A linear MPC algorithm for embedded systems with computational complexity guarantees

In this paper we propose a linear MPC scheme for embedded systems based on the dual fast gradient algorithm for solving the corresponding control problem. We establish computational complexity guarantees for the MPC scheme by appropriately deriving tight convergence estimates of order O(1/k
²
) for an average primal sequence generated by our proposed numerical optimization algorithm. We also show that these estimates rely heavily upon the dual optimal variables (Lagrange multipliers). Since convergence certification in embedded MPC is essential, we also derive tight bounds on the norm of these dual optimal variables. However, computing the norm for the Lagrange multipliers associated to multi-parametric optimization problems can quickly become intractable for high dimension and/or a large set of constraints. Therefore, we recast the problem in a mixed integer formulation by using auxiliary binary variables to characterize the complementarity conditions. We also show that this problem can be solved numerically efficiently.