A Cascade of Systems and the Product of Their $θ$-Symmetric Scaled Relative Graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909830956974080 |
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| author | Yang, Xiaokan Zhang, Ding Chen, Wei Qiu, Li |
| author_facet | Yang, Xiaokan Zhang, Ding Chen, Wei Qiu, Li |
| contents | In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the $θ$-symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of $θ$-symmetric SRG is established, enabling it to handle cyclic interconnections for which conventional graph separation methods are not applicable. By integrating both gain and refined phase information, the $θ$-symmetric SRG provides a unified graphical characterization of the system, which better captures system properties and yields less conservative results. In the scalar case, the $θ$-symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise $θ$-symmetric SRG is more suitable than the standard SRG as a multi-input multi-output extension of the classical Nyquist plot. Illustrative examples are included to demonstrate the effectiveness of the $θ$-symmetric SRG. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Cascade of Systems and the Product of Their $θ$-Symmetric Scaled Relative Graphs Yang, Xiaokan Zhang, Ding Chen, Wei Qiu, Li Systems and Control In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the $θ$-symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of $θ$-symmetric SRG is established, enabling it to handle cyclic interconnections for which conventional graph separation methods are not applicable. By integrating both gain and refined phase information, the $θ$-symmetric SRG provides a unified graphical characterization of the system, which better captures system properties and yields less conservative results. In the scalar case, the $θ$-symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise $θ$-symmetric SRG is more suitable than the standard SRG as a multi-input multi-output extension of the classical Nyquist plot. Illustrative examples are included to demonstrate the effectiveness of the $θ$-symmetric SRG. |
| title | A Cascade of Systems and the Product of Their $θ$-Symmetric Scaled Relative Graphs |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2510.06583 |