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.