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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 plane in conjugate pairs. The frequency-domain characteristics of impedance function approximate to those of ideal fractional calculus operator over a certain frequency range. Given suitable initial impedance and for an arbitrary operational order, it is proved that the impedance function could meet all properties above by studying the zero-pole distribution and analyzing frequency-domain characteristics of the impedance function. Therefore, the impedance function could take on operational performance of the ideal fractional calculus operator and achieve the physical realization. It is of great effectiveness in the generalization of this kind of method in both theory and experiment. The results educed in this paper are the basis for further theoretic research and engineering application in constructing the arbitrary order fractional circuits and systems.
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Keywords:
- fractional calculus /
- fractional operator /
- fractance approximation circuit /
- Carlson rational approximation
[1] Steiglitz K 1964 IEEE Trans. Circuit Theory 11 160
[2] Halijak C A 1964 IEEE Trans. Circuit Theory 11 494
[3] Ren Y, Yuan X 2008 J. Sichuan Univ. (Nat. Sci. Ed.) 45 1100 (in Chinese) [任毅, 袁晓 2008 四川大学学报(自然科学版) 45 1100]
[4] Dutta R S C 1967 IEEE Trans. Circuit Theory 14 264
[5] Krishna B T, Reddy K V V S 2008 Act. Passive Electron.Compon. 2008 369421
[6] Krishna B T 2011 Signal Process. 91 386
[7] Liu Y, Pu Y F, Shen X D, Zhou J L 2012 J. Sichuan Univ. (Eng. Sci. Ed.) 44 153 (in Chinese) [刘 彦, 蒲亦非, 沈晓东, 周激流 2012 四川大学学报(工程科学版) 44 153]
[8] Sun H H, Abdelwahab A A, Onaral B 1984 IEEE Trans. Autom Control 29 441
[9] Zou D, Yuan X 2013 J. Sichuan Univ. (Nat. Sci. Ed.) 50 293 (in Chinese) [邹道, 袁晓 2013 四川大学学报(自然科学版) 50 293]
[10] Carlson G E 1960 M. S. Thesis (Manhattan: Kansas State University)
[11] Carlson G E, Halijak C A 1962 IRE Trans. Circuit Theory 9 302
[12] Carlson G E, Halijak C A 1964 IEEE Trans. Circuit Theory 11 210
[13] Zu Y X, Lu Y Q 2007 Network Analysis and Synthesis (Beijing: China Machine Press) pp111-120 (in Chinese) [俎云霄, 吕玉琴 2007 网络分析与综合(北京:机械工业出版社) 第111-120页]
[14] Liao K, Yuan X, Pu Y F, Zhou J L 2006 J. Sichuan Univ. (Eng. Sci. Ed.) 43 104 (in Chinese) [廖科, 袁晓, 蒲亦非, 周激流 2006 四川大学学报(自然科学版) 43 104]
[15] Pu Y F, Yuan X, Liao K, Zhou J L, Zhang N, Zeng Y 2005 Proceedings of IEEE International Conference on Mechatronics and Automation Niagara Falls, Canada, July 29-August 1, 2005 p1375
[16] Liao K, Yuan X, Pu Y F, Zhou J L 2005 J. Sichuan Univ. (Eng. Sci. Ed.) 37 150 (in Chinese) [廖科, 袁晓, 蒲亦非, 周激流 2005四川大学学报(工程科学版) 37 150]
[17] