Faster and more accurate computation of the $\mathcal{H}_\infty$ norm via optimization

TL;DR

This paper introduces an optimized algorithm for computing the $\\mathcal{H}_\infty$ norm of linear systems that is faster, more precise, and more amenable to parallelization than existing methods.

Contribution

It presents a rebalanced optimization-based approach that reduces eigenvalue computations and enables full-precision calculation with potential for parallel speedup.

Findings

Algorithm is several times faster than existing methods.

Can compute the $\\mathcal{H}_\infty$ norm to full precision efficiently.

Local optimization effectively approximates the norm for large-scale systems.

Abstract

In this paper, we propose an improved method for computing the $\mathcal{H}_\infty$ norm of linear dynamical systems that results in a code that is often several times faster than existing methods. By using standard optimization tools to rebalance the work load of the standard algorithm due to Boyd, Balakrishnan, Bruinsma, and Steinbuch, we aim to minimize the number of expensive eigenvalue computations that must be performed. Unlike the standard algorithm, our modified approach can also calculate the $\mathcal{H}_\infty$ norm to full precision with little extra work, and also offers more opportunity to further accelerate its performance via parallelization. Finally, we demonstrate that the local optimization we have employed to speed up the standard globally-convergent algorithm can also be an effective strategy on its own for approximating the $\mathcal{H}_\infty$ norm of large-scale systems.