Minimal Realization and Dynamic Properties of Optimal Smoothers

Smoothing algorithms of various kinds have been around for several decades. However, some basic issues regarding the dynamical structure and the minimal dimension of the steady-state algorithm are still poorly understood. In this paper, we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state-space description of dimension n. It is shown that the dimension of the smoothing algorithm can vary from n to 2n, depending on the zero structure of the signal model. The dynamics (pole structure) of the steady-state smoother is also characterized explicitly and is related to the zero structure of the model. We use several recent ideas from stochastic realization theory. In particular, a minimal Markovian representation of the smoother is derived, which requires solving a nonsymmetric Wiener-Hopf factorization problem. In this way, the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension, whose state space is a minimal Markovian subspace containing the smoothed estimate x&capped;. This, among other aspects, affords a very simple calculation of the error covariance matrix of the smoother. A reduced-order two-filter implementation of the type due to Mayne (1966) and Fraser (1967) is obtained by solving a Riccati equation of reduced dimension, which is in general smaller than the dimension of the Riccati equations considered in the literature.