Stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

The extended Kalman filter is extensively used in nonlinear state estimation problems. As long as the system characteristics are correctly known, the extended Kalman filter gives the best performance. However, when the system information is partially known or incorrect, the extended Kalman filter may diverge or give biased estimates. An extensive number of works has been published to improve the performance of the extended Kalman filter. Many researchers have proposed the introduction of a forgetting factor, both into the Kalman filter and the extended Kalman filter, to improve the performance. However, there are 2 fundamental problems with this approach: the incorporation of the optimal forgetting factor into the (extended) Kalman filter and the selection of the optimal forgetting factor. These problems have not yet been fully resolved and are still open problems in the field. In this study, we propose a new adaptive fading extended Kalman filter with a matrix forgetting factor, and 2 methods are analyzed for the selection of the optimal forgetting factor. The stability properties of the proposed filter are also investigated. Results of the stability analysis show that the proposed filter is an exponential observer for nonlinear deterministic systems. Additionally, the convergence speed of the filter is simulated.