Optimal self-tuning filtering, prediction, and smoothing for discrete multivariable processes

An algorithm is proposed for self-tuning optimal fixed-lag smoothing or filtering for linear discrete-time multivariable processes. A -transfer function solution to the discrete multivariable estimation problem is first presented. This solution involves spectral factorization of polynomial matrices and assumes knowledge of the process parameters and the noise statistics. The assumption is then made that the signal-generating process and noise statistics are unknown. The problem is reformulated so that the model is in an innovations signal form, and implicit self-tuning estimation algorithms are proposed. The parameters of the innovation model of the process can be estimated using an extended Kalman filter or, alternatively, extended recursive least squares. These estimated parameters are used directly in the calculation of the predicted, smoothed, or filtered estimates. The approach is an attempt to generalize the work of Hagander and Wittenmark.