Soft Constraints and Exact Penalty Functions in Model Predictive Control

One of the strengths of Model Predictive Control (MPC) is its ability to incorporate constraints in the control formulation. Often a disturbance drives the system into a region where the MPC problem is infeasible and hence no control action can be computed. Feasibility can be recovered by softening the constraints using slack variables. This approach does not necessarily guarantee that the constraints will be satisfied, if possible. Results from the theory of exact penalty functions can be used to guarantee constraint satisfaction. This paper describes a method for computing a lower bound for the constraint violation penalty weight of the exact penalty function. One can then guarantee that the soft-constrained MPC solution will be equal to the hard-constrained MPC solution for a bounded subset of initial states, control inputs and reference trajectories.