Vibration isolation control for axially moving strings

High speed transport of continuous materials such as belts, webs, filaments, or bands can cause unwanted vibration. Vibration control for these systems often focuses on restricting the response resulting from external disturbances (e.g. support roller eccentricity or aerodynamic excitation) to areas not requiring high precision positioning. This paper introduces vibration controllers for an axially moving string system consisting of a controlled span coupled to a disturbed span via an actuator. The system model includes a partial differential equation for the two spans and an ordinary differential equation for the actuator. Exact model knowledge and adaptive isolation controllers, based on Lyapunov theory, regulate the controlled span from bounded disturbances in the adjacent, uncontrolled span. Assuming distributed damping in the uncontrolled span, the exact model knowledge and adaptive controllers exponentially and asymptotically drive the controlled span displacement to zero, respectively, while ensuring bounded uncontrolled span displacement and control force.