Robust boundary control of an axially moving string by using a PR transfer function

In this note, a scheme for the vibration suppression of a translating string using a positive real (PR) transfer function is investigated. The transverse vibration of the string is controlled by hydraulic touch-rolls located at the right end of the string. The mathematical model of the system, which consists of a hyperbolic partial differential equation (PDE) describing the dynamics of the moving string and an ordinary differential equation (ODE) for the actuator dynamics, is derived by using Hamilton´s principle for translating continua. The transfer function of the proposed boundary controller is a nonproper but PR function. The asymptotic stability of the closed-loop system in the presence of output disturbance is proved.