Control of second α-order systems using fractal PD and PID controllers

Khalili Golmankhaneh A., Banchuin R., Torres D. F. M., Damián-Adame L.

International Journal of Dynamics and Control, cilt.14, sa.6, 2026 (ESCI, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 14 Sayı: 6
  • Basım Tarihi: 2026
  • Doi Numarası: 10.1007/s40435-026-02116-y
  • Dergi Adı: International Journal of Dynamics and Control
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
  • Anahtar Kelimeler: FPD control, FPID control, Fractal calculus, Fractal time
  • Van Yüzüncü Yıl Üniversitesi Adresli: Evet

Özet

This paper studies second α-order dynamical systems within the framework of fractal calculus and examines their behavior under fractal proportional–derivative (FPD) and fractal proportional–integral–derivative (FPID) controllers. A brief overview of fractal calculus is provided, and fractal initial value and final value theorems are formulated. The structure of FPID control for second α-order systems is presented, and the stability conditions are analyzed within the fractal setting. The response of FPD controllers is also investigated through analytical expressions and numerical simulations for selected values of the fractal order α. As an illustrative example, a series RLC circuit is modeled using fractal derivatives to demonstrate how fractal-order dynamics influence the controlled system response. The results emphasize the role of the fractal order α in shaping transient behavior and steady-state characteristics.