Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function

Cem Civelek; Bedri Sevinc

doi:10.29130/dubited.1448747

EN TR

Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function

Abstract

In this research paper, beginning with the Lagrangian and generalized velocity proportional (Rayleigh) dissipation function of a physical/engineering system, the Lagrange-dissipative model ( {L,D}-model briefly) of the system is initially developed. Upon satisfying the prerequisite condition for a Legendre transform, the Hamiltonian function can be obtained. With the Hamiltonian function and the generalized velocity proportional (Rayleigh) dissipation function, dissipative canonical equations can be derived. Using these dissipative canonical equations as the state-space equations of the system allows for the investigation of observability, controllability, and stability properties. In addition to the equilibrium (or critical or fixed) points of the system, stability properties can also be verified through a Lyapunov function as a residual energy function (REF). Since the proposed method is valid for both linear and nonlinear systems, it has been applied to the Van der Pol oscillator/equation.

Keywords

Observability, Controllability and stability of Van der Pol oscillator/equation, Lyapunov function for stability of Van der Pol oscillator/equation.

Van der Pol Osilatörü, Yitimli Kanonik Denklemler Kullanarak Kontrol Teorik Analizi ve Lyapunov Fonksiyonu

Abstract

Bu araştırma makalesinde, bir fiziksel/mühendislik sisteminin Lagrangian ve genelleştirilmiş hız orantılı (Rayleigh) yitim fonksiyonu ile başlayarak, öncelikle bu sistemin Lagrange-dissipatif modeli (kısaca {L,D}-modeli) oluşturulmuştur. Legendre dönüşümü için gerekli koşulunun sağlanmasıyla Hamilton fonksiyonu elde edilebilir. Hamilton fonksiyonu ve genelleştirilmiş hız orantılı (Rayleigh) yitim fonksiyonuyla, yitimli kanonik denklemler elde edilebilir. Yitimli kanonik denklemlerin sistemin durum uzayı denklemleri olarak kullanılması; gözlemlenebilirlik, kontrol edilebilirlik ve kararlılık özelliklerinin araştırılması için kullanılır. Sistemin denge (veya kritik veya sabit) noktalarının yanı sıra, sistemin kararlılık özellikleri, artık enerji fonksiyonu (REF) olarak bir Lyapunov fonksiyonu aracılığıyla da doğrulanabilir. Önerilen yöntem doğrusal ve doğrusal olmayan sistemler için de geçerli olduğundan, yöntem Van der Pol osilatörüne/denklemine uygulanmıştır.

Keywords

Van der Pol denklemi, Gözlemlenebilirlik, Denetlenebilirlik, Kararlılık, Van der Pol denklemi kararlılığı için Lyapunov fonksiyonu

References

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[3] Van der Pol, B., Van Der Mark, J., "Frequency demultiplicationi" Nature 120.3019,pp. 363-364, 1927.
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[19] Civelek, C., "Observability, controllability and stability of a nonlinear RLC circuit in form of a Duffing oscillator by means of theoretical mechanical approach," Journal of Electrical Engineering 73.2, pp. 140-145, 2022.

Details

Primary Language

English

Subjects

Circuits and Systems

Journal Section

Research Article

Authors

Cem Civelek
0000-0003-0017-8661
Türkiye

Bedri Sevinc ^*
0000-0002-7796-5447
Türkiye

Publication Date

January 30, 2025

Submission Date

March 8, 2024

Acceptance Date

November 15, 2024

Published in Issue

Year 2025 Volume: 13 Number: 1

DOI

https://doi.org/10.29130/dubited.1448747

IZ

https://izlik.org/JA68RG54TC

APA

Civelek, C., & Sevinc, B. (2025). Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function. Duzce University Journal of Science and Technology, 13(1), 479-489. https://doi.org/10.29130/dubited.1448747

AMA

1.Civelek C, Sevinc B. Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function. DUBİTED. 2025;13(1):479-489. doi:10.29130/dubited.1448747

Chicago

Civelek, Cem, and Bedri Sevinc. 2025. “Van Der Pol Oscillator, Its Control Theoretical Analysis Using Dissipative Canonical Equations and Its Lyapunov Function”. Duzce University Journal of Science and Technology 13 (1): 479-89. https://doi.org/10.29130/dubited.1448747.

EndNote

Civelek C, Sevinc B (January 1, 2025) Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function. Duzce University Journal of Science and Technology 13 1 479–489.

IEEE

[1]C. Civelek and B. Sevinc, “Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function”, DUBİTED, vol. 13, no. 1, pp. 479–489, Jan. 2025, doi: 10.29130/dubited.1448747.

ISNAD

Civelek, Cem - Sevinc, Bedri. “Van Der Pol Oscillator, Its Control Theoretical Analysis Using Dissipative Canonical Equations and Its Lyapunov Function”. Duzce University Journal of Science and Technology 13/1 (January 1, 2025): 479-489. https://doi.org/10.29130/dubited.1448747.

JAMA

1.Civelek C, Sevinc B. Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function. DUBİTED. 2025;13:479–489.

MLA

Civelek, Cem, and Bedri Sevinc. “Van Der Pol Oscillator, Its Control Theoretical Analysis Using Dissipative Canonical Equations and Its Lyapunov Function”. Duzce University Journal of Science and Technology, vol. 13, no. 1, Jan. 2025, pp. 479-8, doi:10.29130/dubited.1448747.

Vancouver

1.Cem Civelek, Bedri Sevinc. Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function. DUBİTED. 2025 Jan. 1;13(1):479-8. doi:10.29130/dubited.1448747

Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function

Abstract

Keywords

Van der Pol Osilatörü, Yitimli Kanonik Denklemler Kullanarak Kontrol Teorik Analizi ve Lyapunov Fonksiyonu

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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