A model predictive control framework with robust stability guarantees under unbounded disturbances

To address feasibility issues in model predictive control (MPC), most implementations relax state constraints by using slack variables and adding a penalty to the cost. We propose an alternative strategy: relaxing the initial state constraint with a penalty. Compared to state-of-the-art soft constrained MPC formulations, the proposed formulation has two key features: (i) input-to-state stability and bounds on the cumulative constraint violation for unbounded disturbances; (ii) close-to-optimal performance under nominal operating conditions. To address unbounded disturbances with compact input constraints, we focus on open-loop stable systems; however, the method applies equally to stabilizable systems when soft input constraints are used. In the special case of linear systems, the proposed MPC formulation reduces to a quadratic program, and the offline design and online computational complexity are only marginally increased compared to a nominal design. Numerical examples demonstrate benefits compared to state-of-the-art soft-constrained MPC formulations.