Adaptive second-order backstepping control for a class of 2DoF underactuated systems with input saturation and uncertain disturbances

Abstract

An adaptive second-order backstepping control algorithm is proposed for a kind of two degrees of freedom (2DoF) underactuated systems. The system dynamics is transformed into a nonlinear feedback cascade system with an improved global change of coordinates. Fully taking the cascade structure into consideration and in order to simplify the design process, each step in the backstepping process is designed for a second-order subsystem. Two neural networks are applied to approximate system unknown functions and two adaptive laws are designed to estimate the upper bound of the sum of approximation error and external disturbances. To overcome the explosion problem of complexity, a second-order filter is applied to produce the virtual control and its second-order derivative that is needed in the next backstepping step. Two auxiliary dynamic systems are proposed and integrated into the backstepping process to eliminate the effects of filtering error and input saturation. The system stability is analyzed by the Lyapunov stability theory and verified by numerical simulations with two 2DoF benchmark underactuated systems: the translational oscillator with a rotational actuator (TORA) and the inertial wheel pendulum (IWP).