Study on a fractional-order controllers based on best rational approximation of fractional calculus operators

Abstract

This paper presents a rational approximation method for fractional calculus operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo><mml:mi mathvariant="normal">α</mml:mi></mml:mrow></mml:msup></mml:math>in the given frequency range and the error, which is based on the best rational approximation definition. The fractional integral operator is selected as an example to describe the construction of the rational approximation functions. An application case (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>0.3</mml:mn></mml:mrow></mml:msup></mml:math>) is used to illustrate the effectiveness of the proposed method. The obtained approximation function in the frequency domain is a best rational approximation function, which can further improve the accuracy of the approximation without increasing the orders. On the basis of the presented rational approximation method, a rational approximation equation of a fractional-order PID controller is obtained. Finally, the method for analyzing the optimization and frequency characteristics of the fractional-order controller is implemented to demonstrate the good frequency characteristic and best structure. The results from theoretical analysis and experimental verification show that the proposed method provides a new design idea for the effective application of the fractional-order PID controller in engineering.