Adaptive control of an axially moving string under spatiotemporally varying tension via a hydraulic actuator

This paper presents an adaptive boundary control to suppress the transverse vibrations of an axially moving string by using a hydraulic actuator equipped with a damper, which is located at the right boundary of the string. The tension along the string is assumed to be spatiotemporally varying. The governing equations take the form of coupled partial differential equation and ordinary differential equation (ODE), in which the ODE describes the dynamics of the hydraulic actuator. An adaptive boundary control law is derived using Lyapunov´s direct method and the back stepping technique. Adaptation laws updating unknown system parameters online are included. The effectiveness of the proposed adaptive boundary control is demonstrated through simulations.