Hybrid expansion–contraction: a robust scaleable method for approximating the<i>H</i>_∞norm

Abstract

We present a new scaleable algorithm for approximating the |$H_{\infty }$| norm, an important robust stability measure for linear dynamical systems with input and output. Our spectral-value-set-based method uses a novel hybrid expansion–contraction scheme that, under reasonable assumptions, is guaranteed to converge to a stationary point of the optimization problem defining the |$H_{\infty }$| norm, and, in practice, typically returns local or global maximizers. We prove that the hybrid expansion–contraction method has a quadratic rate of convergence that is also confirmed in practice. In comprehensive numerical experiments, we show that our new method is not only robust but exceptionally fast, successfully completing a large-scale test set 25 times faster than an earlier method by Guglielmi, Gürbüzbalaban & Overton (2013, SIAM J. Matrix Anal. Appl., 34, 709–737), which occasionally breaks down far from a stationary point of the underlying optimization problem.