Ensemble Kalman-Bucy filtering for nonlinear model predictive control

TL;DR

This work addresses the challenge of controlling partially observed nonlinear systems where estimation and control interact nontrivially. It introduces EnKBF-NMPC, a framework that marries the ensemble Kalman-Bucy filter with nonlinear model predictive control, using forward-backward stochastic differential equations derived from Pontryagin's maximum principle to produce linear, receding-horizon control laws that account for uncertainties in both initial states and future observations. In the linear case, the method recovers the classic linear-Gaussian results and yields time-varying gains solved by coupled ODEs; for nonlinear systems, it leverages multiple noise realizations and regression to approximate the backward dynamics and generate tractable controls. A demonstrative inverted pendulum example shows the approach can stabilize an unstable equilibrium under measurement noise and state-estimation uncertainty, highlighting the practical potential for data-assimilation–driven NMPC in real-time applications. Overall, the paper provides a computationally feasible scheme that integrates state estimation and predictive control to improve robustness in partially observed settings.

Abstract

We consider the problem of optimal control for partially observed dynamical systems. Despite its prevalence in practical applications, there are still very few algorithms available, which take uncertainties in the current state estimates and future observations into account. In other words, most current approaches separate state estimation from the optimal control problem. In this paper, we extend the popular ensemble Kalman filter to receding horizon optimal control problems in the spirit of nonlinear model predictive control. We provide an interacting particle approximation to the forward-backward stochastic differential equations arising from Pontryagin's maximum principle with the forward stochastic differential equation provided by the time-continuous ensemble Kalman-Bucy filter equations. The receding horizon control laws are approximated as linear and are continuously updated as in nonlinear model predictive control. We illustrate the performance of the proposed methodology for an inverted pendulum example.

Figures (3)

• Figure 1: Left panels: We display the frequentist forecast uncertainties obtained from simulating (\ref{['eq:SFBSDE']}a) using the optimal control obtained from the previous iteration of (\ref{['eq:SFBSDE']}) with the initial control being $u\equiv 0$. We perform three iterations. Right panels: We display the corresponding entries of $G^{\rm T}\Lambda_t^\ast$, which have been obtained by minimizing (\ref{['eq:cost 2']}). We find that the control terms have essentially converged after two iteration.
• Figure 2: Ensemble mean from an EnKBF filter implementation with controls computed using the proposed EnKBF-NMPC methodology. The control laws were computed over time intervals of length $T = 0.5$ with the computed control then applied over intervals of length $\Delta \tau = 0.05$. The process was repeated 5000 times. One can clearly identify the stabilizing effect of the control.
• Figure 3: Same setting as used for Figure \ref{['fig2']} except for a reduced measurement with $R = 0.1$. The reduction in variance of the computed solution relative to the unstable $q=0$ equilibrium is clearly visible.