Nonlinear control via uniform system approximation

An approach for designing nonlinear controllers for single-input nonlinear systems that do not satisfy the involutivity conditions required for input-to-state linearization but are linearly controllable in a neighborhood of a set of operating points is presented. For such systems, it is shown how to construct a nonlinear system that is input-to-state linearizable and that approximates the given nonlinear system uniformly with respect to the equilibrium manifold. Therefore, in a sense, the linearizing transformation (feedback and coordinate change) for the approximate system approximately transforms the given system into phase coordinates (Brunovsky canonical form). Then, using a control law designed to exactly track the canonical output of the approximate system, the system can be operated in a stable manner in a neighborhood of the equilibrium manifold. A simple example is given to illustrate the approach