Dynamic backstepping control for pure-feedback non-linear systems

A dynamic backstepping control method is proposed for non-linear systems in the pure-feedback form, for which the traditional backstepping method suffers from solving the implicit non-linear algebraic equation. This method treats the implicit algebraic equation directly via a dynamic way, by augmenting the (virtual) controls as states during each recursive step. Compared with the traditional backstepping method, one more Lyapunov design is executed in each step. As new dynamics are included in the design, the resulting control law is in the dynamic feedback form. Under appropriate assumptions, the proposed control scheme achieves the uniformly asymptotic stability and the closed-loop system is local input-to-state stable for various disturbance. Moreover, the control law may be simplified to the inverse-free form by setting large gains, which will alleviate the problem of 'explosion of terms'. The effectiveness of this method is illustrated by the stabilization and tracking numerical examples.