Please use this identifier to cite or link to this item: http://ea.donntu.edu.ua:8080/jspui/handle/123456789/22817
Title: О ВЫБОРЕ ПАРАМЕТРОВ ПРЕОБРАЗОВАНИЯ МЕБИУСА ПРИ КОНСТРУИРОВАНИИ СТАБИЛИЗИРУЮЩИХ РЕГУЛЯТОРОВ
Other Titles: Selecting Parameters of Möbius Transformation for Stabilizing Controller Design
Про вибір параметрів перетворення Мебіусу при конструюванні стабілізуючих регуляторів
Authors: Хорхордин, А.В.
Батыр, С.С.
Безрук, А.А.
Khorkhordin, A.V.
Batyr, S.S.
Bezruk, A.A.
Хорхордін, А.В.
Безрук, О.А.
Батир, С.С.
Keywords: перетворення Мебіусу
регулятор
Q-параметризація
математична модель
передавальна функція
робастна стійкість
Möbius transformation
controller
Q-parameterization
mathematical model
transfer function
robust stability
преобразование Мебиуса
Q-параметризация
математическая модель
передаточная функция
робастная устойчивость
Issue Date: 2013
Publisher: Донецький національний технічний університет
Citation: Наукові праці Донецького національного технічного університету. Серія: Обчислювальна техніка та автоматизація. Випуск 2 (25). - Донецьк, ДонНТУ, 2013. С - 152-159
Abstract: Выполнен анализ влияния параметров преобразования Мебиуса на качество системы управления неустойчивыми объектами. Рассмотрена последовательность расчета стабилизирующих регуляторов на основе Q-параметризации. Проведен выбор структуры регулятора и даны рекомендации по выбору его параметров для объекта, представляющего собой последовательное соединение двух интеграторов. Выполнена проверка робастной устойчивости замкнутой системы регулирования.
Description: The problem of unstable objects stabilization is effectively solved by Q-parametrization (Youla-parametrization). The Q-parameterization procedure uses Möbius transformation. Its parameters, along with purposeful choice of bounded above and stable transfer function Q(s) directly affect the values of controls and, therefore, the quality of control system. The paper considers the influence of parameters of Möbius transformation on the quality of control system. The dependence of the closed loop system transfer function’s poles on the parameters of Möbius transformations and conditions of robust stability of the system is studied. In the most general case, the transfer function of the controller is calculated by formula W (s) (X MQ) /(Y NQ) R    in which the transfer functions N(s) , M (s) , X (s) and Y (s) can be obtained as a result of the generalized Euclidean algorithm for polynomials. The algorithm starts with the fact that the transfer function of the object is a permutation s  (a  b) /(c  d) . a,b,c, d  C and ad  bc  0 . The result is a transfer function of the object W ( )  N( ) / M ( ) as a ratio of polynomials. The algorithm of Euclid was applied to the obtained polynomials. Rational functions X ( ) and Y ( ) were received from the relationship NX  MY  1 . Polynomials N(s) , M (s) , X (s) and Y (s) were obtained by reverse Möbius transformations. The regulator’s transfer function was defined in general way and the transfer functions of the closed loop system were obtained, which includes Q(s) as a parameter. Q(s) was selected to provide the necessary quality of the control system. Taking into account Q(s) we calculated the transfer function of regulator WR(s), which will provide stability and set quality of the control system. With the use of the controller built with the help of Q-parameterization, all the poles of the transfer function of the closed system are "pulled together" at one point s  a / c . Duration of the transition process (system performance) depends only on the parameters a and c of Möbius transformations: it decreases with a increasing and increases with c . The poles of the transfer function of the closed loop system do not depend on the parameters b and d of Möbius transformation. The simplest solution is the choice Q(s)  0 , then the transfer function of the controller W (s) X (s) / Y (s) R  . The results obtained in this study can be used to calculate stabilizing controllers to provide the necessary performance and robustness of the closed system.
URI: http://ea.donntu.edu.ua/handle/123456789/22817
ISSN: 2075-4272
Appears in Collections:Випуск 2 (25)'2013
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