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Statistical learning theory and high dimensional statistics have had atremendous impact on Machine Learning theory and have impacted a variety ofdomains including systems and control theory. Over the past few years we havewitnessed a variety of applications of such theoretical tools to help answerquestions such as: how many state-action pairs are needed to learn a staticcontrol policy to a given accuracy? Recent results have shown that continuouslydifferentiable and stabilizing control policies can be well-approximated usingneural networks with hard guarantees on performance, yet often even thesimplest constrained control problems are not smooth. To address this void, inthis 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 theexponential stability properties of the original non-smooth MPC policy. Using acareful analysis of the proposed barrier MPC, we show that its smoothnessconstant can be carefully controlled, thereby paving the way for new samplecomplexity results for approximating MPC policies from sampled state-actionpairs.

Smooth Model Predictive Control with Applications to Statistical Learning