Balanced Truncation of Linear Time-Invariant Systems over Finite-frequency Ranges

TL;DR

This paper introduces two new frequency-dependent balanced truncation methods for model reduction of LTI systems over finite frequency ranges, providing error bounds and demonstrating improved in-band approximation.

Contribution

The paper develops two novel frequency-dependent balanced truncation techniques with derived error bounds for finite-frequency model reduction.

Findings

Methods achieve good in-band approximation performance

Error bounds accurately estimate approximation error

Examples demonstrate effectiveness of the proposed methods

Abstract

This paper discusses model order reduction of LTI systems over limited frequency intervals within the framework of balanced truncation. Two new \emph{frequency-dependent balanced truncation} methods were developed, one is \emph{SF-type frequency-dependent balanced truncation} to copy with the cases that only a single dominating point of the operating frequency interval is pre-known, the other is \emph{interval-type frequency-dependent balanced truncation} to deal with the cases that both of the upper and lower bound of frequency interval are known \emph{a priori}. SF-type error bound and interval-type error bound are derived for the first time to estimate the desired approximation error over pre-specified frequency interval. We show that the new methods generally lead to good in-band approximation performance, at the same time, provide accurate error bounds under certain conditions. Examples are included for illustration.