Smooth Model Predictive Control with Applications to Statistical Learning

Statistical learning theory and high dimensional statistics have had a tremendous impact on Machine Learning theory and have impacted a variety of domains including systems and control theory. Over the past few years we have witnessed a variety of applications of such theoretical tools to help answer questions such as: how many state-action pairs are needed to learn a static control policy to a given accuracy? Recent results have shown that continuously differentiable and stabilizing control policies can be well-approximated using neural networks with hard guarantees on performance, yet often even the simplest constrained control problems are not smooth. To address this void, in this paper we study smooth approximations of linear Model Predictive Control (MPC) policies, in which hard constraints are replaced by barrier functions, a.k.a. barrier MPC. In particular, we show that barrier MPC inherits the exponential stability properties of the original non-smooth MPC policy. Using a careful analysis of the proposed barrier MPC, we show that its smoothness constant can be carefully controlled, thereby paving the way for new sample complexity results for approximating MPC policies from sampled state-action pairs.