A Unified Control Theory Derivation of Discrete-Time Linear Ensemble Kalman Filters

Abstract: The ensemble Kalman filter (EnKF) has become a standard methodology for state estimation in high-dimensional systems, yet its various stochastic and deterministic formulations often appear conceptually disconnected. In this paper, a unified derivation framework for EnKF algorithms are established by leveraging the classical duality between estimation and optimal control, which is the key concept in deriving Kalman filter. By recasting the minimum variance estimation problem into second order moment for the ensembles, we demonstrate that seemingly distinct EnKF variants -- both with or without perturbed observation -- can be systematically classified. Specifically, the duality based framework reveals that the operational differences among these variety of EnKF algorithms reduce to a specific choice of hyperparameters. Ultimately, this perspective not only covers existing EnKF variants but also provides a systematic foundation for designing novel hybrid filters using control theory approach.