分数阶运算符的Newton方法优化与应用

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"分数阶运算符的近似方法与应用" 分数阶运算符是数学和工程领域中的一个重要概念，尤其在控制系统、信号处理和物理建模中具有广泛的应用。分数阶运算扩展了传统的整数阶微积分，允许我们处理非局部性和历史依赖性问题。与整数阶微积分不同，分数阶运算符可以捕捉系统的复杂动态行为，如记忆和非线性特性。本文主要关注第一阶隐式分数阶传递函数的近似。这种类型的传递函数在频率限定的分数阶微分器或积分器中起到关键作用。分数阶微分器可以实现非线性滤波和系统辨识，而分数阶积分器则有助于消除高频噪声并捕获长期依赖性。作者Aleksei Tepljakov, Eduard Petlenkov和Juri Belikov提出了一个基于牛顿法的算法来迭代逼近第一阶隐式分数阶传递函数的根。牛顿法是一种常用的数值分析方法，用于寻找函数的零点，即找到使得函数值为零的点。在本研究中，该方法被用来精确估计分数阶运算的参数。分数阶运算的近似对于实际应用至关重要，因为精确计算分数阶导数或积分通常涉及复杂的数学运算。通过改进现有技术，作者的算法提升了性能和准确性之间的平衡，使得模型的构建和控制设计更加有效。特别是，他们强调了这种方法在分数阶领先-滞后补偿器设计中的应用。领先-滞后补偿器是控制系统设计中的一个重要工具，它可以改善系统的稳定性和响应速度。在指数术语方面，文章涵盖了分数微积分、牛顿法、Carlson's方法（一种解决特殊类型的代数方程的方法）、MATLAB（一种广泛用于数值计算的软件环境）、隐式分数阶传递函数以及分数幂零极点。MATLAB的使用表明，这些算法和方法可以通过编程实现，为实际工程问题提供了解决方案。这篇论文为理解和应用分数阶运算提供了新的视角，特别是通过优化的近似方法提高了计算效率和精度。这不仅有助于理论研究，也对实际控制系统的设计和分析产生了积极影响。通过利用分数阶运算符的特性，工程师可以设计出更灵活、更具适应性的控制系统，以应对复杂动态环境中的挑战。

Application of the Newton Method to First-order

Implicit Fractional Transfer Function Approximation

Aleksei Tepljakov, Eduard Petlenkov, and Juri Belikov

Department of Computer Control

Tallinn University of Technology

Ehitajate tee 5, 19086, Tallinn, Estonia

{aleksei.tepljakov, eduard.petlenkov, juri.belikov}@dcc.ttu.ee

Abstract—In this paper, a method for approximating a ﬁrst-

order implicit fractional transfer function, that corresponds to

a frequency-bounded fractional differentiator or integrator, is

presented. The proposed method is based on the well-known

Newton’s method for iterative root approximation. First-order

implicit fractional transfer functions have several applications

in modeling and control. This type of transfer function is the

basis for the fractional lead-lag compensator. In the following,

we provide the description of our algorithm, that enhances the

existing technique, and illustrate its use in modeling and control

with relevant comments.

Index Terms—fractional calculus, Newton’s method, Carlson’s

method, Matlab, implicit fractional transfer function, fractional

power zero-pole

I. INTRODUCTION

Today, fractional-order calculus is a rapidly evolving scien-

tiﬁc ﬁeld. It allows for more accurate modeling of complex

systems, such as those that possess memory and hereditary

properties [1]. The beneﬁts of using fractional calculus in

control are also evident. New types of controllers have been

developed [2], [3], [4] based on the added ﬂexibility of the

fractional-order models.

However, many problems arise in the implementation of

fractional-order controllers. Since fractional models are inher-

ently complex, which follows from the fact that they describe

inﬁnite-dimensional systems [5], deducing an effective direct

realization method is a difﬁcult task. Therefore, methods for

approximating the fractional operators have been developed,

including both continuous and discrete approximations.

In this paper, we focus on one particular continuous

integer-order approximation, derived from Newton’s method.

In Section II the underlying method and its modiﬁcation are

summarized and applications to fractional-order modeling are

presented. In Section III our method used for implicit ﬁrst-

order fractional transfer function approximation is proposed

and discussed. Applications of this method to controller im-

plementation are also presented and a MATLAB realization

is described. Illustrative examples follow in Section IV. Some

issues and limitations of the proposed method are discussed

in Section V. Finally, conclusions are drawn in Section VI.

II. A

PPLICATION OF NEWTON’S METHOD TO FRACTIONAL

CALCULUS

Newton’s method, also known as Newton-Rhapson method

[6], is a numerical algorithm for ﬁnding a real root of a

function f(x). It suggests that in order to solve a general

nonlinear equation f(x)=0, the following iterative formula

can be used, given an initial estimate x

k+1

= x

−

f(x

)



)

. (1)

A modiﬁed algorithm is proposed in [7], [8] such that the

convergence of the sequence {x

} is more rapid than that

resulting from using formula (1). The corresponding formula

is called Halley’s formula:

k+1

= x

−

f(x

)



) −

f(x



)



)

. (2)

Consider now a problem of ﬁnding an nth root of a real

number. The corresponding function is f(x)=x

− A and

using (2) the following particular iteration formula is obtained:

k+1

= x

(n − 1)(x

)

+(n +1)A

(n +1)(x

)

+(n − 1)A

. (3)

This formula is considered by Carlson [9] and more recently

in [5], [10], [11]. In his paper, Carlson has shown, that this

formula holds for both even n =2m and odd n =2m +1

roots. The method can be applied to approximation of frac-

tional capacitors of the form (1/s)

1/n

in the following way:

k+1

(s)=G

(s)

(n − 1) (G

(s)) + (n +1)(H(s))

(n +1)(G

(s)) + (n − 1) (H(s))

, (4)

H(s)=

(s)=1.

Since in this case the real variable A is replaced by the

transfer function H(s), convergence and rate of convergence

cannot be evaluated in the same way as in the case of a real-

valued function.

Consider now an example. Using equation (4) we shall

obtain an approximation of a fractional capacitor



1/s. With

MIXED

DESIGN

MIXDES 2012, 19

International Conference

"Mixed Design of Integrated Circuits and Systems"

, May 24-26, 2012, Warsaw, Poland

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