Modeling and Control of a Nonuniform Vibrating String Under Spatiotemporally Varying Tension and Disturbance

In this paper, robust adaptive boundary control is developed for a class of flexible string systems under unknown spatiotemporally varying distributed disturbance and time-varying boundary disturbance. The vibrating string is nonuniform since the spatiotemporally varying tension applied to the system. The nonuniform vibrating string system is represented by a nonlinear nonhomogeneous partial differential equation (PDE) and two ordinary differential equations (ODEs). Model-based control is first proposed at the right boundary of the string to suppress the vibration of the flexible nonuniform string system. To compensate for the system parametric uncertainties, robust adaptive boundary control is developed. With the proposed control, the uniformly ultimate boundness of the closed-loop system is demonstrated via Lyapunov´s direct method. The state of the nonuniform string system is proven to converge to a small neighborhood of zero by appropriately choosing the design parameters. Simulations are provided to illustrate the effectiveness of the proposed control.