Convergence Analysis of the Unscented Kalman Filter for Filtering Noisy Chaotic Signals

The unscented Kalman filter (UKF) has recently been proposed for filtering noisy chaotic signals. Though computationally advantageous, the UKF has not been thoroughly analyzed in terms of its convergence property. In this paper, non-periodic oscillatory behavior of the UKF when used to filter chaotic signals is reported. We show both theoretically and experimentally that the gain of the UKF may oscillate aperiodically. More precisely, when applied to periodic signals generated from nonlinear systems, the Kalman gain and error covariance of the UKF converge to zero. However, when the system being considered is chaotic, the Kalman gain either converges to a fixed point with a magnitude larger than zero or oscillates aperiodically.