Robust Kalman Filtering via Correntropy-Based Higher-Order Moment Adaptation and Variable Structure Gains

This paper proposes two novel filtering strategies as sub-optimal robust solutions for state estimation in systems affected by non-Gaussian noise, outliers, or modeling uncertainties. The moments-based Kalman filter (MKF) and moments-based innovation filter (MIF) replace the mean squared error criterion with a correntropy-based cost function that incorporates higher-order statistical moments of the innovation sequence. Through Taylor series expansion of the Gaussian kernel, correntropy inherently captures all even-order moments-including variance, kurtosis, and higher-order statistics-providing natural robustness to heavy-tailed and asymmetric noise distributions. An adaptive kernel bandwidth mechanism uses real-time estimates of innovation skewness and kurtosis to automatically balance efficiency and robustness. The MIF augments this framework with variable structure control theory, incorporating a saturation-based gain that bounds corrective action during large disturbances. Both methods employ fixed-point iteration with correntropy-weighted covariance matrices in their predictor-corrector algorithms. Mathematical derivations and stability proofs are provided for both filters. The approaches extend to nonlinear systems through first-order Taylor series linearization, yielding the extended MKF (EMKF) and extended MIF (EMIF). To validate their robustness relative to the conventional Kalman filter, the proposed methods are applied to both linear and nonlinear representations of a simulated electrohydrostatic actuator (EHA) experiencing leakage faults. Computational experiments demonstrate that the MKF and MIF achieve superior estimation accuracy compared to the KF under non-Gaussian conditions, more faithfully representing faulty system behavior.