Approximate dynamic tracking and feedback linearization

In this paper approximate feedback linearization is revisited. It is shown that, under mild assumptions, a nonlinear system with state of dimension n can be immersed into an extended system that comprises a chain of n integrators, hence with linear input/output behavior, which 'contains' all the components of the state of the original nonlinear system. This result is achieved systematically and without resorting to the solution of any partial differential equation. Moreover, it is not required that the nonlinear system be linearly controllable, hence feedback linearizable in the classical sense. The construction is then specialized to provide a linear design technique to define control laws that enforce (local) asymptotic stability of a desired equilibrium point or asymptotic tracking of reference signals. The performance of the design methodology are assessed by means of two numerical examples, encompassing a locally uncontrollable planar system with a non-hyperbolic equilibrium point and the Ball and Beam model.