Upper bounds for mixed H^2/H^∞ control

Abstract

We consider the mixed H^2/H^∞ control problem of choosing a controller to minimize the H^2 norm of a given closed-loop map, subject to the H^∞ norm of another closed-loop map being less than a prescribed value γ. Let d_2 and γ_2 denote the H^2 and H^∞ norms for the pure H^2-optimal solution (without any H^∞ constraint), and let d_c and γ_c < γ denote the H^2 and H^∞ norms for any solution that yields an H^∞ norm strictly less than γ (such as, say, the central solution). Then if d_m denotes the optimal H2 norm that can be achieved in the mixed H^2/H^∞ control problem, we show that (d^2_m - d^2_2)/(d^_c - d^2_2) ⩽ ((γ_2 - γ)/(γ_2 - γ_c))^2 < ((γ^2_2 - γ^2)/(γ^2_2 - γ^2_c))^2 < 1.