The inclusion of output constraints in the on-line optimization problem solved by quadratic dynamic matrix control (QDMC) can result in a unstable, closed-loop system, even when the corresponding unconstrained algorithm is stable. The presence of constraints in the optimization problem produces a nonlinear, closed-loop system, although the plant and model dynamics are assumed linear. This article quantifies the effect of output constraints on both nominal and robust stability for single-input/single-output processes. Stability conditions provided can be used to select the QDMC parameters and constraint window. Examples demonstrate that these conditions can capture the nonlinear effect of the constraints and that tuning rules developed for the unconstrained case should not be used with output constraints. An example with dead time error illustrates the use of this framework to guarantee the robustness of constrained QDMC with respect to modeling error.