Boundary control of an axially moving string: Actuator dynamics included

In this paper, an active vibration control of a translating tensioned string with the use of an electro-hydraulic servo mechanism at the right boundary is investigated. The equations of motion of the string are derived by using Hamilton's principle for the systems with changing mass. The control objective is to suppress the transverse vibrations of the string via a right-boundary control. An energy-based right-boundary control law, generating a specific current input to the servo-valve, is derived. It is revealed that a time-varying boundary force, as a function of the slope of the string at the right end and a suitably chosen damping coefficient of the actuator, can successfully suppress the transverse vibrations. The exponential stability of the closed loop system is proved. The effectiveness of the proposed control law is demonstrated via simulations.