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

Yıl 2025, Cilt: 13 Sayı: 1, 479 - 489, 30.01.2025

Cem Civelek , Bedri Sevinc

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

Öz

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.

Anahtar Kelimeler

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

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

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