Abstract
The proportional Integral Derivative (PID) controller has three basic parameters: Proportional gain (Kp), Integral gain (Ki) and Derivative gain (Kd). In a conventional PID controller, integral and derivative operators are integer order. The researchers proposed a fractional order PID (PIλDµ) controller by using the fractional integral and derivative operators instead of the integer order integral and derivative operators in the traditional PID controller because it improves the control performance. The PIλDµ controller has an additional fractional integrator degree (λ) and fractional derivative degree (µ). In this study, the focus is on the design of a fractional-order PID controller according to a reference model in the time domain. Bode's ideal transfer function was used as the reference model. It is aimed to obtain PIλDµ parameters by minimizing the error between the time domain response of Bode's ideal transfer function model and the output of the system to be controlled by using the optimization method. Genetic Algorithm (GA) optimization was used as the optimization method. The study was carried out as a simulation study on an inverted pendulum system with a single-input multiple-output (SIMO) structure.
Keywords
Fractional order PID controller model reference design inverted pendulum single input multi output system.
References
- Astrom KJ. (1995). PID controllers: theory, design and tuning. Instrt Soc Am.
- Azarmi R., Tavakoli-Kakhki M., Sedigh A. K. & Fatehi A. (2015). Analytical design of fractional order PID controllers based on the fractional set-point weighted structure: Case study in twin rotor helicopter, Mechatronics Volume 31, 222-233.
- Barbosa R. S., Machado J. A. T. & Ferreıra I. M., (2004). Tuning Of Pıd Controllers Based On Bode’s Ideal Transfer Function, Nonlinear Dynamics 38: 305–321.
- Bouafoura M. K. & Braiek N. B. (2010). PIλDµ controller design for integer and fractional plants using piecewise orthogonal functions. Communications in Nonlinear Science and Numerical Simulation, 15 (5), 1267-1278.
- Carlson G. & Halijak C. (1964). Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process, IEEE Transactions on Circuit Theory, 11:2, 210-213.
- Castillo F.J., Feliu V., Sánchez L., Rivas R. & Jaramillo V.H. (2010). Robust PDα Time-Domain Tuning Rules for Controlling DC-Motors. The 4th IFAC Workshop Fractional Differentiation and its Applications, Badajoz, Spain.
- Castillo F.J., Feliu V., Sánchez L., Rivas R. & L.Sánchezc. (2010). Design of a class of fractional controllers from frequency specifications with guaranteed time domain behavior. Computers & Mathematics with Applications, 59(5), 1656-1666. Charef A., Sun H., Tsao Y. & Onaral B. (1992). Fractal system as represented by singularity function, IEEE Transactions on Automatic Control, 37:9, 1465-1470.
- Chao H., Luo Y., Di L. & Chen Y.Q. (2010). Roll-channel fractional order controller design for a small fixed-wing unmanned aerial vehicle. Control Eng Prac 18(7), 761–72.
- Deniz F. N., Yüce A. & Tan N. (2019). Tuning of PI-PD Controller Based on Standard Forms for Fractional Order, 8 (1), 5-21.
- Deniz F. N., Yüce A., Tan N. & Atherton D. P. (2017). Tuning of Fractional Order PID Controllers Based on Integral Performance Criteria Using Fourier Series Method, IFAC PapersOnLine 50-1, 8561–8566.
- Doğruer T., Yüce A. & Tan N. (2017). PID Controller Design Using Optimization Method for Fractional Order Control Systems with Time Delay, Gaziosmanpasa Journal of Scientific Research, 6, Special Issue, (ISMSIT2017), 30-39.
- Doğruer T., Yüce A. & Tan N. (2017). PID Controller Design Based on Reference Model in Fractional Order Control Systems, Bilge International Journal of Science and Technology Research, 1 (1), 52-58.
- Doğruer T. & Tan N., (2020). Real Tıme Control of Twin Rotor Mimo System With PID and Fractional Order PID Controller, Mugla Journal Of Science And Technology 6, 1-9.
- El-Khazali R. (2013). Fractional-order PIλDµ controller design, Computers & Mathematics with Applications Volume 66, Issue 5, 639-646.
- Hamamci S. E & Koksal M. (2010). Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems. Computers & Mathematics with Applications, 59 (5), 1621-1629.
- Keyser R. D., Muresan C. I. & Ionescu C. M. (2016). A novel auto-tuning method for fractional order PI/PD controllers, ISA Transactions Volume 62, 268-275.
- Keyser R. D., Muresan C. I. & Ionescu C. M. (2018). “Autotuning of a Robust Fractional Order PID Controller”, IFAC PapersOnLine 51-25, 466–471.
- Luo Y. & Chen Y.Q. (2009). Fractional-order [Proportional Derivative] Controller for Robust Motion Control: Tuning Procedure and Validation, 2009 American Control Conference, St. Louis, Missouri, USA.
