Algebraic Riccati Equation and J-Spectral Factorization for Hinfty Smoothing and Deconvolution

Abstract

This paper deals with a general steady-state estimation problem in the $\mathcal{H}_{\infty}$ setting. The existence of the stabilizing solution of the related algebraic Riccati equation (ARE) and of the solution of the associated J-spectral factorization problem is investigated. The existence ofsuch solutions is well established if the prescribed attenuation level $\gamma$ is larger than $\gamma_{_f}$ (the infimum of the values of $\gamma$ for which a {\em causal} estimator with attenuation level $\gamma$ exists). We consider the case when $\gamma\leq\gamma_{_f}$ and show that the stabilizing solution of the ARE still exists (except for a finite number of values of $\gamma$) as long as a fixed-lag acausal estimator (smoother) does. The stabilizing solution of the ARE may be employed to derive a state-space realization of a minimum-phase J-spectral factor of the J-spectrum associated with the estimation problem. This J-spectral factor may be used, in turn, to compute the minimum-lag smoothing estimator. Some of the aspects of the J-spectral factorization problem and the properties of its solutions are discussed in correspondence to the (finite number of) values of $\gamma$ for which the stabilizing solution of the ARE does not exist.