On-line identification of linear dynamic systems with applications to Kalman filtering

Kalman gave a set of recursive equations for estimating the state of a linear dynamic system. However, the Kalman filter requires a knowledge of all the system and noise parameters. Here it is assumed that all these parameters are unknown and therefore must be identified before use in the Kalman filter. A correlation technique which identifies a system in its canonical form is presented. The estimates are shown to be asymptotically normal, unbiased, and consistent. The scheme is capable of being implemented on-line and can be used in conjunction with the Kalman filter. A technique for more efficient estimation by using higher order correlations is also given. A recursive technique is given to determine the order of the system when the dimension of the system is unknown. The results are first derived for stationary processes and are then extended to nonstationary processes which are stationary in the
q
th increment. An application of the results to a practical problem is presented.