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Genelleştirilmiş sınırlı kararlılık bölgesi ile PI ve PID denetleyici tasarımı

PI and PID controller design based on generalized stability boundary locus

Journal Name:

  • Dicle Üniversitesi Mühendislik Fakültesi Mühendislik Dergisi

Publication Year:

  • 2017

Key Words:

Keywords (Original Language):

Author NameUniversity of AuthorFaculty of Author
Abstract (2. Language): 
The paper introduces a generalized approach to identify all stabilizing PI and PID controllers. For this purpose, a stable first order plus dead time (FOPDT) model is used to model higher order plant transfer functions. In order to estimate the process transfer function parameters, relay feedback identification method given by Kaya and Atherton (2001) is used. The identification method, assuming no measurement errors and disturbances, results in exact estimations. After obtaining the plant transfer function model, normalized form of the process transfer function model and controller is used to plot stability boundary locus in  , (T/ T ) c c i KK KK plane for the PI controller and  (T / T), (T/ T ) c d c i KK KK plane for the PID controller for a certain value of normalized dead time,   /T , where  and T are, respectively, the time delay and time constant of the FOPDT model. By doing so, the need to compute the stability boundary locus for different plant transfer functions has been removed. If the actual and the model transfer functions match exactly, then the proposed approach will give exact solutions. However, in the case of a mismatch between the actual and the model transfer functions, the approach will result in approximate solutions, especially for the frequencies near the critical frequency. Therefore, if the actual plant transfer function is a higher order one, then, one must pay attention with the points selected in the stability region. Here two suggestions are provided in order to ensure a stable closed loop response while selecting the points to be used in determining PI and PID controller parameters: 1) The points towards the centre of the stability region must be selected. 2) The stability region that corresponds to a previous larger normalized dead time than the current normalized dead time should be used. The proposed approach can be extended to PID controllers as well. For the all stabilizing PID controller design the following procedure can be used. In the controller design, it is more usual to assume a controller gain and then to calculate the remaining other two tuning parameters, namely, i T and d T . Stability region in the ( (T/ T ), (T / T)) c i c d KK KK plane for a fixed value of c KK can be obtained by using the stability regions obtained in ( , (T / T)) c c d KK KK and ( , (T/ T )) c c i KK KK planes. Once the above cited stability boundary locus are obtained, plotting stability boundary locus in the ( (T/ T ), (T / T)) c i c d KK KK plane for different values of normalized dead time ratios can be carried out. The proposed approach brings the advantage of not requiring to plot the stability boundary locus each time as the process transfer function changes, which is the case for the so far reported studies in the literature. Simulation examples are provided to illustrate the usefulness of the proposed approach.
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Abstract (Original Language): 
Bu çalışma, zaman gecikmeli kontrol sistemlerinin kararlılığı için tüm PI ve PID denetleyici parametre değerlerinin hesaplanmasında genelleştirilmiş bir yaklaşım önermektedir. Bu yaklaşımda yüksek mertebeden transfer fonksiyonlarının, birinci derece artı zaman gecikmeli transfer fonksiyonları ile modellenmesi gerekir. Elde edilen model ve denetleyici transfer fonksiyonları normalize edilerek PI denetleyici tasarımı için  , (T/ T ) c c i KK KK düzleminde sınırlı kararlılık bölgesi oluşturulur. Benzer şekilde PID denetleyici tasarımı için  , (T / T) c c d KK KK ,  , (T/ T ) c c i KK KK ve  (T / T), (T/ T ) c d c i KK KK düzlemlerinde sınırlı kararlılık bölgeleri oluşturulur. PI ve PID denetleyici parametre değerleri, elde edilen sınırlı kararlılık bölgeleri ile belirlenir. Bu yaklaşım sayesinde transfer fonksiyonunun her değişmesi ile sınırlı kararlılık bölgelerinin yeniden oluşturulmasına ihtiyaç duyulmaz. Elde edilen genelleştirilmiş sınırlı kararlılık bölgeleri ile tüm PI ve PID denetleyici parametre değerleri hesaplanabilir. Böylelikle bu yaklaşım şimdiye kadar literatürde bildirilmiş çalışmalara göre avantaj sağlar. Önerilen yaklaşımın kullanışlılığını açıklamak için örnek benzetimler verilmiştir.
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