A Cyclic Small Phase Theorem for MIMO LTI Systems

Chao Chen; Wei Chen; Di Zhao; Jianqi Chen; Li Qiu

doi:10.1016/j.ifacol.2023.10.1906

A Cyclic Small Phase Theorem for MIMO LTI Systems

Chao Chen^*, Wei Chen, Di Zhao, Jianqi Chen^*, Li Qiu^*

^*Corresponding author for this work

Department of Electronic and Computer Engineering

Research output: Chapter in Book/Conference Proceeding/Report › Conference Paper published in a book › peer-review

3 Citations (Scopus)

Abstract

This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between −π and π. The proposed theorem complements the celebrated small gain theorem.

Original language	English
Title of host publication	IFAC-PapersOnLine
Editors	Hideaki Ishii, Yoshio Ebihara, Jun-ichi Imura, Masaki Yamakita
Publisher	Elsevier B.V.
Pages	1883-1888
Number of pages	6
Edition	2
ISBN (Electronic)	9781713872344
DOIs	https://doi.org/10.1016/j.ifacol.2023.10.1906
Publication status	Published - 1 Jul 2023
Event	22nd IFAC World Congress - Yokohama, Japan Duration: 9 Jul 2023 → 14 Jul 2023

Publication series

Name	IFAC-PapersOnLine
Number	2
Volume	56
ISSN (Electronic)	2405-8963

Conference

Conference	22nd IFAC World Congress
Country/Territory	Japan
City	Yokohama
Period	9/07/23 → 14/07/23

Bibliographical note

Publisher Copyright:
Copyright © 2023 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords

cyclic feedback systems
lead-lag compensation
segmental phase
Small phase theorem
stability analysis

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10.1016/j.ifacol.2023.10.1906

Cite this

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title = "A Cyclic Small Phase Theorem for MIMO LTI Systems",

abstract = "This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between −π and π. The proposed theorem complements the celebrated small gain theorem.",

keywords = "cyclic feedback systems, lead-lag compensation, segmental phase, Small phase theorem, stability analysis",

author = "Chao Chen and Wei Chen and Di Zhao and Jianqi Chen and Li Qiu",

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A Cyclic Small Phase Theorem for MIMO LTI Systems. / Chen, Chao; Chen, Wei; Zhao, Di et al.
IFAC-PapersOnLine. ed. / Hideaki Ishii; Yoshio Ebihara; Jun-ichi Imura; Masaki Yamakita. 2. ed. Elsevier B.V., 2023. p. 1883-1888 (IFAC-PapersOnLine; Vol. 56, No. 2).

Research output: Chapter in Book/Conference Proceeding/Report › Conference Paper published in a book › peer-review

TY - GEN

T1 - A Cyclic Small Phase Theorem for MIMO LTI Systems

AU - Chen, Chao

AU - Chen, Wei

AU - Zhao, Di

AU - Chen, Jianqi

AU - Qiu, Li

PY - 2023/7/1

Y1 - 2023/7/1

N2 - This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between −π and π. The proposed theorem complements the celebrated small gain theorem.

AB - This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between −π and π. The proposed theorem complements the celebrated small gain theorem.

KW - cyclic feedback systems

KW - lead-lag compensation

KW - segmental phase

KW - Small phase theorem

KW - stability analysis

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DO - 10.1016/j.ifacol.2023.10.1906

M3 - Conference Paper published in a book

AN - SCOPUS:85180575253

T3 - IFAC-PapersOnLine

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EP - 1888

BT - IFAC-PapersOnLine

A2 - Ishii, Hideaki

A2 - Ebihara, Yoshio

A2 - Imura, Jun-ichi

A2 - Yamakita, Masaki

PB - Elsevier B.V.

T2 - 22nd IFAC World Congress

Y2 - 9 July 2023 through 14 July 2023

ER -

A Cyclic Small Phase Theorem for MIMO LTI Systems

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Bibliographical note

Keywords

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