Statistical Results on Filtering and Epi-convergence for Learning-Based Model Predictive Control

TL;DR

This paper provides theoretical proofs on the stochastic convergence and epi-convergence of statistical estimators used in learning-based model predictive control, enhancing understanding of robustness and performance guarantees.

Contribution

It offers new proofs and analysis on the convergence properties of statistical estimators within LBMPC, supporting improved model identification and control robustness.

Findings

Proves stochastic convergence of LBMPC

Establishes epi-convergence of statistical estimators

Analyzes properties of a nonparametric estimator

Abstract

Learning-based model predictive control (LBMPC) is a technique that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance. This technical note provides proofs that elucidate the reasons for our choice of measurement model, as well as giving proofs concerning the stochastic convergence of LBMPC. The first part of this note discusses simultaneous state estimation and statistical identification (or learning) of unmodeled dynamics, for dynamical systems that can be described by ordinary differential equations (ODE's). The second part provides proofs concerning the epi-convergence of different statistical estimators that can be used with the learning-based model predictive control (LBMPC) technique. In particular, we prove results on the statistical properties of a nonparametric estimator that we have designed to have the correct deterministic and stochastic properties for numerical implementation when used in conjunction with LBMPC.