We introduce a model predictive controller for input constrained linear systems based on a novel barrier function. As the barrier has fixed weight, the controller may be implemented in a highly efficient and numerically robust manner. We prove that the resulting closed loop behaviour is stable and converges to any specified target steady-state solution on the interior of the constraint set. In the case where the optimal steady-state solution may lie on the boundary of the constraint set, we propose a sub-optimal target computed with a traditional logarithmic barrier, but again with fixed weight. This guarantees a solution on the interior of the constraint set. The duality gap allows us to pre-specify a bound on the resulting degradation in system performance. In such a case the maximal invariant set may become arbitrarily small and this may be problematic in terms of feasibility. Such remarks are pertinent to any model predictive controller where stability results are based on invariant sets.