H/sub /spl infin// fixed-lag smoothing and prediction for linear continous-time systems

This paper addresses the H/sub /spl infin// fixed-lag smoothing and prediction problems for linear continuous-time systems. We first present a solution to the optimal H/sub 2/ estimation problem for linear continuous-time systems with instantaneous and delayed measurements. It is then shown that the H/sub /spl infin// fixed-lag smoothing and prediction problems can be converted to the latter problem in Krein space. Therefore, the H/sub 2/ estimation is extended to give conditions on the existence of H/sub /spl infin// fixed-lag smoother and predictor based on innovation analysis and projection in Krein space and a solution for H/spl infin/ smoother or predictor is given in terms of a Riccati differential equation and matrix differential equations.

1 Introduction

The smoothing problem is to estimate a linear combination of system states at time instant based on measurements up to for any . It is usually classified into three categories, namely the fixed-point smoothing, fixed interval smoothing and fixed-lag smoothing. Note that in the discrete-time case where the delay dynamics are finite dimensional, the smoothing problem may be converted into standard filtering through state augmentation [8] or can be solved through a recently developed direct approach [10]. However, for the continuous-time case, the delay dynamics are infinite dimensional, the smoothing problem has been proven to be difficult. The problem of fixed-interval smoothing has been solved in [6] for linear continuous-time systems and the optimal fixed interval smoother has been known to be the same as the optimal smoother [1]. However, for fixed point smoothing and fixed-lag smoothing, the and smoothing will be different. Recently, Mirkin [5] has tackled the infinite horizon fixed-lag smoothing problem using a spectral factorization approach. The finite horizon fixed lag smoothing and prediction for continuous-time systems remains open.