On noncausal estimation, stochastic realization, and the Riccati inequality

The study of a noncausal state estimation problem associated with a Markovian representation of a stationary increment process brings up interesting facts about the structure of the solution set P of the associated Riccati inequality. Given one p∈P, i.e. one minimal realization, there is a unique tightest sublattice of P for which the minimum element p_0- and the maximum element p₀₊ satisfy the algebraic Riccati equation and define two filters which provide a representation of the noncausal state estimate. (Therefore, for the first time, all solutions of the algebraic Riccati equation are given direct systems-theoretical interpretations). It is shown that the structure of each sublattice is completely determined by the zeros of the minimal spectral factor corresponding to p