Vibration Control and Boundary Tension Constraint of an Axially Moving String System

In this chapter, it is concerned with vibration control and boundary tension constraint of an axially moving string system. The main control objectives are to suppress the vibration of the string system and ensure the boundary tension in a constrained region. To this end, a boundary control strategy is developed via constructing a proper barrier Lyapunov function. In addition, the disturbance observer including the barrier term is introduced to tackle the effects of unknown boundary disturbance. With the proposed control strategy, the states of the closed-loop system are proven to be uniformly ultimately bounded through rigorous Lyapunov analysis and the boundary tension constraint is not violated.