Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals
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arXiv
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| Format: | Preprint |
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2024
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| author | Song, Qiu-Yan Zulfiqar, Umair Xiao, Zhi-Hua Uddin, Mohammad Monir Sreeram, Victor |
| author_facet | Song, Qiu-Yan Zulfiqar, Umair Xiao, Zhi-Hua Uddin, Mohammad Monir Sreeram, Victor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals Song, Qiu-Yan Zulfiqar, Umair Xiao, Zhi-Hua Uddin, Mohammad Monir Sreeram, Victor Systems and Control 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. |
| title | Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2402.11445 |