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

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

References

1. 1. Bertsias, P., Psychalinos, C., Maundy, B. J., Elwakil, A. S. & Radwan, A. G. (2019) Partial fraction expansion–based realizations of fractional‐order differentiators and integrators using active filters, International Journal of Circuit Theory and Applications, 47(4), 513–531. https://doi.org/10.1002/cta.2598
2. 2. Bingi, K., Ibrahim, R., Karsiti, M. N., Hassam, S. M. & Harindran, V. R. (2019) Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers, International Journal of Applied Mathematics and Computer Science, 29(2), 311–326. https://doi.org/10.2478/amcs-2019-0023
3. 3. Caponetto, R., Dongola, G., Fortuna, L. & Petráš, I. (2010). Fractional Order Systems. In Advances in Industrial Control (Vol. 72, Issue 9781849963343). WORLD SCIENTIFIC. https://doi.org/10.1142/7709
4. 4. Chen, Y., Petráš, I. & Xue, D. (2009) Fractional order control-a tutorial, 2009 American Control Conference, 1397–1411. https://doi.org/10.1109/ACC.2009.5160719
5. 5. Colín-Cervantes, J. D., Sánchez-López, C., Ochoa-Montiel, R., Torres-Muñoz, D., Hernández-Mejía, C. M., Sánchez-Gaspariano, L. A. & González-Hernández, H. G. (2021) Rational Approximations of Arbitrary Order: A Survey, Fractal and Fractional, 5(4), 267. https://doi.org/10.3390/fractalfract5040267
6. 6. Delghavi, M. B., Shoja-Majidabad, S. & Yazdani, A. (2016) Fractional-Order Sliding-Mode Control of Islanded Distributed Energy Resource Systems, IEEE Transactions on Sustainable Energy, 7(4), 1482–1491. https://doi.org/10.1109/TSTE.2016.2564105
7. 7. Deniz, F. N., Alagoz, B. B., Tan, N. & Atherton, D. P. (2016) An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators, ISA Transactions, 62, 154–163. https://doi.org/10.1016/j.isatra.2016.01.020
8. 8. Deniz, F. N., Alagoz, B. B., Tan, N. & Koseoglu, M. (2020) Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation, time and frequency response matching analyses, Annual Reviews in Control, 49, 239–257. https://doi.org/10.1016/j.arcontrol.2020.03.003

