Multiscale systems, Kalman filters, and Riccati equations

An algorithm analogous to the Rauch-Tung-Striebel algorithm-consisting of a fine-to-coarse Kalman filter-like sweep followed by a coarse-to-fine smoothing step-was developed previously by the authors (ibid. vol.39, no.3, p.464-78 (1994)). In this paper they present a detailed system-theoretic analysis of this filter and of the new scale-recursive Riccati equation associated with it. While this analysis is similar in spirit to that for standard Kalman filters, the structure of the dyadic tree leads to several significant differences. In particular, the structure of the Kalman filter error dynamics leads to the formulation of an ML version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the Hamiltonian for the smoothing problem. In addition, the notion of stability for dynamics requires some care as do the concepts of reachability and observability. Using these system-theoretic constructs, the stability and steady-state behavior of the fine-to-coarse Kalman filter and its Riccati equation are analysed