Stability analysis of the Kalman filter under practical conditions

The Kalman filter is optimal in the sense of minimum variance under the assumptions of white noises and perfect model-system matching. If these assumptions are not satisfied, the filter loses its optimality. Does it, however, remain a good state estimator with stable error dynamics? The answer to this question cannot rely on the classical stability proof of the Kalman filter, which was based on its optimality. It is shown in this paper that, without the restrictive assumptions of white noises and perfect model-system matching, the Kalman filter has an exponentially stable error dynamics under observability and controllability conditions. Moreover, the second moment of its state estimation error has an upper bound linearly depending on the discrepancies between model and system. In this sense, the Kalman filter is stable and remains a good state estimator under practical conditions.