  To address feasibility issues in model predictive control (MPC), most
implementations relax hard state constraints using additional slack variables
with a suitable penalty. We propose an alternative strategy for open-loop
asymptotically/Lyapunov stable nonlinear systems by relaxing the initial state
constraint with a suitable penalty. The proposed MPC framework is globally
feasible, ensures (semi-)global asymptotic stability, and (approximately)
recovers the closed-loop properties of the nominal MPC on the feasible set. The
proposed framework can be naturally combined with a robust formulation to
ensure robustness subject to bounded disturbances while retaining
input-ot-state stability in case of arbitrarily large disturbances. We also
show how the overall design can be simplified in case the nonlinear system is
exponentially stable. In the special case of linear systems, the proposed MPC
formulation reduces to a quadratic program and the offline design and online
computational complexity is only marginally increased compared to anominal
design. Benefits compared to classical soft contrained MPC formulations are
demonstrated with numerical examples.
