On Optimal Solutions to Two-Block H∞ Problems

Abstract

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.

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