Notes on Convergence and Modeling for the Extended Kalman Filter

The goal of this work is to provide an understanding of estimation technology for both linear and nonlinear dynamical systems. A critical analysis of both the Kalman filter (KF) and the extended Kalman filter (EKF) will be provided, along with examples to illustrate some important issues related to filtering convergence due to system modeling. A conceptual explanation of the topic with illustrative examples provided in the paper can help the readers capture the essential principles and avoid making mistakes while implementing the algorithms. Adding fictitious process noise to the system model assumed by the filter designers for convergence assurance is being investigated. A comparison of estimation accuracy with linear and nonlinear measurements is made. Parameter identification by the state estimation method through the augmentation of the state vector is also discussed. The intended readers of this article may include researchers, working engineers, or engineering students. This article can serve as a better understanding of the topic as well as a further connection to probability, stochastic process, and system theory. The lesson learned enables the readers to interpret the theory and algorithms appropriately and precisely implement the computer codes that nicely match the estimation algorithms related to the mathematical equations. This is especially helpful for those readers with less experience or background in optimal estimation theory, as it provides a solid foundation for further study on the theory and applications of the topic.