  A dynamic backstepping method is proposed to design controllers for nonlinear
systems in the pure-feedback form, for which the traditional backstepping
method suffers from solving the implicit nonlinear algebraic equation. The idea
of this method is to augment the (virtual) controls as states during each
recursive step. As new dynamics are included in the design, the resulting
controller is in the dynamic feedback form. Procedure of deriving the
controller is detailed, and one more Lyapunov design is executed in each step
compared with the traditional backstepping method. Under appropriate
assumptions, the proposed control scheme achieves the uniformly asymptotically
stability. The effectiveness of this method is illustrated by the stabilization
and tracking numerical examples.
