Stability of Model Predictive Control with Soft Constraints

We derive stability conditions for Model Predictive Control (MPC) with hard constraints on the inputs and "soft" constraints on the outputs for an infinitely long output horizon. We show that with state feedback MPC is globally asymptotically stabilizing if and only if all the eigenvalues of the open loop system are in the closed unit disk. With output feedback the eigenvalues must be strictly inside the unit circle. The on-line optimization problem defining MPC can be posed as a finite dimensional quadratic program even though the output constraints are specified over an infinite horizon.