Nonlinear state estimation, indistinguishable states, and the extended Kalman filter

It has been shown by Judd and Smith that it is impossible to determine the state of a nonlinear dynamical system from noisy observations of the system, even with perfect knowledge of the system dynamics and unlimited prior observation. There is always a set of states indistinguishable from the true state. However, a new, simple method to assimilate data into a model and estimate the state is suggested. This method is related to a dynamical systems approach to nonlinear filtering, that is, the use of shadowing trajectories in nonlinear noise reduction. In this paper the performance of this new method of state estimation is compared with that of the extended Kalman filter. It is found that the new method performs better, largely owing to it taking into account the nonlinearity of the system.