Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions

Morgan Jones, Matthew Peet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinant-like objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.

Original languageEnglish (US)
Title of host publication2019 American Control Conference, ACC 2019
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages2084-2091
Number of pages8
ISBN (Electronic)9781538679265
StatePublished - Jul 1 2019
Event2019 American Control Conference, ACC 2019 - Philadelphia, United States
Duration: Jul 10 2019Jul 12 2019

Publication series

NameProceedings of the American Control Conference
Volume2019-July
ISSN (Print)0743-1619

Conference

Conference2019 American Control Conference, ACC 2019
CountryUnited States
CityPhiladelphia
Period7/10/197/12/19

Fingerprint

Nonlinear systems

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Cite this

Jones, M., & Peet, M. (2019). Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions. In 2019 American Control Conference, ACC 2019 (pp. 2084-2091). [8814848] (Proceedings of the American Control Conference; Vol. 2019-July). Institute of Electrical and Electronics Engineers Inc..

Using SOS for optimal semialgebraic representation of sets : Finding minimal representations of limit cycles, chaotic attractors and unions. / Jones, Morgan; Peet, Matthew.

2019 American Control Conference, ACC 2019. Institute of Electrical and Electronics Engineers Inc., 2019. p. 2084-2091 8814848 (Proceedings of the American Control Conference; Vol. 2019-July).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Jones, M & Peet, M 2019, Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions. in 2019 American Control Conference, ACC 2019., 8814848, Proceedings of the American Control Conference, vol. 2019-July, Institute of Electrical and Electronics Engineers Inc., pp. 2084-2091, 2019 American Control Conference, ACC 2019, Philadelphia, United States, 7/10/19.
Jones M, Peet M. Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions. In 2019 American Control Conference, ACC 2019. Institute of Electrical and Electronics Engineers Inc. 2019. p. 2084-2091. 8814848. (Proceedings of the American Control Conference).
Jones, Morgan ; Peet, Matthew. / Using SOS for optimal semialgebraic representation of sets : Finding minimal representations of limit cycles, chaotic attractors and unions. 2019 American Control Conference, ACC 2019. Institute of Electrical and Electronics Engineers Inc., 2019. pp. 2084-2091 (Proceedings of the American Control Conference).
@inproceedings{04ec26a588e24e9083f415de1c04bdd0,
title = "Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions",
abstract = "In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinant-like objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.",
author = "Morgan Jones and Matthew Peet",
year = "2019",
month = "7",
day = "1",
language = "English (US)",
series = "Proceedings of the American Control Conference",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "2084--2091",
booktitle = "2019 American Control Conference, ACC 2019",

}

TY - GEN

T1 - Using SOS for optimal semialgebraic representation of sets

T2 - Finding minimal representations of limit cycles, chaotic attractors and unions

AU - Jones, Morgan

AU - Peet, Matthew

PY - 2019/7/1

Y1 - 2019/7/1

N2 - In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinant-like objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.

AB - In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinant-like objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.

UR - http://www.scopus.com/inward/record.url?scp=85072299272&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072299272&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85072299272

T3 - Proceedings of the American Control Conference

SP - 2084

EP - 2091

BT - 2019 American Control Conference, ACC 2019

PB - Institute of Electrical and Electronics Engineers Inc.

ER -