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<p>This paper studies second <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> in shaping transient behavior and steady-state characteristics.</p>

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Control of second \(\alpha \)-order systems using fractal PD and PID controllers

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Alireza Khalili Golmankhaneh,
  • Rawid Banchuin,
  • Delfim F. M. Torres,
  • Lucero Damián-Adame

摘要

This paper studies second \(\alpha \) 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 \(\alpha \) 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 \(\alpha \) 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 \(\alpha \) α in shaping transient behavior and steady-state characteristics.