In this paper, a Linear Parameter Varying (LPV) model of the Furuta pendulum is derived. Based on this model, a balancing controller is designed using robust predictive control techniques. Invariant set theory is used to accurately characterise the region of the state space in which the balancing controller is effective. An energy-based control law is used to swing up the pendulum. Invariant sets calculated for the balancing controller can be exploited to systematically determine the switching condition between swing up and balancing controllers. In practice, the pendulum may be swung towards the upright position with varying speed and the speed of the rotating arm may also vary. Based on this physical insight, the speed of the rotating arm is chosen as gain scheduling variable. It is shown that this strategy is effective in achieving a more consistent swing up and balancing behaviour which is not sensitive to the performance of the swing up controller.