A symmetric group method for controllability characterization of bilinear systems on the special euclidean group

Wei Zhang; Jr Shin Li

doi:10.1016/j.ifacol.2019.11.815

A symmetric group method for controllability characterization of bilinear systems on the special euclidean group

Wei Zhang, Jr Shin Li

Research output: Contribution to journal › Conference article › peer-review

1 Scopus citations

Abstract

Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.

Original language	English
Pages (from-to)	412-417
Number of pages	6
Journal	IFAC-PapersOnLine
Volume	52
Issue number	16
DOIs	https://doi.org/10.1016/j.ifacol.2019.11.815
State	Published - Sep 2019
Event	11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019 - Vienna, Austria Duration: Sep 4 2019 → Sep 6 2019

Keywords

Bilinear systems
Controllability
Lie groups and Lie algebras
Symmetric groups

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10.1016/j.ifacol.2019.11.815

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title = "A symmetric group method for controllability characterization of bilinear systems on the special euclidean group",

abstract = "Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.",

keywords = "Bilinear systems, Controllability, Lie groups and Lie algebras, Symmetric groups",

author = "Wei Zhang and Li, {Jr Shin}",

note = "Funding Information: This work was supported in part by the National Science Foundation under the awards CMMI-1462796, ECCS-1509342, and ECCS-1810202, and by the Air Force Office of Scientific Research under the award FA9550-17-1-0166. Publisher Copyright: {\textcopyright} 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.; 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019 ; Conference date: 04-09-2019 Through 06-09-2019",

year = "2019",

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doi = "10.1016/j.ifacol.2019.11.815",

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T1 - A symmetric group method for controllability characterization of bilinear systems on the special euclidean group

AU - Zhang, Wei

AU - Li, Jr Shin

N1 - Funding Information: This work was supported in part by the National Science Foundation under the awards CMMI-1462796, ECCS-1509342, and ECCS-1810202, and by the Air Force Office of Scientific Research under the award FA9550-17-1-0166. Publisher Copyright: © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

PY - 2019/9

Y1 - 2019/9

N2 - Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.

AB - Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.

KW - Bilinear systems

KW - Controllability

KW - Lie groups and Lie algebras

KW - Symmetric groups

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U2 - 10.1016/j.ifacol.2019.11.815

DO - 10.1016/j.ifacol.2019.11.815

M3 - Conference article

AN - SCOPUS:85077431729

SN - 2405-8963

VL - 52

SP - 412

EP - 417

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

IS - 16

T2 - 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019

Y2 - 4 September 2019 through 6 September 2019

ER -

A symmetric group method for controllability characterization of bilinear systems on the special euclidean group

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Keywords

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