On the nonlinear dynamics of fast filtering algorithms

A fundamental open problem in linear filtering and estimation is addressed, i.e. what is the steady-state or asymptotic behavior of the Kalman filter, or the Kalman gain, when the observed stationary stochastic process is not generated by a finite-dimensional stochastic system, or when it is generated by a stochastic system having higher dimensional unmodeled dynamics? For a scalar observation process, necessary and sufficient conditions are derived for the Kalman filter to converge, using methods from stochastic systems and from nonlinear dynamics, especially the use of stable, unstable and center manifolds. It is shown that, in nonconvergent cases, there exist periodic points of every period p, p⩾3 which are arbitrarily close to initial conditions having unbounded orbits. This rigorously demonstrates that the Kalman filter can also be sensitive to initial conditions