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

In this paper, a vibration suppression scheme of an axially moving string using a PR(positive real)-like transfer function is investigated. The transverse vibration of the axially moving string is controlled by the hydraulic touch-roll actuator located at the right boundary of the string system. The mathematical model of the system, which consists of a hyperbolic partial differential equation describing the dynamics of the moving string and an ordinary differential equation describing the actuator dynamics, is derived by using the Hamilton´s principle of changing mass. The transfer function of the robust boundary controller proposed is a PR-like function which contains a positive real function of the complex variables. The PR-like transfer function has a pair of complex conjugate poles on the imaginary axis, provided that the residues corresponding to these poles are nonnegative. The asymptotic stability of the closed loop system under the robust boundary controller is proved through the use of semigroup theory and invariance principle. In the case where the output of the controller is corrupted by a disturbance, it may be possible to attenuate the effect of the disturbance at the output if the controller transfer function is appropriately chosen.