On optimal solutions to two-block H∞ problems

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 one to find explicit expressions for the optimal H^∞ norm in several important cases: the equalization problem, and the problem of filtering signals in additive noise. Furthermore, it leads one 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