Examples of zero robustness in constrained model predictive control

Nominal robustness of model predictive control for nonlinear systems is considered. It is shown, by examples, that when the optimization problem involves state constraints, or terminal constraints coupled with short optimization horizons, the asymptotic stability of the closed loop may have absolutely no robustness. Namely, it is possible for arbitrarily small disturbances to keep the closed loop strictly inside the interior of the feasibility region of the optimization problem and, at the same time, far from the desired set point. This phenomenon does not occur when using model predictive control for linear systems with convex constraint sets. It is emphasized that a necessary condition for the absence of nominal robustness in nonlinear model predictive control is that the value function and feedback law are discontinuous at some point(s) in the interior of the feasibility region.