Optimal H∞ interpolation: A new approach

Abstract

Explicitly computable solutions to the problem of L∞ sensitivity minimization for (a possibly infinite-dimensional plant represented by) an inner function M ∈ H∞, subject to a rational weighting W ∈ H∞, are obtained. This is equivalent to the problem of best approximation of M*W ∈ L∞ by Q ∈ H∞ (more generally, Q ∈ H[∞] ∈). The main new idea involves the representation of the Hankel operator Γ of M*W as a finite-rank perturbation of the multiplication operator M*W. The perturbation takes the form of a "Complementary Hankel Operator" determined by W. This idea is exploited to obtain explicit formulas for: (a) All discrete eigenvalues and eigenvectors of Γ*Γ; (b) all S-numbers of Γ*Γ; (c) the optimal H[∞] ĸ approximations; (d) the essential spectrum of Γ*Γ. The formulas obtained are surprisingly simple when the order of W is small, even for infinite-dimensional M, and therefore appear to be particularly well suited to control-sensitivity problems. The case of 1st-order W is worked out in detail.