Predictability and unpredictability in Kalman filtering

The authors study the dynamical behavior of the Kalman filter when the given parameters are allowed to vary in a way which does not necessarily correspond to an underlying stochastic system. This may correspond to situations in which the basic parameters are chosen incorrectly through estimates. The authors show that, as has been suggested by Kalman, the filter equations converge to a limit (corresponding to a steady-state filter) for a subset of the parameter space which is much larger than that corresponding to bona fide stochastic systems. More surprisingly, in the complement of this subset, the filtering equations behave in both a regular and an unpredictable manner, representative of some of the basic aspects of chaotic dynamics. This interesting dynamical behavior occurs already for one-dimensional filters, and a complete phase portrait in this case is given.>