Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

Bernard M.J. Maschke; Romeo Ortega; Arjan J. van der Schaft

doi:10.1109/CDC.1998.761738

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

Bernard M.J. Maschke, Romeo Ortega, Arjan J. van der Schaft

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

Abstract

We propose a constructive procedure to modify the total energy function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for non-zero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions (i.e., first integrals) can be easily constructed. For linear systems the resulting Lyapunov function is the incremental energy, thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given.

Original language	English
Title of host publication	Proceedings 37th IEEE Conference on Decision and Control, CDC 1998
Place of Publication	Los Alamitos, CA
Publisher	IEEE
Pages	3599-3604
Number of pages	6
ISBN (Print)	0-7803-4394-8
DOIs	https://doi.org/10.1109/CDC.1998.761738
Publication status	Published - 16 Dec 1998
Event	37th IEEE Conference on Decision and Control, CDC 1998 - Hyatt Regency Westshore, Tampa, United States Duration: 16 Dec 1998 → 18 Dec 1998 Conference number: 37

Publication series

Name	IEEE Conference on Decision and Control, CDC
Publisher	IEEE
Volume	1998
ISSN (Print)	0191-2216

Conference

Conference	37th IEEE Conference on Decision and Control, CDC 1998
Abbreviated title	CDC
Country/Territory	United States
City	Tampa
Period	16/12/98 → 18/12/98

Access to Document

10.1109/CDC.1998.761738

Cite this

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title = "Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation",

abstract = "We propose a constructive procedure to modify the total energy function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for non-zero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions (i.e., first integrals) can be easily constructed. For linear systems the resulting Lyapunov function is the incremental energy, thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given.",

author = "Maschke, {Bernard M.J.} and Romeo Ortega and {van der Schaft}, {Arjan J.}",

year = "1998",

month = dec,

day = "16",

doi = "10.1109/CDC.1998.761738",

language = "English",

isbn = "0-7803-4394-8",

series = "IEEE Conference on Decision and Control, CDC",

publisher = "IEEE",

pages = "3599--3604",

booktitle = "Proceedings 37th IEEE Conference on Decision and Control, CDC 1998",

address = "United States",

note = "37th IEEE Conference on Decision and Control, CDC 1998, CDC ; Conference date: 16-12-1998 Through 18-12-1998",

}

Maschke, BMJ, Ortega, R & van der Schaft, AJ 1998, Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. in Proceedings 37th IEEE Conference on Decision and Control, CDC 1998. IEEE Conference on Decision and Control, CDC, vol. 1998, IEEE, Los Alamitos, CA, pp. 3599-3604, 37th IEEE Conference on Decision and Control, CDC 1998, Tampa, Florida, United States, 16/12/98. https://doi.org/10.1109/CDC.1998.761738

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. / Maschke, Bernard M.J.; Ortega, Romeo; van der Schaft, Arjan J.

Proceedings 37th IEEE Conference on Decision and Control, CDC 1998. Los Alamitos, CA : IEEE, 1998. p. 3599-3604 (IEEE Conference on Decision and Control, CDC; Vol. 1998).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

TY - GEN

T1 - Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

AU - Maschke, Bernard M.J.

AU - Ortega, Romeo

AU - van der Schaft, Arjan J.

N1 - Conference code: 37

PY - 1998/12/16

Y1 - 1998/12/16

N2 - We propose a constructive procedure to modify the total energy function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for non-zero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions (i.e., first integrals) can be easily constructed. For linear systems the resulting Lyapunov function is the incremental energy, thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given.

AB - We propose a constructive procedure to modify the total energy function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for non-zero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions (i.e., first integrals) can be easily constructed. For linear systems the resulting Lyapunov function is the incremental energy, thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given.

U2 - 10.1109/CDC.1998.761738

DO - 10.1109/CDC.1998.761738

M3 - Conference contribution

SN - 0-7803-4394-8

T3 - IEEE Conference on Decision and Control, CDC

SP - 3599

EP - 3604

BT - Proceedings 37th IEEE Conference on Decision and Control, CDC 1998

PB - IEEE

CY - Los Alamitos, CA

T2 - 37th IEEE Conference on Decision and Control, CDC 1998

Y2 - 16 December 1998 through 18 December 1998

ER -

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

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