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**Robust adaptive boundary control of an axially moving string under a spatiotemporally varying tension.**
*(English)*
Zbl 1236.93100

Summary: A vibration suppression scheme for an axially moving string under a spatiotemporally varying tension and an unknown boundary disturbance is investigated. The lower bound of the tension variation is assumed to be sufficiently larger than the derivatives of the tension. The axially moving string system is divided into two spans, i.e., a controlled span and an uncontrolled span, by a hydraulic touch-roll actuator which is located in the middle section of the string. The transverse vibration of the controlled span part of the string is controlled by the hydraulic touch-roll actuator, and the position of the actuator is considered as the right boundary of the controlled span part. 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. The Lyapunov method is employed to design a robust boundary control law and adaptation laws for ensuring the vibration reduction of the controlled span part. The asymptotic stability of the closed loop system under the robust adaptive boundary control scheme is proved through the use of semigroup theory. Simulation results verify the effectiveness of the robust adaptive boundary controller proposed.