The model predictive control problem with guaranteed stability for a class of linear distributed parameter systems with simultaneous amplitude and rate control constraints is investigated. The approach is developed using a finite dimensional, input-output modelling technique for distributed parameter systems based on a distributed output projection onto a finite basis set. The paper employs a dual mode prediction strategy in conjunction with the projection-based state space system description and proposes algorithms for the design of polyhedral positive invariant and feasible sets which serve as terminal target sets for predicted states. An extension to predictive control with integral action is described. The proposed algorithms are shown to work well by means of both simulations and experimental results.