Upper bounds for mixed H2/H∞ control

We consider the mixed H-2/H-infinity control problem of choosing a controller to minimize the H-2 norm of a given closed-loop map, subject to the H-infinity norm of another closed-loop map being less than a prescribed value gamma. Let d(2) and gamma(2) denote the H-2 and H-infinity norms for the pure H-2-optimal solution (without any H-infinity constraint), and let d(c) and gamma(c) < gamma denote the H-2 and H-infinity norms for any solution that yields an H-infinity norm strictly less than gamma (such as, say, the central solution). Then if d(m) denotes the optimal H-2 norm that can be achieved in the mixed H-2/H-infinity control problem, we show that d(m)(2)-d(2)(2)/d(c)(2)-d(2)(2) less than or equal to(gamma(2)-gamma/gamma(2)-gamma(c))(2) < (gamma(2)(2)-gamma(2)/gamma(2)(2)-gamma(c)(2))(2) < 1.