Approximate Feedback Linearization: A 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. First, we show that if the system is close to being involutive, then it is also close to being linearizable. Rather than working directly with the characteristic distribution of the system, we work with characteristic one-forms, i.e., with the one-forms annihilating the characteristic distribution. We show that homotopy operators can be used to decompose a given characteristic one-form into an exact and an 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 is 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. One can design control for the original system by designing it for the approximating linearizable system and applying it to the original one. We apply this approach for design of locally stabilizing feedback laws for nonlinear systems that are close to being linearizable.