Design of recursive least-squares fixed-lag smoother using covariance information in linear continuous stochastic systems

Abstract

This paper newly designs the recursive least-squares (RLS) fixed-lag smoother and filter using the covariance information in linear continuous-time stochastic systems. It is assumed that the signal is observed with additive white observation noise and the signal is uncorrelated with the observation noise. The estimators require the covariance information of the signal and the variance of the observation noise. The auto-covariance function of the signal is expressed in the semi-degenerate kernel form.

RLS fixed-lag smoother using covariance information in linear continuous stochastic systems

“…The current fixedlag smoother is designed based on the linear least-squares criterion (5) as a natural consequence. The current fixed-lag smoothing algorithm, which is different from that in [15], is shown in Theorem 1.…”

“…The semi-degenerate kernel can express the auto-covariance function of the stationary or non-stationary stochastic signal processes generally by a finite sum of nonrandom functions. It is a characteristic that the estimators in [14,15] do not use the state-space model for the signal.…”

“…In this paper, under the same assumptions for the signal and the observations noise with [15], the RLS fixed-lag smoother is newly designed in linear continuous-time stochastic systems. It is assumed that the signal is observed with additive white observation noise and the signal is uncorrelated with the observation noise.…”

“…The auto-covariance function of the signal is expressed in the semi-degenerate kernel form. In [15], the fixed-lag smoothing estimate is given as a sum of the filtering estimate and the correction term based on the effect of the smoothing. The impulse response function h 1 (t, s), 0 6 s 6 t À D 6 t, for the filtering estimate satisfies (11).…”

“…Hence, the fixed-lag smoother with the auto-covariance function in the degenerate kernel form cannot be applied to the estimation of the stochastic signal process whose auto-covariance function is not expressed in the degenerate kernel form. In [15], by assuming that the fixed-lag smoothing estimate is given as a sum of the filtering estimate of the signal and correction term to the filtering estimate, the recursive least-squares (RLS) fixed-lag smoother using the covariance information is designed in linear continuous-time stochastic systems. Here, the auto-covariance function of the signal is expressed in the semidegenerate kernel form.…”