Asymptotic Behavior of Kalman-Type Filters Applied to Chaotic Plants

This paper discusses tX application of the linearized and extended Kalman filters to plants with chaotic dynamics. The behavior of the filter is explored by an covariance analysis and numerical experiment. A linearized Kalman filter operating over a nominal, noise-free, state trajectory is considered. Under certain sufficient conditions, the covariance estimate and gains of this approximation become, at long times, a unique, continuous function of the plant´s state. It is pointed out that the existence of such asymptotic gains allows the construction of an approximate extended Kalman filter using gains and covariance estimate that are functions of state estimate only. A generalized steady-state Riccati equation for the steady-state covariance is given. It is shown under certain conditions that its solution is determined by the solutions of periodic matrix Riccati equations for linearized Kalman filters operating over the unstable periodic orbits that lie in the plant´s attractor or stochastic region. Numerical experiments with the Henon quadratic map are used as demonstrate the linearized filter covariance analysis and to check its applicability to the extended Kalman filter.