H-infinity deconvolution filtering: a Krein space approach in state-space setting

Abstract

This note is concerned with the H-infinity deconvolution filtering problem for linear time-varying discretetime systems described by state space models. The H-infinity deconvolution filter is derived by proposing a new approach in Krein space. With the new approach, it is clearly shown that the central deconvolution filter in an H-infinity setting is the same as the one in an H 2 setting associated with one constructed stochastic state-space model. This insight allows us to calculate the complicated H-infinity deconvolution filter in an intuitive and simple way. The deconvolution filter is calculated by performing Riccati equation with the same order as that of the original system.

On the H ∞ deconvolution fixed-lag smoothing

“…Same as [13], we introduce the following Krein-space stochastic system model associated with (1), (10) and…”

“…In the past decades, much attention has been paid to the deconvolution problem, see [1,2,5,10] and references therein.…”

“…It should be noted that the J-spectral factorization for the H ∞ fixedlag smoothing and prediction are more complicated than for the filtering as they are usually related with system augmentation [6]. Under the state space model, [3,10] studied the H ∞ finite-horizon deconvolution filtering. The deconvolution filter is calculated in terms of the solution to one standard H ∞ Riccati equation in [3], and Krein-space is used in [10].…”

“…Under the state space model, [3,10] studied the H ∞ finite-horizon deconvolution filtering. The deconvolution filter is calculated in terms of the solution to one standard H ∞ Riccati equation in [3], and Krein-space is used in [10]. With the help of LMI (Linear Matrix Inequality), [8] proposed an H ∞ deconvolution filter and [9] obtained a mixed 2 H H ∞ / filter.…”