Formulation of Closed-Loop Min–Max MPC as a Quadratically Constrained Quadratic Program

In this note, we show that min-max model predictive control (MPC) for linearly constrained polytopic systems with quadratic cost can be cast as a quadratically constrained quadratic program (QCQP). We use the rigorous closed loop formulation of min-max MPC, and show that any such min-max MPC problem with convex costs and constraints can be cast as a finite dimensional convex optimization problem, with the QCQP arising from quadratic costs as a special case. At the base of the proof is a lemma showing the convexity of the dynamic programming cost-to-go, which implies that the worst case on an infinite polytopic set is assumed on one of its finitely many vertices. As the approach is based on a scenario tree formulation, the number of variables in this problem grows exponentially with the horizon length. Fortunately, the QCQP is tree structured, and can thus be efficiently solved by specially tailored interior-point methods whose computational costs are linear in the number of variables. The new formulation as a tree sparse QCQP promises to facilitate online solution of the rigorous min-max MPC problem with quadratic costs