分数阶微分器的实现及其阶次对方法选择的影响

分数阶微分的计算是分析分数阶系统和设计分数阶控制器的一个重要环节,不同方法设计的分数阶微分器其性能不同。为了在应用时选择或设计合适的分数阶微分器,本文总结了时域中14种分数阶微分器的实现方法,同时对它们的性能进行了分析和比较,包括:各种分数阶微分器的阶数对其零极点位置、冲击响应和频率响应的影响。结果表明,采用连分式展开的近似方法要比采用级数展开的近似方法好,其中一阶、二阶和三阶向后有限差分公式在两种方式下都能得到较好的近似结果,Tustin公式和Simpson公式不适宜于级数展开方式。

Digital Implementation of Fractional Order Differentiators and Its Order Influence on Choice of Methods

The computation of the fractional-order differentiation is necessary for analyzing fractional-order systems and designing fractional-order controllers.Performances of fractional-order differentiators in different designed ways are different. In order to select an appropriate fractional-order differentiator for a particular case,14 fractional-order differentiators realized in the time domain are summarized and their performances are analyzed including placement of zeros and poles;impulse responses and frequency responses for different parameters of fractional-order differentiators.Results show that continued fraction expansion methods are better than series expansion methods.The first-order difference formula,the second-order difference formula and the third-order difference formula are be suited for two kinds of methods.The Tustin and the Simpson formulae are not be suited for the series expansion method.