This paper introduces a tube-based model predictive control (MPC) for linear parameter-varying (LPV) systems which exploits knowledge about bounds on the parameters' rate of change to extrapolate its admissible values over the prediction horizon. This information is used to construct state tubes to which the future trajectories of the state are confined. The tubes are consequently used for constraint tightening. Then, an MPC optimization problem subject to tightened sets for the state and control constraints is solved for only a nominal system corresponding to a nominal trajectory of the scheduling parameter starting from its current value. Recursive feasibility and asymptotic stability are proven and two numerical examples are given to demonstrate the effectiveness of the proposed approach. (C) 2019 Elsevier Ltd. All rights reserved.