Fast computation of achievable feedback performance in mixed sensitivityH^{∞}design

The computational bottleneck of the
H^{\infty}
design has been recognized to be the "ε-iteration," a computationally demanding direct search of the minimum achievable
H^{\infty}
performance. Verma and Jonckheere showed that the optimal
H^{\infty}
performance can be characterized as the spectral radius of the so-called "Toeplitz plus Hankel" operator. Even before the appearance of the "Toeplitz plus Hankel" operator in the
H^{\infty}
setting, the same operator had already been shown to play a crucial role in the spectral theory of the linear-quadratic problem developed by Jonckheere and Silverman. In this paper, we exploit this common "Toeplitz plus Hankel" operator structure shared by the seemingly unrelated linear-quadratic and
H^{\infty}
problems, and we construct fast state-space algorithms for evaluating the spectral radius of the "Toeplitz plus Hankel" operator. The salient feature of the algorithm is that the spectral radius can be evaluated, with an accuracy predicted by an identifiable error bound, from the antistabilizing solution of the algebraic Riccati equation of the linear-quadratic problem associated with the
H\infty
design.