Reducing the computational effort of MPC with closed-loop optimal sequences of affine laws

Abstract We consider the classical infinite-horizon constrained linear-quadratic regulator (CLQR) problem and its receding-horizon variant used in model predictive control (MPC). If the terminal constraints are inactive for the current initial condition, the optimal input signal sequence that results for the open-loop CLQR problem is equal to the closed-loop optimal sequence that results for MPC. Consequently, the closed-loop optimal solution is available from solving only one CLQR problem instead of the usual infinite number of CLQR problems solved on the receding horizon. In the presence of disturbances or because of plant-model mismatch, the system will eventually leave the predicted optimal trajectory. Consequently, the solution of the single open-loop CLQR problem is no longer optimal, and the receding horizon problem must resume. We show, however, that the open-loop solution is also robust. Robustness essentially is given, because the solution of the CLQR problem not only provides the sequence of nominally optimal input signals, but a sequence of optimal affine laws along with their polytopes of validity. We analyze the degree of robustness by computational experiments. The results indicate the degree of robustness is practically relevant.