Analysis of discrete-time Kalman filtering under incorrect noise covariances

Analysis tools are developed that can be effectively used to study the performance degradation of a filter when incorrect models of the state and measurement noise covariances are used. For a linear time-variant system with stationary noise processes, it is shown that under certain stability conditions on the system model, the one-step prediction error covariance matrix will converge to a steady-state solution even when the filter gain is not optimal. On the other hand, if the state transition matrix has an unreachable mode outside a unit circle, then the modeling errors in the noise covariances may cause the filter to diverge. Bounds on the asymptotic filter performance are computed when the range of errors in the noise covariance matrices are known. Using simple examples, insights into the behavior of a Kalman filter under nonideal conditions are provided