- Matsuda K. & Fujii H. (1993). H (infinity) optimized wave-absorbing controlAnalytical and experimental results, Journal of Guidance, Control, and Dynamics, 16:6, 1146-1153.
- Michigan. (2022). Author. Retrieved from https://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum§ion=SystemModeling
- Muresan C. I., Dulf E. H. & Both R. (2016). Vector-based tuning and experimental validation of fractional-order PI/PD controllers, Nonlinear Dyn, 84, 179–188.
- Muresan C. I. & Keyser R. D., (2022). Revisiting Ziegler–Nichols. A fractional order approach, ISA Transactions Volume 129, Part A, 287-296.
- O’Dwyer A.. (2006). Handbook of PI and PID Controller Tuning Rules, Imperial College Press, http://dx.doi.org/10.1142/9781860949104
- Oustaloup A., Levron F., Mathieu B. & Nanot F. M.. (2000). Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47:1, 25-39.
- Ozyetkin M. M, Onat C. & Tan N. (2020). PI‐PD controller design for time delay systems via the weighted geometrical center method. Asian Journal of Control, 22 (5), 1811-1826.
- Podlubny I. (1999). Fractional-Order Systems and PIλDµ-Controllers, IEEE Transactions on Automatic Control, 44(1), 208 - 214.
- Podlubny I., Petráš I., Vinagre B. M., O'leary P. & Dorčák Ľ. (2002). Analogue realizations of fractional-order controllers, Nonlinear dynamics, 29,1-4, 281-296.
- Shah P. & Agashe S. (2016). Review of fractional PID controller, Mechatronics 38, 29–41,
- Shankaran V. P., Azid S. I. & Mehta U., (2022).Fractional-order PI plus D controller for second-order integrating plants: Stabilization and tuning method, ISA Transactions Volume 129, Part A, 592-604.
- Valério D. & da Costa JS. (2011). Introduction to single-input, single-output fractional control. IET Control Theory Appl 5(8), 1033–57.
- Yeroglu C. & Tan N.. (2011). Note on fractional-order proportional–integral–differential controller design, Volume 5, Issue 17, 1978 – 1989.
Abstract
Oransal İntegral Türev (PID) denetleyicisinin üç temel parametresi vardır: Oransal kazanç (Kp), İntegral kazanç (Ki) ve Türev kazancı (Kd). Geleneksel bir PID denetleyicide, integral ve türev operatörleri tamsayı derecelidir. Araştırmacılar, kontrol performansını iyileştirdiği için geleneksel PID denetleyicideki tamsayı sıralı integral ve türev operatörleri yerine kesirli integral ve türev operatörlerini kullanarak kesir dereceli PID (PIλDµ) denetleyici önermişlerdir. PIλDµ denetleyici, ek bir kesirli entegratör derecesine (λ) ve kesirli türev derecesine (µ) sahiptir. Bu çalışmada, zaman domeninde referans modele göre kesir dereceli PID denetleyici tasarımı üzerinde durulmuştur. Referans model olarak Bode'nin ideal transfer fonksiyonu kullanılmıştır. Optimizasyon yöntemi kullanılarak Bode'nin ideal transfer fonksiyon modelinin zaman domeni yanıtı ile kontrol edilecek sistemin çıkışı arasındaki hata minimize edilerek PIλDµ parametrelerinin elde edilmesi amaçlanmıştır. Optimizasyon yöntemi olarak Genetik Algoritma (GA) optimizasyonu kullanılmıştır. Çalışma, tek girişli çoklu çıkış (SIMO) yapısına sahip ters sarkaç sistemi üzerinde simülasyon çalışması olarak gerçekleştirilmiştir.
Keywords
Kesir dereceli PID denetleyici model referans tasarım ters sarkaç tek girişli çoklu çıkış sistem.
References
- Astrom KJ. (1995). PID controllers: theory, design and tuning. Instrt Soc Am.
- Azarmi R., Tavakoli-Kakhki M., Sedigh A. K. & Fatehi A. (2015). Analytical design of fractional order PID controllers based on the fractional set-point weighted structure: Case study in twin rotor helicopter, Mechatronics Volume 31, 222-233.
- Barbosa R. S., Machado J. A. T. & Ferreıra I. M., (2004). Tuning Of Pıd Controllers Based On Bode’s Ideal Transfer Function, Nonlinear Dynamics 38: 305–321.
- Bouafoura M. K. & Braiek N. B. (2010). PIλDµ controller design for integer and fractional plants using piecewise orthogonal functions. Communications in Nonlinear Science and Numerical Simulation, 15 (5), 1267-1278.
- Carlson G. & Halijak C. (1964). Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process, IEEE Transactions on Circuit Theory, 11:2, 210-213.
- Castillo F.J., Feliu V., Sánchez L., Rivas R. & Jaramillo V.H. (2010). Robust PDα Time-Domain Tuning Rules for Controlling DC-Motors. The 4th IFAC Workshop Fractional Differentiation and its Applications, Badajoz, Spain.
