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Programs are available on request from the authors.

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Abstract-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.Index Terms-Dynamic multiscale system (DMS), Kalman filtering, optimal estimation, wavelet transform.

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Abstract-This paper presents a multiresolution multisensor data fusion scheme for dynamic systems to be observed by several sensors of different resolutions. A state projection equation is introduced to associate the states of a system at each resolution with others. This projection equation together with the state transition equation and the measurement equations at each of the resolutions construct a continuous-time model of the system. The model meets the requirements of Kalman filtering. In discrete time, the state transition is described at the finest resolution and the sampling frequencies of sensors decrease successively by a factor of two in resolution. We can build a discrete model of the system by using a linear projection operator to approximate the state space projection. This discrete model satisfies the requirements of discrete Kalman filtering, which actually offers an optimal estimation algorithm of the system. In time-invariant case, the stability of the Kalman filter is analyzed and a sufficient condition for the filtering stability is given. A Markov-process-based example is given to illustrate and evaluate the proposed scheme of multiresolution modeling and estimation with multiple sensors.

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