Terminal constraints provide for stable predicted trajectories and thus form the basis of predictive control algorithms with guaranteed stability. In the presence of system input constraints, however, stability is dependent on a feasibility assumption, namely that it is possible to meet the terminal constraints with the allowable inputs, The terminal constraints used to date are only sufficient and lead to highly tuned controllers with high input activity which may cause infeasibility with a consequent loss of the guarantee of stability, Here we overcome this problem by deriving : conditions which are both necessary and sufficient. Use of these leads to an increase in the degrees of freedom which can be used to either improve performance or reduce the relevant control horizon thereby affording significant computational advantages. The efficacy of the proposed algorithms are illustrated by means of numerical examples.