Abstract
This paper designs the new fixed-lag smoother by the covariance information of the signal and observation noise processes. Two kinds of observation noises are considered here, which are the stationary white noise and the stationary white plus coloured-noise. We design the optimal fixed-lag smoother in the meaning of minimum variance of estimation error for each noise. The new smoother has two advantages: 1) The linear nonstationary system state can be estimated, with a simple initial-value system, directly by the curve-fitted, functions of the covariance information, and by the observed values. So, we do not have any trouble of finding an approximating state-space model by the spectral factorization method, using covariance information. 2) The calculation of the algorithm for the fixed-lag smoothing estimate is easy. One reason for that is that the covariance function of the signal process is assumed to be degenerate. Degenerate kernels can express the general kind of covariance information of nonstationary process in the form of finite sums of products of nonrandom functions. The other reason for that is that we can approximate the covariance information of the coloured-noise process by semi-degenerate kernels. Semi-degenerate kernels also can express general kind of covariance information of nonstationary process by a curve-fitting method.
The effectiveness of the new fixed-lag smoother was assured by a numerical example.
