Learning a convex cost-to-go for single step model predictive control

Abstract: For large uncertain systems, solving model predictive control problems online can be computationally taxing. Using a shorter prediction horizon can help, but may lead to poor performance and instability without appropriate modifications. This work focuses on learning convex objective terms to enable a single-step control horizon, reducing online computational costs. We consider two surrogates for approximating the cost-to-go: (1) a convex interpolating function and (2) an input-convex neural network. Regardless of the surrogate choice, its behavior near the origin and its ability to describe the feasible region are crucial for the closed-loop performance of the new MPC problem. We address this by tailoring the surrogate to ensure good performance in both aspects. We conclude with numerical examples, in which we compare the convex surrogates to using a standard neural network in the objective, solely using an LQR cost-to-go, and to using a neural network to learn a control policy. The proposed approaches are shown to achieve better performance with less data.