# A Subspace Framework for ${\mathcal H}_\infty$-Norm Minimization

## TL;DR

This paper introduces a subspace-based iterative method for efficiently minimizing the ${\mathcal H}_\infty$-norm of large-scale parameter-dependent systems, with proven superlinear convergence and demonstrated effectiveness on large systems.

## Contribution

It proposes a novel greedy interpolatory subspace framework for ${\mathcal H}_\infty$-norm minimization, extending previous work to parameter-dependent systems with convergence guarantees.

## Key findings

- Framework achieves superlinear convergence under smoothness assumptions.
- Method effectively handles large-scale systems in practice.
- Numerical experiments demonstrate fast convergence and accuracy.

## Abstract

We deal with the minimization of the ${\mathcal H}_\infty$-norm of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values. Subspace frameworks are proposed for such minimization problems where the involved systems are of large order. The proposed algorithms are greedy interpolatory approaches inspired by our recent work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38(4):1496--1516, 2017] for the computation of the ${\mathcal H}_\infty$-norm. In this work, we minimize the ${\mathcal H}_\infty$-norm of a reduced-order parameter-dependent system obtained by two-sided restrictions onto certain subspaces. Then we expand the subspaces so that Hermite interpolation properties hold between the full and reduced-order system at the optimal parameter value for the reduced order system. We formally establish the superlinear convergence of the subspace frameworks under some smoothness assumptions. The fast convergence of the proposed frameworks in practice is illustrated by several large-scale systems.

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Source: https://tomesphere.com/paper/1905.04208