Nonlinear Kalman Filtering for Systems under the Influence of State-Dependent Noises

Abstract

Kalman Filtering (KF) theory stands as a cornerstone in the field of dynamic state estimation, but it continues to encounter persistent challenges, particularly with respect to nonlinear systems and state-dependent noise. While established variants such as the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) have achieved considerable prominence for their utility in complex estimation problems, their foundational assumption of zero-mean, Gaussian noise is often at odds with some physical systems and their observations. Indeed, practical engineering applications — ranging from autonomous vehicles navigating uncertain environments to financial models tracking volatile markets, and robotic sensors operating under fluctuating conditions — reveal the prevalence of noise that is not only non-Gaussian and biased, but also intricately linked to the system state. Such conditions can significantly undermine estimation accuracy and may even precipitate filter divergence. Accordingly, there is a pressing need for filtering methodologies that are both more resilient and more attuned to the nuances of real-world systems that are influenced by state-dependent noises.

This dissertation seeks to address these gaps through several key contributions. First, it introduces a novel nonlinear Kalman Filtering approach that explicitly accommodates non-zero-mean and state-dependent noise within both process and measurement models. Second, it introduces a structured framework for noise modeling, seamlessly integrating these characteristics into a revised prediction-correction paradigm. Third, the methodology is extended to encompass systems lacking direct measurement-to-state correspondences and is shown to be compatible with arbitrary nonlinear transformations, thereby broadening its practical scope. Fourth, rigorous theoretical guarantees are established, demonstrating that the proposed filter achieves unbiasedness and minimum variance under well-defined conditions. Fifth, the principle underlying state-dependent noise Kalman filtering is extended to improve performance when stronger nonlinearities exist. The proposed Kalman filtering schemes preserve the well-known recursive structure of Kalman filters while maintaining computational tractability.

A comprehensive suite of empirical evaluations attests to the efficacy of the proposed approach. Across a spectrum of test scenarios, the proposed filters demonstrate the ability to give reliable state estimates by reducing estimation errors and improving robustness. These empirical findings not only reinforce the theoretical developments presented herein but also illustrate the filter's capacity to adapt to nonlinear systems characterized by intricate, state-dependent noise. Furthermore, this work draws attention to enduring limitations in current Kalman Filtering methodologies, including the need for more comprehensive convergence analyses and the development of robust strategies for handling systems influenced by state-dependent noise.

Opportunities for future research emerge in several promising directions, including the design of adaptive filters leveraging machine learning for dynamic noise model adaptation, and the incorporation of supplementary sensing modalities to enhance error detection and mitigation. The formulation of a unified theoretical framework capable of accommodating a wide array of noise structures would represent a significant advancement for real-time state estimation. Addressing these open challenges promises not only to advance the field of nonlinear filtering theory but also to broaden its applicability to areas such as autonomous systems, sensor networks, economics, and healthcare. This dissertation strengthens the foundational theory of Kalman filtering and creates a path forward for sustained scholarly innovation in the modeling and estimation of systems that do not typically satisfy the assumptions required for the implementation of traditional Kalman filtering.