Adaptive Stabilization Control for a Class of Non-Strict Feedback Underactuated Nonlinear Systems by Backstepping

Abstract

Inspired by backstepping method, this paper proposes a novel adaptive stabilizing control method for a class of uncertain underactuated nonlinear systems, which is named as underactuated adaptive backstepping. Firstly, based on a virtual controller and a partial differential equation (PDE), a residual system is constructed. Then, a novel coordinate transformation is introduced to analyse the adaptive stabilizing control problem of the residual system, which breaks through the restriction of strict feedback form and achieves the estimation of unknown parameters. The proposed method first applies backstepping to form a residual system, and then uses backstepping again to design an adaptive controller for the residual system. The stability of the proposed method is proven based on Lyapunov’s theorems. Finally, two actual underactuated examples with physical damping are provided to demonstrate the effectiveness of the proposed underactuated adaptive backstepping. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper aims to develop a novel underactuated adaptive scheme called underactuated adaptive backstepping for underactuated mechanical systems. This scheme can effectively address the parameter uncertainty problem and achieve good control effect, even if there is uncertain physical damping which may change the equilibrium point in the system. Compared with existing methods, this scheme provides a novel design approach without linearization and approximation, which fully promotes the development of backstepping in the underactuated field. Moreover, the simulation experiments are carried out on the inertia wheel pendulum and ball and beam system with physical damping, thus effectively proving the feasibility of this scheme. In the future, the scheme will be applied to practical mechanical systems.