Adaptive Boundary Control of a Nonlinear Flexible String System

In this brief, the vibration control problem is investigated for a flexible string system in both transverse and longitudinal directions. The vibrating string is nonlinear due to the coupling between transverse and longitudinal displacements. Using the Hamilton's principle, the dynamics of the nonlinear string are presented by two partial and four ordinary differential equations. With the Lyapunov's direct method, adaptive boundary control is developed to suppress the string's vibration and the adaptive law is designed to compensate for the system parametric uncertainties. With the proposed control, the states of the system eventually converge to a compact set. Numerical simulations are carried out to verify the effectiveness of the proposed control.