Improved stabilising conditions for model predictive control

It is known that stability of a model predictive control system is ensured if the terminal conditions of the optimal control problem solved online satisfy certain criteria. The usual requirement is that the terminal cost function is a control Lyapunov function defined on the terminal constraint set. Conventionally the terminal cost function is chosen, when the system being controlled is linear, to be the value function for the infinite horizon unconstrained optimal control problem and the terminal constraint set is chosen to be the output admissible set for the closed-loop system using the optimal unconstrained controller u=-Kx. The purpose of the paper is to relax these terminal conditions thereby facilitating online solution of the optimal control problem. Using some earlier results, we present alternative conditions that employ, as the terminal cost, the infinite horizon cost resulting from a nonlinear controller u=-sat(Kx) and, as the terminal constraint set, the set in which this controller is optimal for the infinite horizon constrained optimal control problem. It is shown that this solution provides a considerably larger terminal constraint set