Real-Time MPC with Input Constraints Using the Fast Gradient Method

Abstract

This paper considers the application of Nesterov's fast gradient method in the context of linear quadratic model predictive control (MPC) with input constraints. The main focus is on the method's a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre-specified suboptimality is obtained for any possible state of the dynamic system. Cold- and warm-starting strategies for MPC are defined and the corresponding problems to obtain the bounds are discussed. Moreover, an easy to compute upper bound for cold-starting is given and an asymptotic characterization for warm-starting is presented. We also investigate the choice of the suboptimality level in the context of MPC and propose an optimal preconditioning technique to increase the convergence rate of the method. Finally, we characterize the set of MPC problems for which small iteration bounds and thus short solution times are expected in terms of the system and weight matrices and underpin the theoretical investigations as well as the practical applicability by various numerical examples, which indicate that extremely short certified solution times can be obtained.