An approach for practical realization of fractional-order oscillators

Year 2024, Volume: 13 Issue: 4, 1342 - 1346, 15.10.2024

İbrahim Ethem Saçu

https://doi.org/10.28948/ngumuh.1336490

Abstract

In this study, an approach for the practical implementation of fractional-order oscillators has been proposed. In this approach, a first-order approximation function is derived in the s-domain to satisfy the fractional integral operator, then this function is discretized and transferred to the z-domain. This allows the continuous-time dynamic equations of the oscillator to be transformed into discrete-time difference equations, making it feasible for realization on digital platforms. The advantage of difference equations lies in the ease with which their coefficients can be updated based on changes in the fractional order. The feasibility of the proposed approach is tested on the Van der Pol (VdP) oscillator. The difference equations of the fractional-order VdP oscillator have been obtained by using the proposed method. These difference equations are used in numerical simulations. Furthermore, the equations are tested on a microcontroller application board, and the proposed method is experimentally validated also.

Keywords

Fractional calculus, Dynamic system, Van der Pol oscillator, Fractional derivative, Stability

Fraksiyonel dereceli osilatörlerin pratik gerçekleştirimi için bir yaklaşım

Abstract

Bu çalışmada, fraksiyonel dereceli osilatörlerin pratik gerçekleştirimi için bir yaklaşım önerilmektedir. Bu yaklaşımda, fraksiyonel integral operatörünü sağlamak için s-domeninde birinci dereceden bir yaklaşıklık fonksiyonu türetilmekte daha sonra bu fonksiyon ayrıklaştırma işlemine tabi tutularak z-domenine aktarılmaktadır. Böylece ilgili osilatörün sürekli zamanlı dinamik denklemleri fark denklemlerine dönüştürülerek dijital platformlarda gerçekleştirilebilme imkanı elde edilmektedir. Fark denklemlerinin avantajı, katsayılarının fraksiyonel derecenin değişimine bağlı olarak hızlıca güncellenebilmesidir. Önerilen yaklaşımın uygulanabilirliği Van der Pol (VdP) osilatüründe test edilmiştir. Önerilen yöntem kullanılarak fraksiyonel VdP osilatörünün fark eşitlikleri elde edilmiştir. Nümerik benzetimlerde bu fark denklemlerinden faydalanılmıştır. Ayrıca denklemler bir mikroişlemci uygulama kartında test edilerek önerilen yöntem deneysel olarak da doğrulanmıştır.

Keywords

Fraksiyonel hesaplama, Dinamik sistem, Van der Pol osilatörü, Fraksiyonel türev, Kararlılık

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