Leveraging the Hankel norm approximation and block-AAA algorithms in reduced order modeling

TL;DR

This paper introduces a two-stage reduction method for large-scale LTI systems using a modified AAA algorithm and Hankel norm approximation, improving stability, efficiency, and accuracy over existing techniques.

Contribution

It presents a novel combination of AAA and Hankel norm approximation for stable, efficient model reduction of large-scale LTI systems from transfer function samples.

Findings

The proposed method is more efficient than SVD-based algorithms.

It achieves higher accuracy than moment-matching methods.

The approach stabilizes the reduction process by addressing numerical issues in HNA.

Abstract

Large-scale linear, time-invariant (LTI) dynamical systems are widely used to characterize complicated physical phenomena. We propose a two-stage algorithm to reduce the order of a large-scale LTI system given samples of its transfer function for a target degree $k$ of the reduced system. In the first stage, a modified adaptive Antoulas--Anderson (AAA) algorithm is used to construct a degree $d$ rational approximation of the transfer function that corresponds to an intermediate system, which can be numerically stably reduced in the second stage using ideas from the theory on Hankel norm approximation (HNA). We also study the numerical issues of Glover's HNA algorithm and provide a remedy for its numerical instabilities. A carefully computed rational approximation of degree $d$ gives us a numerically stable algorithm for reducing an LTI system, which is more efficient than SVD-based algorithms and more accurate than moment-matching algorithms.