Effects of hard constraints in the stability of model-predictive control (MPC) are reviewed. Assuming a fixed active set, the optimal solution can be expressed in a general state-feedback closed form, which corresponds to a piecewise linear con-troller for the linear model case. Changes in the original unconstrained solution by the active constraints and other effects related to the loss of degrees of freedom are depicted in this analysis. In addition to modifications in the unconstrained feedback gain, we show that the presence of active output constraints can introduce extra feedback terms in the predictive controller. This can lead to instability of the constrained closed-loop system with certain active sets, independent of the choice of tuning parameters. To cope with these problems and extend the constraint handling capabilities of MPC, we introduce the consideration of soft constraints. We compare the use of the l2-(quadratic), l1- (exact), and l(infinity)-norm penalty formulations. The analysis reveals a strong similarity between the control laws, which allows a direct extrapolation of the unconstrained tuning guidelines to the constrained case. In particular, the exact penalty treatment has identical stability characteristics to the correspondent unconstrained case and therefore seems well suited for general soft constraint handling, even with nonlinear models. These extensions are included in the previously developed Newton control framework, allowing the use of the approach within a consistent framework for both linear and nonlinear process models, increasing the scope of applications of the method. Process examples illustrate the capabilities of the proposed approaches.