- Castillo F.J., Feliu V., Sánchez L., Rivas R. & L.Sánchezc. (2010). Design of a class of fractional controllers from frequency specifications with guaranteed time domain behavior. Computers & Mathematics with Applications, 59(5), 1656-1666. Charef A., Sun H., Tsao Y. & Onaral B. (1992). Fractal system as represented by singularity function, IEEE Transactions on Automatic Control, 37:9, 1465-1470.
- Chao H., Luo Y., Di L. & Chen Y.Q. (2010). Roll-channel fractional order controller design for a small fixed-wing unmanned aerial vehicle. Control Eng Prac 18(7), 761–72.
- Deniz F. N., Yüce A. & Tan N. (2019). Tuning of PI-PD Controller Based on Standard Forms for Fractional Order, 8 (1), 5-21.
- Deniz F. N., Yüce A., Tan N. & Atherton D. P. (2017). Tuning of Fractional Order PID Controllers Based on Integral Performance Criteria Using Fourier Series Method, IFAC PapersOnLine 50-1, 8561–8566.
- Doğruer T., Yüce A. & Tan N. (2017). PID Controller Design Using Optimization Method for Fractional Order Control Systems with Time Delay, Gaziosmanpasa Journal of Scientific Research, 6, Special Issue, (ISMSIT2017), 30-39.
- Doğruer T., Yüce A. & Tan N. (2017). PID Controller Design Based on Reference Model in Fractional Order Control Systems, Bilge International Journal of Science and Technology Research, 1 (1), 52-58.
- Doğruer T. & Tan N., (2020). Real Tıme Control of Twin Rotor Mimo System With PID and Fractional Order PID Controller, Mugla Journal Of Science And Technology 6, 1-9.
- El-Khazali R. (2013). Fractional-order PIλDµ controller design, Computers & Mathematics with Applications Volume 66, Issue 5, 639-646.
- Hamamci S. E & Koksal M. (2010). Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems. Computers & Mathematics with Applications, 59 (5), 1621-1629.
- Keyser R. D., Muresan C. I. & Ionescu C. M. (2016). A novel auto-tuning method for fractional order PI/PD controllers, ISA Transactions Volume 62, 268-275.
- Keyser R. D., Muresan C. I. & Ionescu C. M. (2018). “Autotuning of a Robust Fractional Order PID Controller”, IFAC PapersOnLine 51-25, 466–471.
- Luo Y. & Chen Y.Q. (2009). Fractional-order [Proportional Derivative] Controller for Robust Motion Control: Tuning Procedure and Validation, 2009 American Control Conference, St. Louis, Missouri, USA.
- Matsuda K. & Fujii H. (1993). H (infinity) optimized wave-absorbing controlAnalytical and experimental results, Journal of Guidance, Control, and Dynamics, 16:6, 1146-1153.
- Michigan. (2022). Author. Retrieved from https://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum§ion=SystemModeling
- Muresan C. I., Dulf E. H. & Both R. (2016). Vector-based tuning and experimental validation of fractional-order PI/PD controllers, Nonlinear Dyn, 84, 179–188.
- Muresan C. I. & Keyser R. D., (2022). Revisiting Ziegler–Nichols. A fractional order approach, ISA Transactions Volume 129, Part A, 287-296.
- O’Dwyer A.. (2006). Handbook of PI and PID Controller Tuning Rules, Imperial College Press, http://dx.doi.org/10.1142/9781860949104
- Oustaloup A., Levron F., Mathieu B. & Nanot F. M.. (2000). Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47:1, 25-39.
- Ozyetkin M. M, Onat C. & Tan N. (2020). PI‐PD controller design for time delay systems via the weighted geometrical center method. Asian Journal of Control, 22 (5), 1811-1826.
- Podlubny I. (1999). Fractional-Order Systems and PIλDµ-Controllers, IEEE Transactions on Automatic Control, 44(1), 208 - 214.
- Podlubny I., Petráš I., Vinagre B. M., O'leary P. & Dorčák Ľ. (2002). Analogue realizations of fractional-order controllers, Nonlinear dynamics, 29,1-4, 281-296.
- Shah P. & Agashe S. (2016). Review of fractional PID controller, Mechatronics 38, 29–41,
- Shankaran V. P., Azid S. I. & Mehta U., (2022).Fractional-order PI plus D controller for second-order integrating plants: Stabilization and tuning method, ISA Transactions Volume 129, Part A, 592-604.
- Valério D. & da Costa JS. (2011). Introduction to single-input, single-output fractional control. IET Control Theory Appl 5(8), 1033–57.
- Yeroglu C. & Tan N.. (2011). Note on fractional-order proportional–integral–differential controller design, Volume 5, Issue 17, 1978 – 1989.
Details
| Primary Language | English |
|---|---|
| Subjects | Electrical Engineering |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | July 11, 2023 |
| Publication Date | July 14, 2023 |
| Submission Date | May 17, 2023 |
| Published in Issue | Year 2023 Volume: 15 Issue: 2 |
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