9. 9. Dolai, S. K., Mondal, A. & Sarkar, P. (2022) Discretization of Fractional Order Operator in Delta Domain, Gazi University Journal of Science Part A: Engineering and Innovation, 9(4), 401–420. https://doi.org/10.54287/gujsa.1167156
10. 10. Elwakil, A. S. (2010) Fractional-order circuits and systems: An emerging interdisciplinary research area, IEEE CIRCUITS AND SYSTEMS MAGAZINE, 10(4), 40–50. https://doi.org/10.1109/MCAS.2010.938637
11. 11. Homaeinezhad, M. R. & Shahhosseini, A. (2020) Fractional order actuation systems: Theoretical foundation and application in feedback control of mechanical systems, Applied Mathematical Modelling, 87, 625–639. https://doi.org/10.1016/j.apm.2020.06.030
12. 12. Kartci, A., Agambayev, A., Farhat, M., Herencsar, N., Brancik, L., Bagci, H. & Salama, K. N. (2019) Synthesis and Optimization of Fractional-Order Elements Using a Genetic Algorithm, IEEE Access, 7, 80233–80246. https://doi.org/10.1109/ACCESS.2019.2923166
13. 13. Koseoglu, M. (2022) Time response optimal rational approximation: Improvement of time responses of MSBL based approximate fractional order derivative operators by using gradient descent optimization, Engineering Science and Technology, an International Journal, 101167. https://doi.org/10.1016/j.jestch.2022.101167
14. 14. Koseoglu, M., Deniz, F. N., Alagoz, B. B. & Alisoy, H. (2021) An effective analog circuit design of approximate fractional-order derivative models of M-SBL fitting method, Engineering Science and Technology, an International Journal. https://doi.org/10.1016/j.jestch.2021.10.001
15. 15. Koseoglu, M., Deniz, F. N., Alagoz, B. B., Yuce, A. & Tan, N. (2021) An experimental analog circuit realization of Matsuda’s approximate fractional-order integral operators for industrial electronics, Engineering Research Express. https://doi.org/10.1088/2631-8695/ac3e11
16. 16. Krishna, B. T. (2011) Studies on fractional order differentiators and integrators: A survey, Signal Processing, 91(3), 386–426. https://doi.org/10.1016/j.sigpro.2010.06.022
17. 17. Matlab-R2020b (2020) MATLAB Release 2020b, The MathWorks, Inc., Natick, Massachusetts, United States.
18. 18. Monje, C. A., Chen, Y., Vinagre, B. M., Xue, D. & Feliu, V. (2010) Fractional-order Systems and Controls, Springer London. https://doi.org/10.1007/978-1-84996-335-0
19. 19. NI-Multisim-14.1 (2017) National Instruments, Electronics Workbench Group, NI Multisim14.1, Available: http://www.ni.com/multisim/.
20. 20. Radwan, A. G., Khanday, F. A. & Said, L. A. (Eds.). (2021) Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing and Control. Elsevier. https://doi.org/10.1016/C2020-0-03165-8
21. 21. Shah, Z. M., Kathjoo, M. Y., Khanday, F. A., Biswas, K. & Psychalinos, C. (2019) A survey of single and multi-component Fractional-Order Elements (FOEs) and their applications, Microelectronics Journal, 84, 9–25. https://doi.org/10.1016/j.mejo.2018.12.010
22. 22. Sidhardh, S., Patnaik, S. & Semperlotti, F. (2020) Geometrically nonlinear response of a fractional-order nonlocal model of elasticity, International Journal of Non-Linear Mechanics, 125, 103529. https://doi.org/10.1016/j.ijnonlinmec.2020.103529
23. 23. Silva-Juárez, A., Tlelo-Cuautle, E., de la Fraga, L. G. & Li, R. (2020) FPAA-based implementation of fractional-order chaotic oscillators using first-order active filter blocks, Journal of Advanced Research, 25, 77–85. https://doi.org/10.1016/j.jare.2020.05.014
24. 24. Sun, H., Zhang, Y., Baleanu, D., Chen, W. & Chen, Y. (2018) A new collection of real world applications of fractional calculus in science and engineering, Communications in Nonlinear Science and Numerical Simulation, 64, 213–231 https://doi.org/10.1016/j.cnsns.2018.04.019
25. 25. Swarnakar, J., Sarkar, P. & Singh, L. J. (2019) Direct Discretization Method for Realizing a Class of Fractional Order System in Delta Domain – a Unified Approach, Automatic Control and Computer Sciences, 53(2), 127–139. https://doi.org/10.3103/S014641161902007X
26. 26. Tapadar, A., Khanday, F. A., Sen, S. & Adhikary, A. (2022) Fractional calculus in electronic circuits: a review, In A. G. Radwan, F. A. Khanday & L. A. Said (Eds.), Fractional Order Systems: An Overview of Mathematics, Design, and Applications for Engineers (pp. 441–482). Academic Press. https://doi.org/10.1016/B978-0-12-824293-3.00016-8
27. 27. Tepljakov, A. (2017) Fractional-order Modeling and Control of Dynamic Systems, Springer International Publishing. https://doi.org/10.1007/978-3-319-52950-9
28. 28. Tepljakov, A., Alagoz, B. B., Yeroglu, C., Gonzalez, E. A., Hosseinnia, S. H., Petlenkov, E., Ates, A. & Cech, M. (2021) Towards Industrialization of FOPID Controllers: A Survey on Milestones of Fractional- Order Control and Pathways for Future Developments, IEEE Access, 9, 21016–21042. https://doi.org/10.1109/ACCESS.2021.3055117
29. 29. Tufenkci, S., Senol, B., Alagoz, B. B. & Matušů, R. (2020) Disturbance rejection FOPID controller design in v-domain, Journal of Advanced Research, 25, 171–180. https://doi.org/10.1016/j.jare.2020.03.002
30. 30. Tzounas, G., Dassios, I., Murad, M. A. A. & Milano, F. (2020) Theory and Implementation of Fractional Order Controllers for Power System Applications, IEEE Transactions on Power Systems, 35(6), 4622–4631. https://doi.org/10.1109/TPWRS.2020.2999415
31. 31. Vigya, Mahto, T., Malik, H., Mukherjee, V., Alotaibi, M. A. & Almutairi, A. (2021) Renewable generation based hybrid power system control using fractional order-fuzzy controller, Energy Reports, 7, 641– 653. https://doi.org/10.1016/j.egyr.2021.01.022
32. 32. Vinagre, B., Podlubny, I., Hernández, A. & Feliu, V. (2000) Some approximations of fractional order operators used in control theory and applications, Fractional Calculus and Applied Analysis, 3(3), 231–248.
33. 33. Yang, B., Zhu, T., Zhang, X., Wang, J., Shu, H., Li, S., He, T., Yang, L. & Yu, T. (2020) Design and implementation of Battery/SMES hybrid energy storage systems used in electric vehicles: A nonlinear robust fractional-order control approach, Energy, 191, 116510. https://doi.org/10.1016/j.energy.2019.116510
34. 34. Yüce, A. & Tan, N. (2020) Electronic realisation technique for fractional order integrators, The Journal of Engineering, 2020(5), 157–167. https://doi.org/10.1049/joe.2019.1024