Algebraic Riccati equation and J-spectral factorization for H infinity smoothing and deconvolution

This paper deals with a general steady-state estimation problem in the H∞ 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 of such solutions is well established if the prescribed attenuation level γ is larger than γf (the infimum of the values of γ for which a causal estimator with attenuation level γ exists). We consider the case when γ ≤ γf and show that the stabilizing solution of the ARE still exists (except for a finite number of values of γ) 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 γ for which the stabilizing solution of the ARE does not exist.