The discrete Kalman filtering of a class of dynamic multiscale systems

This paper discusses the optimal estimation of a class of dynamic multiscale systems (DMS), which are observed by several sensors at different scales. The resolution and sampling frequencies of the sensors are supposed to decrease by a factor of two. By using the Haar wavelet transform to link the state nodes at each of the scales within a time block, we generalize the DMS into the standard state-space model, for which the Kalman filtering can be employed as the optimal estimation algorithm. The stochastic controllability and observability of time invariant DMS are analyzed and the stability of the Kalman filter is then discussed. Despite that the DMS model maybe become incompletely controllable and observable, it is proved that as long as the DMS is completely controllable and observable at the finest scale, the associated Kalman filter will be asymptotically stable. The scheme is illustrated with a two-scale Markov process.