Carlson迭代与任意阶分数微积分算子的有理逼近  

Carlson iterating and rational approximation of arbitrary order fractional calculus operator

作　　者：何秋燕[1] 袁晓[1] 

机构地区：[1]四川大学电子信息学院,成都610065

出　　处：《物理学报》2016年第16期25-34,共10页Acta Physica Sinica

基　　金：成都市科技计划(批准号:12DXYB255JH-002)资助的课题~~

摘　　要：将针对1/n阶微积分算子有理逼近的经典Carlson正则牛顿迭代法拓展到任意阶分数微积分算子的有理逼近.为了构造一个有理函数序列收敛于无理的分数微积分算子函数,将分数微积分算子有理逼近问题转换为二项方程的算术根代数迭代求解.并引入预矫正函数,使用牛顿迭代公式求解算术根,获得任意阶分数微积分算子的有理逼近阻抗函数.对n从2到5变化的九种不同运算阶,针对特定的运算阶,选择八种不同的初始阻抗,通过研究阻抗函数的零极点分布和频域特征,分析阻抗函数是否同时满足计算有理性、正实性原理和运算有效性.证明对任意的运算阶,在选择合适的初始阻抗的情况下,阻抗函数具有物理可实现性,在一定频率范围内具有分数微积分算子的运算特性.Carlson正则牛顿迭代法的推广为进一步的理论研究和构造任意分数阶电路与系统提供一种新思路.With the development of factional calculus theory and applications in different fields in recent years, the rational approximation problem of fractional calculus operator has become a hot spot of research. In the early 1950s and 1960s, Carlson and Halijak proposed regular Newton iterating method to implement rational approximation of the one-nth calculus operator. Carlson regular Newton iterating method has a great sense of innovation for the rational approximation of fractional calculus operator, however, it has been used only for certain calculus operators. The aim of this paper is to achieve rational approximation of arbitrary order fractional calculus operator. The realization is achieved via the generalization of Carlson regular Newton iterating method. To construct a rational function sequence which is convergent to irrational fractional calculus operator function, the rational approximation problem of fractional calculus operator is transformed into the algebra iterating solution of arithmetic root of binomial equation. To speed up the convergence, the pre-distortion function is introduced. And the Newton iterating formula is used to solve arithmetic root. Then the approximated rational impedance function of arbitrary order fractional calculus operator is obtained. For nine different operational orders with n changing from 2 to 5, the impedance functions are calculated respectively through choosing eight different initial impedances for a certain operational order. Considering fractional order operation characteristics of the impedance function and the physical realization of network synthesis, the impedance function should satisfy these basic properties simultaneously： computational rationality, positive reality principle and operational validity. In other words, there exists only rational computation of operational variable s in the expression of impedance function. All the zeros and poles of impedance function are located on the negative real axis of s complex plane or the left-half plane of s complex

关 键 词：分数微积分 分数算子 分抗逼近电路 Carlson有理逼近 

分 类 号：O174.41[理学—数学]