Adaptive boundary control of an axially moving string under the effect of boundary disturbance

In this paper, an active vibration control of a tensioned axially moving string under the effect of boundary disturbance is investigated. The dynamics of the translating string are described by a non-linear partial differential equation coupled with an ordinary differential equation. Disturbance estimation at the boundary is performed. A time varying control in the form of right boundary transverse motions is proposed in stabilizing the transverse vibrations of the translating continuum. A control law based on Lyapunov method is derived. Exponential stability of the closed-loop system is verified. The effectiveness of the proposed controller is shown through simulations.