The effect of erroneous models on the Kalman filter response

The optimal filtering equations, as derived by Kalman [1], [2], require the specification of a number of models for a given application. This paper concerns itself with the effect of errors in the assumed models on the filter response. The types of errors considered are those in the covariance of the initial state vector, the covariance of the stochastic inputs to the system, and the covariance of the uncorrelated measurement noise. Presented here is a derivation of a recursive equation for the actual covariance matrix of the estimation error when the filter design is based upon erroneous models. The derived equation can also be used to obtain the covariance matrix of the estimation error when the optimal filter gains are approximated by simple functions of time to be used in a real-time filtering application. A numerical example illustrates the use of the derived equations.