On Optimal Solutions to Two - Block H ” Problems

In this paper we obtain a new formula for the minimum achievable disturbance attenuation in two-block H” problems. This new formula has the same structure as the optimal H” norm formula for noncausal problems, except that doubly-infinite (so-called Laurent) operators must be replaced by semi-infinite (so-called Toeplitz) operators. The benefit of the new formula is that it allows us to find explicit expressions for the optimal H” norm in several important cases: the equalization problem (or its dual, the tracking problem), and the problem of filtering signals in additive noise. Furthermore, it leads us to the concepts of “worst-case non-estimability” , corresponding to when causal filters cannot reduce the H” norms from their a priori values, and “worst-case complete estimability” , corresponding to when causal filters offer the same H” performance as noncausal ones. We also obtain an explicit characterization of worst-case non-estimability and study the consequences to the problem of equalization with finite delay.