Boundary control of a vibrating string under unknown time-varying disturbance

In this paper, robust adaptive boundary control for a vibrating string under unknown time-varying disturbance is developed to suppress the string´s vibration. The dynamics of the string is represented by a partial differential equation (PDE) and several ordinary differential equations (ODEs) involving functions of space and time. To deal with the system parametric uncertainty and stabilize the string, robust adaptive boundary control is developed at the tip of the string based on the Lyapunov´s direct method. With the proposed boundary control, uniform ultimate boundedness of the closed loop system is achieved. The state of the string system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Simulations are presented to illustrate the effectiveness of the proposed control.