A crane control system is presented where the crane load can track a position trajectory with ultimately bounded pendulum motions of the load. This is done with an inner loop using a Lyapunov-based damping controller which is designed to damp out the pendulum motions of the load, and an outer loop where a nonlinear MPC (model predictive controller) controls the position of the load. The Lyapunov-based damping controller is designed to be exponentially stable, which means that the pendulum motion will be ultimately bounded in the presence of a bounded perturbation. Then this is achieved by constraining the control action of the nonlinear MPC so that the perturbations to the Lyapunov-based damping controller are sufficiently bounded. The suspension point of the load, which is the tip of the crane, can then be controlled to track a time-varying desired trajectory in the plane with ultimately bounded pendulum motions. The control system is validated in simulations in a simulation study.