Robust adaptive boundary control of a vibrating string with time-varying constraints

In this paper, we present the vibration control design for a string with the boundary time-varying output constraints. To prevent the constraint violations, a novel barrier Lyapunov function is proposed for the control design and stability analysis. The dynamics of the string is a distributed parameter system described by a partial differential equations (PDE) and two ordinary differential equations (ODEs). Firstly, model-based barrier control is employed to suppress the vibration of a nonlinear string under the boundary time-varying output constraints. Subsequently, adaptive control is designed to handle the system parametric uncertainties. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed design.