HYBRID APPROXIMATION METHOD FOR TIME RESPONSE IMPROVEMENT OF CFE BASED APPROXIMATE FRACTIONAL ORDER DERIVATIVE MODELS BY USING GRADIENT DESCENT ALGORITHM

Year 2023, Volume: 28 Issue: 2, 403 - 416, 31.08.2023

Murat Köseoğlu , Furkan Nur Deniz , Barış Baykant Alagöz

https://doi.org/10.17482/uumfd.1148882

Abstract

Due to its high computational complexity, fractional order (FO) derivative operators have been widely implemented by using rational transfer function approximation methods. Since these methods commonly utilize frequency domain approximation techniques, their time responses may not be prominent for time-domain solutions. Therefore, time response improvements for the approximate FO derivative models can contribute to real-world performance of FO applications. Recent works address the hybrid use of popular frequency-domain approximation methods and time-domain approximation methods to deal with time response performance problems. In this context, this study presents a hybrid approach that implements Continued Fraction Expansion (CFE) method as frequency domain approximation and applies the gradient descent optimization (GDO) for step response improvement of the CFE-based approximate model of FO derivative operators. It was observed that GDO can fine-tune coefficients of CFE-based rational transfer function models, and this hybrid use can significantly improve step and impulse responses of CFE-based approximate models of derivative operators. Besides, we demonstrate analog circuit realization of this optimized transfer function model of the FO derivative element according to the sum of low pass active filters in Multisim and Matlab simulation environments. Performance improvements of hybrid CFE-GDO approximation method were demonstrated in comparison with the stand-alone CFE method.

Keywords

CFE approximation method, FO realization, Optimization, Time response improvement

Gradyan İniş Algoritması Kullanarak CFE Tabanlı Yaklaşık Kesirli Dereceli Türev Modellerinin Zaman Cevabının İyileştirilmesi İçin Hibrit Yaklaşım Yöntemi

Abstract

Yüksek hesaplama karmaşıklığı nedeniyle, kesirli dereceli (KD) türev operatörleri, yaygın olarak rasyonel transfer fonksiyonu yaklaşım yöntemleri kullanılarak gerçekleştirilmektedir. Bu yöntemler genelde frekans alanı yaklaşım tekniklerini kullandığından, zaman cevapları zaman bölgesi çözümleri için yeterince iyi olmayabilir. Bu nedenle, yaklaşık KD türev modellerinin zaman cevaplarının iyileştirilmesi, KD uygulamaların gerçek hayattaki kullanım performanslarına katkıda bulunabilir. Son zamanlardaki çalışmalar, zaman cevabı performans problemlerinin üstesinden gelebilmek için popüler frekans alanı yaklaşımı yöntemlerinin ve zaman alanı yaklaşım yöntemlerinin hibrit kullanımını ele almaktadır. Bu bağlamda, bu çalışma, frekans alanı yaklaşımı olarak Sürekli Kesir Açılımı (SKA) yöntemini uygulayan ve KD türev operatörlerinin SKA tabanlı yaklaşık modelinin basamak cevabı iyileştirmesi için gradyan iniş optimizasyonunu (GİO) uygulayan hibrit bir yaklaşım sunmaktadır. GİO'nun SKA tabanlı rasyonel transfer fonksiyonu modelinin katsayılarını hassas şekilde değiştirebildiği ve bu hibrit kullanımın, SKA tabanlı yaklaşık türev operatör modellerinin birim basamak ve impuls cevaplarını önemli ölçüde iyileştirebildiği gözlemlenmiştir. Ayrıca, KD türevin optimize edilmiş transfer fonksiyonu, Multisim ve Matlab simülasyon ortamlarında alçak geçiren aktif filtrelerin toplamı şeklinde analog devre olarak gerçekleştirilmesini göstermekteyiz. Hibrit SKA-GİO yaklaşımının performans iyileştirmesi klasik SKA yöntemi ile karşılaştırmalı olarak gösterilmiştir.

Keywords

SKA yaklaşım yöntemi, KD gerçekleştirme, Optimizasyon, Zaman cevabı iyileştirme

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