Some studies on state estimation of linear and nonlinear systems

Abstract

This thesis investigates the performance of linear and nonlinear estimation methods, focusing on Kalman Filter (KF), Extended Kalman Filter (EKF), and Adaptive Kalman Filter (AKF) in estimating dynamic system states. A comprehensive MATLAB implementation is developed for 1D, 2D, 3D, and N-Dimensional Kalman Filters, based on established algorithms, to analyze and validate these filters' performance across a range of linear and nonlinear systems. This work provides a comparative study between the EKF and the recently proposed AKF in nonlinear environments, with an emphasis on their responses to measurement noise covariance. The study primarily evaluates the filter’s ability to minimize estimation error under varying conditions of measurement noise covariance, 𝑅. It is observed that the EKF demonstrates robust performance in systems with mild nonlinearities, given that the measurement noise covariance is accurately known or estimated. However, in scenarios where measurement noise covariance is uncertain or unknown, the AKF exhibits superior adaptability and precision, thus outperforming the EKF. This adaptability of the AKF makes it a preferable choice for applications where measurement noise characteristics cannot be precisely determined. Results indicate that when the actual measurement noise covariance is known or reasonably approximated, both AKF and EKF offer comparable accuracy. Nevertheless, when covariance knowledge is limited, AKF’s performance advantage becomes significant. While the MATLAB code developed for this study successfully supports these analyses, further optimization could reduce computational demands, particularly for extensive Monte Carlo simulations. Additionally, refining the AKF algorithm could enhance its applicability to systems with stronger nonlinear behaviors, potentially expanding its practical usage across more complex, real-world systems. In conclusion, this thesis underscores the importance of adaptive filtering in scenarios of measurement uncertainty and highlights potential areas for advancing Kalman Filter implementations in both theoretical and applied contexts.