Robust Adaptive Boundary Control of a Vibrating String Under Unknown Time-Varying Disturbance

In this paper, robust adaptive boundary control is developed for a class of flexible string-type systems under unknown time-varying disturbance. The dynamics of the string system is represented by a nonhomogeneous hyperbolic partial differential equation (PDE) and two ordinary differential equations. Boundary control is proposed at the right boundary of the string based on the original distributed parameter system model (PDE) to suppress the vibration excited by the external unknown disturbance. Adaptive control is designed to compensate the system parametric uncertainty. With the proposed robust adaptive boundary control, all the signals in the closed-loop system are guaranteed to be uniformly ultimately bounded. The state of the string system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Simulations are provided to illustrate the effectiveness of the proposed control.