PID Controller Design Based on Reference Model in Fractional Order Control Systems

Abstract

In the modeling of physical systems, fractional order systems are known to perform a more successful modelling than integer order systems. In this paper, PID controller design was performed according to a reference model for a fractional order system. The main purpose of the study is to obtain PID controller parameters according to a desired time response in the output signal. It is aimed to obtain the optimum PID parameters by minimizing the error between the reference model and controlled system. Integral performance criteria were used to minimize the error. The reference model is a transfer function of a second order system. This transfer function has two parameters that need to be set. These parameters are natural frequency (ω_n) and damping ratio (ζ). By setting these two parameters, desired unit step response curve can be obtained. PID controller parameters were obtained by optimization method. Optimization describes problem-solving processes in a systematic way by minimizing or maximizing a real function and placing values in the function. PID controller parameters are obtained by optimizing according to model transfer functions. By applying the calculated PID controller parameters to the fractional order control system, the unit step responses are obtained. The success of the optimization method can be seen from the graphs obtained and from the given tables.

Keywords

• PID Controller,Optimization,Fractional order systems

Kesir Dereceli Kontrol Sistemlerinde Referans Modele Dayalı PID Kontrolör Tasarımı

Abstract

Fiziksel sistemlerin modellenmesinde, kesir dereceli sistemler tamsayı dereceli sistemlere göre daha başarılı bir modelleme gerçekleştirebilmektedir. Çalışmada, kesir dereceli bir sistem için referans bir modele göre PID kontrolör tasarımı gerçekleştirilmiştir. Çalışmanın temel amacı, çıkış sinyalinde istenen bir zaman cevabına göre PID kontrolör parametrelerini elde etmektir. Referans model ile denetlenen sistem arasındaki hata minimize edilerek optimum PID parametrelerinin elde edilmesi amaçlanmıştır. Hatayı minimize etmek için sıklıkla integral performans kriterleri kullanılır. Referans olarak alınan sistem ikinci mertebeden bir sistemin transfer fonksiyonudur. Bu transfer fonksiyonu ayarlanması gereken iki parametreye sahiptir. Bu parametreler doğal frekans (ω_n) ve sönüm oranıdır (ζ). Bu iki parametre ayarlanarak istenilen birim basamak cevap eğrisi elde edilebilir. PID kontrolör parametrelerinin elde edilmesi optimizasyon yöntemiyle gerçekleştirilmiştir. Optimizasyon bir gerçel fonksiyonu minimize ya da maksimize etmek amacı ile fonksiyona değerler yerleştirerek sistematik bir şekilde problem çözüm işlemlerini tanımlar. Bu model transfer fonksiyonlarına göre optimizasyon işlemi gerçekleştirilerek, PID kontrolör parametreleri elde edilebilir. Bulunan PID kontrolör parametrelerinin kesir dereceli kontrol sistemine uygulanmasıyla, denetlenen sistem çıkışında birim basamak cevapları elde edilir. Optimizasyon yönteminin başarısı elde edilen grafiklerden ve oluşturulan tablolardan görülmektedir.

Keywords

• PID Kontrolör,Optimizasyon,Kesir dereceli sistemler

References

1. Åström, K. J., Hägglund, T. (2001). The future of PID control. Control engineering practice, 9(11), 1163-1175.
2. Atherton, D. (2009). Control engineering: Bookboon.
3. Battula, K., Reddy, K. (2008). Active and Passive Realization of Fractance Device of Order 1/2 , pp.1-6.
4. Bohannan, G. W. (2008). Analog Fractional Order Controller in Temperature and Motor Control Applications. Journal of Vibration and Control, 14(9-10),1487-1498.
5. 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.
6. Cervera, J., Baños, A. (2006). Automatic loop shaping in qft by using crone structures. IFAC Proceedings Volumes, 39(11), 207-212.
7. Chen, Y., Petras, I., Xue, D. (2009). Fractional order control-A tutorial. Paper presented at the American Control Conference, ACC'09, pp.1397-1411.
8. Gutiérrez, R. E., Rosário, J. M., Tenreiro Machado, J. (2010). Fractional order calculus: basic concepts and engineering applications. Mathematical Problems in Engineering, 2010.

9. Katsuhiko, O. (2010). Modern control engineering.
10. Kuo, B. C. (1987). Automatic control systems: Prentice Hall PTR.
11. Lokenath, D. (2003). Recent applications of fractional calculus to science and engineering (Vol. 2003).
12. Manabe, S. (1961). The noninteger integral and its application to control systems. English Translation Journal Japan, 6, 83-87.
13. Manabe, S. (1963). The system design by the use of a model consisting of a saturation and noninteger integral. English Translation Journal Japan, 47-150.
14. Matsuda, K., Fujii, H. (1993). H(infinity) optimized wave-absorbing control - Analytical and experimental results. Journal of Guidance, Control, and Dynamics, 16(6), 1146-1153. Monje, C. A., Chen, Y., Vinagre, B. M., Xue, D., Feliu-Batlle, V. (2010). Fractional-order systems and controls: fundamentals and applications: Springer Science & Business Media.
15. 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.
16. Panda, R., Dash, M. (2006). Fractional generalized splines and signal processing. Signal Processing, 86(9), 2340-2350.
17. Pu, Y.-F., Yuan, X., Liao, k., Zhou, J., Ni, Z., Pu, X., Zeng, Y. (2006). A recursive two-circuits series analog fractance circuit for any order fractional calculus - art. no. 60271Y (Vol. 6027).
18. Tustin, A., Allanson, J., Layton, J., Jakeways, R. (1958). The design of systems for automatic control of the position of massive objects. Proceedings of the IEE-Part C: Monographs, 105(1S), 1-57.
19. Xue, D., Zhao, C., Chen, Y. (2006, 25-28 June 2006). A Modified Approximation Method of Fractional Order System. Paper presented at the 2006 International Conference on Mechatronics and Automation.
20. Ziegler, J. G., Nichols, N. B. (1942). Optimum settings for automatic controllers. trans. ASME, 64(11).