Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals

TL;DR

This paper extends balanced truncation model order reduction to linear systems with quadratic outputs, focusing on accuracy within specific time and frequency intervals by defining limited interval Gramians and deriving related Lyapunov equations.

Contribution

It introduces generalized limited-interval Gramians and algorithms for time- and frequency-limited balanced truncation of quadratic-output systems, ensuring high accuracy in specified intervals.

Findings

Algorithms outperform traditional methods outside the targeted intervals.

Numerical examples demonstrate improved accuracy within specified time and frequency ranges.

Proposed methods preserve quadratic-output structure effectively.

Abstract

Model order reduction involves constructing a reduced-order approximation of a high-order model while retaining its essential characteristics. This reduced-order model serves as a substitute for the original one in various applications such as simulation, analysis, and design. Often, there's a need to maintain high accuracy within a specific time or frequency interval, while errors beyond this limit can be tolerated. This paper addresses time-limited and frequency-limited model order reduction scenarios for linear systems with quadratic outputs, by generalizing the recently introduced structure-preserving balanced truncation algorithm. To that end, limited interval system Gramians are defined, and the corresponding generalized Lyapunov equations governing their computation are derived. Additionally, low-rank solutions for these equations are investigated. Next, balanced truncation algorithms are proposed for time-limited and frequency-limited scenarios, each utilizing its corresponding limited-interval system Gramians. The proposed algorithms ensure accurate results within specified time and frequency intervals while preserving the quadratic-output structure. Two benchmark numerical examples are presented to demonstrate the effectiveness of the algorithms, showcasing their ability to achieve superior accuracy within the desired time or frequency interval.