Kesir Dereceli Kontrol Sistemlerinde Referans Modele Dayalı PID Kontrolör Tasarımı

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 (ωn) 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.

Anahtar Kelimeler:

PID Kontrolör, Optimizasyon, Kesir dereceli sistemler

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

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 (ωn) 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,

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