Backstepping control for a class of underactuated nonlinear mechanical systems with a novel coordinate transformation in the discrete-time setting

Abstract

In this technical note, a combined discrete-time controller, consisting of a partial feedback linearization controller and a backstepping controller, is proposed to manage the stabilization for a class of underactuated nonlinear mechanical systems. The traditional backstepping control method is impracticable to directly apply it to underactuated systems since they are generally not in the lower triangular form. First, a feedback controller that partially linearizes the underactuated mechanical systems is derived to facilitate the backstepping controller design. In the design, discrete-time system dynamics obtained utilizing the Euler approximation are considered. Afterward, a novel coordinate transformation is introduced for a class of underactuated systems that transform the mathematical model of the partially linearized underactuated mechanical systems into the strict-feedback form. After these steps, the conventional backstepping approach is structured in discrete-time setting. The stability of the closed-loop system dynamics with the proposed combined discrete-time controller is shown utilizing Lyapunov theory. Several computer-based simulations are executed to depict the performance and the applicability of the proposed method on the Cart-Pole system and the Inertia Wheel Pendulum. Moreover, the effectiveness of the proposed nonlinear controller is verified by numerical simulation results in comparison with a discrete-time chattering-free sliding-mode controller.