Analytical fractional PID controller design based on Bode's ideal transfer function plus time delay

Erhan Yumuk, Müjde Güzelkaya*, İbrahim Eksin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

68 Citations (Scopus)


In this paper, a fractional order PID controller cascaded with a fractional filter is proposed for higher order processes. In this analytical design methodology, one or two reduced fractional orders plus time delay models are used to represent higher order system transfer functions. The controller parameters are determined so as to meet certain frequency domain specifications. A unity feedback reference model is employed where Bode's ideal loop transfer function plus time delay of the fractional order model is placed in the forward path. The addition of this time delay provides the exact determination of frequency domain specifications if the system either intrinsically owns a time delay or a time delay is injected by its reduced order model. The proposed methodology is compared with two other related methodologies and it has been observed that the proposed controller performs much better than the others. Moreover, some empirical formulas for time domain characteristics of the reference model are numerically derived in terms of certain frequency domain specifications and time delay of the fractional reduced order model. The accuracy of these formulas is tested by simulations. The iso-damping, noise attenuation and load disturbance suppression performances of the proposed controller are also considered and compared with those of other related controllers.

Original languageEnglish
Pages (from-to)196-206
Number of pages11
JournalISA Transactions
Publication statusPublished - Aug 2019

Bibliographical note

Publisher Copyright:
© 2019 ISA


This study was supported by Istanbul Technical University Research Support Program (ITU-BAP) (Project Code: MDK-2018-41344).

FundersFunder number
Istanbul Teknik ÜniversitesiMDK-2018-41344


    • Bode's ideal loop transfer function
    • Fractional order PID controllers
    • Frequency domain design
    • Reduced fractional order models
    • Time delay


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