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

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.

💡 Research Summary

This technical note provides a rigorous theoretical foundation for Learning‑Based Model Predictive Control (LBMPC) by addressing two intertwined challenges: the design of a measurement model that enables simultaneous state estimation and learning of unmodeled dynamics, and the statistical convergence properties of the learned model when it is embedded in the MPC optimization.
In the first part the authors consider continuous‑time dynamical systems described by ordinary differential equations and derive a discrete‑time observation model suitable for digital implementation. Recognizing that a standard linear Kalman filter would introduce bias in the presence of nonlinearities, they propose an extended‑Kalman‑filter‑like scheme that incorporates a learning‑based correction term. This term is updated online using the most recent measurements and the current state estimate, thereby compensating for the residual nonlinear dynamics. The resulting estimator yields a probabilistically bounded error, which is essential for guaranteeing the robustness of the subsequent MPC law.
The second part focuses on the statistical estimator that supplies the predictive model for MPC. The authors introduce a non‑parametric kernel‑regression estimator specifically engineered to satisfy the deterministic requirements of MPC (boundedness, continuity) while also possessing strong stochastic properties. The central theoretical contribution is a proof of epi‑convergence of the estimated cost function to the true cost function as the sample size grows. Under standard assumptions—i.i.d. data, a bounded symmetric kernel, and a bandwidth sequence that shrinks to zero while satisfying (n h_n^d \to \infty)—they show that both bias and variance vanish, guaranteeing that the estimator’s mean‑squared error converges to zero. Consequently, the optimal control sequence computed with the learned model converges to the optimal sequence that would be obtained with the exact model, preserving the robust guarantees of LBMPC.
To make the non‑parametric estimator practical for real‑time control, the authors develop an algorithm that combines random Fourier features with dimensionality reduction, dramatically lowering memory and computational demands. A normalized weight‑update rule ensures numerical stability, and the algorithm can process streaming data without batch recomputation.
Simulation studies on a nonlinear robotic arm and an aircraft attitude control problem illustrate the benefits. Compared with a conventional LBMPC that relies on a linear nominal model, the proposed approach reduces the average control cost by roughly 15 % while maintaining disturbance rejection capabilities. As the learning progresses, control input variability diminishes and response times improve, confirming the theoretical predictions.
In summary, the paper establishes that (1) a carefully constructed measurement model can deliver accurate, online state estimates even for nonlinear ODE systems, and (2) a suitably designed non‑parametric estimator enjoys epi‑convergence, ensuring that the learned model integrates seamlessly with MPC without sacrificing robustness. These results broaden the applicability of LBMPC to complex, uncertain systems where richer models are essential for high performance.