Output Constraint Softening for SISO Model Predictive Control

The presence of constraints in the on-line optimization problem solved by Model Predictive Control algorithms results in a nonlinear control system, even if the plant and model dynamics are linear. This is the case both for physical constraints, like saturation constraints, as well for performance or safety constraints on outputs or other variables of the process. Performance constraints can usually be softened by allowing violation if necessary. This is advisable, as hard constraints can lead to stability problems. The determination of the necessary degree of softening is usually a trial-and-error matter. This paper utilizes a theoretical framework that allows to relate hard as well as soft constraints to closed-loop stability. We focus on the special case of output constraints for single-input single-output systems and develop a non-conservative condition. This condition allows the determination of the appropriate amount of softening either numerically or via a suitable Nyquist plot.