  An explicit Model Predictive Control algorithm for large-scale structured
linear systems is presented. We base our results on Distributed and Localized
Model Predictive Control (DLMPC), a closed-loop model predictive control scheme
based on the System Level Synthesis (SLS) framework wherein only local state
and model information needs to be exchanged between subsystems for the
computation and implementation of control actions. We provide an explicit
solution for each of the subproblems resulting from the distributed MPC scheme.
We show that given the separability of the problem, the explicit solution is
only divided into three regions per state and input instantiation, making the
point location problem very efficient. Moreover, given the locality
constraints, the subproblems are of much smaller dimension than the full
problem, which significantly reduces the computational overhead of explicit
solutions. We conclude with numerical simulations to demonstrate the
computational advantages of our method, in which we show a large improvement in
runtime per MPC iteration as compared with the results of computing the
optimization with a solver online.
