Application of adaptive backstepping to the stabilization of the inverted pendulum

In this paper we propose a nonlinear Lyapunov-based controller to stabilize the famous cart-pole system. The novelty is in the use of a two-loop cascade controller. The inner loop uses an adaptive nonlinear controller, obtained by the backstepping recursive approach. It ensures the stabilization and the convergence towards zero of the angle tracking error and the (unknown) rod length estimation error. The reference signal to be tracked by the angle is generated by the outer loop linear controller. The controlled part of the system (rod angle) has thus a quasi-linear dynamics, which can be modeled by a double-integration with a variable gain. An indirect MRA controller is used to compensate the outer loop (cart position)