On asymptotic closed-loop performance for linear MPC with terminal constraints

Employing Lyapunov theorems, under mild conditions, to establish asymptotic stability of the origin, is now common practice in the field of regulatory Model Predictive Control (MPC). However, feasibility and stability does not necessarily constitute towards optimality. By optimality we imply the closed-loop optimality of a receding-horizon control law with respect to the infinite-horizon cost function. Recent results in literature has quantified the sub-optimal closed-loop performance of stabilizing, receding-horizon control laws for MPC formulations with stabilizing terminal state regions. In this work we extend on the aforementioned results on performance, by stipulating necessary conditions under which one can compare asymptotic closed-loop performance of different MPC formulations with terminal regions of varying size. In particular, we present results for linear MPC where one can apply inverse optimality to some degree; hence, conclude results for asymptotic closed-loop performance. The developed concepts are illustrated on two numerical linear case studies.