Vibration control of the flexible manipulator with input constraints and external disturbances based on Nussbaum function

In this paper, a boundary control scheme based on the partial differential equation (PDE) model is proposed for the vibration control problem of the flexible manipulator with input constraints and external disturbances. Based on the backstepping method, two boundary controllers are designed to stabilize the position loop subsystem and the attitude loop subsystem, respectively, and auxiliary systems based on the smooth hyperbolic tangent function and Nussbaum function are designed in the controllers to deal with the input saturation and external disturbances. The Nussbaum function can overcome the difficulties in controller design and stability analysis caused by the derivatives of smooth hyperbolic tangent functions. The well-posedness of the closed-loop system is proven by employing the semigroup theory, and the uniformly bounded stability is proved by Lyapunov direct method. Finally, the performance of the proposed control laws is verified by numerical simulations.