Robust model predictive control (RMPC) adopts either nominal or worst-case cost as the performance index to be minimised online. The former leads to efficient implementations but has the disadvantage that model uncertainty may lead to an over-optimistic strategy that ignores the sensitivity of the cost to the effects of model uncertainty. The introduction of dynamics into the prediction structure of RMPC through the use of a Youla parameter provides extra degrees of freedom with which to desensitise the cost to model uncertainty. This paper develops a methodology that allows this idea to be used in the presence of constraints. To avoid limitations concerning the systematic design of the Youla parameter, more general prediction dynamics are considered and an efficient control algorithm with guaranteed feasibility and stability is presented. A simple modification is also proposed that minimises the worst case cost while retaining the same control theoretical properties and efficiency of implementation.