depend continuously on the antiexact part and they vanish whenever the antiexact part does. Thus, the antiexact part of a given characteristic one-form defines a measure of nonlinearizability of the system. If the nonlinear terms are small, by neglecting them we obtain a linearizable system approximating the original system.