On the two-block H∞ problem for a class of unstable distributed systems

Abstract

This paper deals with the two-block H ∞ control problem for distributed plants with finitely many unstable modes. We assume that weighting filters in the H ∞ mixed-sensitivity problem are finite-dimensional. Then the corresponding optimal two-block problem can be solved by finding the Schmidt pairs of a Hankel operator whose symbol is of the form m∗ 2(m∗ 1u + u ̂ ) where u ∈ R H ∞ , u ̂ ∈ H ∞ , and m 2 ∈ RH ∞ and m 1 ∈ H ∞ are inner; and the suboptimal two-block problem can be solved by finding the solutions of certain functional equations very similar to the ones satisfied by the Schmit pairs of the above-mentioned Hankel operator. In this paper a unified approach is proposed for solving both the optimal and suboptimal two-block problems. We obtain two systems of linear equations, expressed in terms of state-space realizations of u and m 2, whose solutions give the Schmidt pairs of the associated Hankel operator and the functions needed for the parametrization of all the suboptimal solutions, respectively.