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.