Kontrol Teorisinde Sık Kullanılan Bazı Fonksiyonların Kesirli Dereceden Çeşitli Türevlerinin Farklı Yöntemlere Göre Hata Değerlendirilmesi

Yıl 2021, Sayı: 28, 381 - 385, 30.11.2021

Mehmet Korkmaz

https://doi.org/10.31590/ejosat.1001249

Öz

Bu çalışma kesirli dereceden türev işlemlerinin farklı kontrol fonksiyonları ve farklı kesirli türev yöntemlerine göre bir analizi üzerinedir. Kontrol fonksiyonları olarak sıklıkla kullanılan birim basamak ve rampa fonksiyonları seçilmiştir. Kesirli dereceden türev hesaplaması için analitik yöntemlerin yanı sıra literatürde sıklıkla kullanılan Grünwald-Letnikov ve Riemann-Liouville yöntemleri tercih edilmiştir. Fonksiyonların farklı dereceden türevleri hesaplanarak analitik çözümle karşılaştırılmıştır. Elde edilen sonuçların hata değerlendirilmesi yapılmış ve ilgili eğrileri çizdirilmiştir.

Anahtar Kelimeler

Gama fonksiyonu, kontrol teorisi, kesirli dereceden türevli sistemler, kesirli dereceden türev hesabı

Error Evaluation of Some Frequently Used Control Functions in Terms of Different Order Fractional Derivatives According to Several Calculation Methods

Öz

This study is an analysis of the calculation of fractional order derivatives of some frequently used control functions according to the different calculation methods. A unit step and ramp functions are used as control functions which are often preferred to assess the system outputs. Besides analytical solutions, Grünwald-Letnikov and Riemann-Liouville methods are chosen as calculation procedures of fractional derivatives. Analytical solutions of functions at different orders are compared with different definitions methods for the same orders. Obtained results are analysed in terms of error values and relevant curves are plotted.

Anahtar Kelimeler

Gamma function, control theory, fractional order derivative systems, fractional order derivative calculation

Kaynakça

• Al-Dhaifallah, M., Nassef, A. M., Rezk, H., & Nisar, K. S. (2018). Optimal parameter design of fractional order control based INC-MPPT for PV system. Solar Energy, 159, 650-664.
• Chen, D., Sun, S., Zhang, C., Chen, Y., & Xue, D. (2013). Fractional-order TV-L 2 model for image denoising. Central European Journal of Physics, 11(10), 1414-1422.
• Cruz–Duarte, J. M., Rosales–Garcia, J., Correa–Cely, C. R., Garcia–Perez, A., & Avina–Cervantes, J. G. (2018). A closed form expression for the Gaussian–based Caputo–Fabrizio fractional derivative for signal processing applications. Communications in Nonlinear Science and Numerical Simulation, 61, 138-148.
• Kumar, S., Saxena, R., & Singh, K. (2017). Fractional Fourier transform and fractional-order calculus-based image edge detection. Circuits, Systems, and Signal Processing, 36(4), 1493-1513.
• Ma, C. Y., Shiri, B., Wu, G. C., & Baleanu, D. (2020). New fractional signal smoothing equations with short memory and variable order. Optik, 218, 164507.
• Machado, J. T., & Lopes, A. M. (2019). Fractional-order modeling of a diode. Communications in Nonlinear Science and Numerical Simulation, 70, 343-353.
• Muñoz-Vázquez, A. J., Gaxiola, F., Martínez-Reyes, F., & Manzo-Martínez, A. (2019). A fuzzy fractional-order control of robotic manipulators with PID error manifolds. Applied soft computing, 83, 105646.
• Saçu, İ. E., & Korkmaz, N. (2021). Fraksiyonel Dereceli Kaotik Lorenz Sistemi’nin Devre Sentezi. Avrupa Bilim ve Teknoloji Dergisi, (24), 42-46.
• Soukkou, A., Belhour, M. C., & Leulmi, S. (2016). Review, design, optimization and stability analysis of fractional-order PID controller. International Journal of Intelligent Systems and Applications, 8(7), 73.
• Tepljakov, A. (2011). Fractional-order calculus based identification and control of linear dynamic systems. Tallinn University of Technology.
• Tepljakov, A., Alagoz, B. B., Yeroglu, C., Gonzalez, E. A., Hosseinnia, S. H., Petlenkov, E., ... & 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.
• Valsa, J. (2011). Numerical inversion of Laplace transforms in MATLAB. MATLAB Central File ID, 32824. Wang, B., Li, S. E., Peng, H., & Liu, Z. (2015). Fractional-order modeling and parameter identification for lithium-ion batteries. Journal of Power Sources, 293, 151-161.
• Xue, D., & Chen, Y. (2015). Modeling, analysis and design of control systems in MATLAB and Simulink. World Scientific Publishing.
• Xue, D. (2017). Fractional-order control systems. de Gruyter.