Some reduced-order non-Riccati equations for linear least-squares estimation : the stationary, single-output case†

Abstract

The problem of determining the Kalman—Bucy filter for an n-dimensional single-output model is the topic of this paper. Both the discrete-time case and continuous-time case are considered. The model processes are assumed to be stationary. It is shown that, under certain regularity conditions, only n first-order difference or differential equations are required for determining the error covariance function, and hence also the filter gain, rather than 1/2n(n + 1) equations as with the Riccati approach or 2n as in the previous non-Riccati algorithm. This reduction is achieved by constructing a system of simple integrals for the 2n non-Riccati equations. The reduced-order algorithms have non-trivial steady-state versions, which are equivalent to the algebraic equations obtained by spectral factorization. The stationary and single-output assumptions are for convenience. In fact, the basic method works also in a more general setting.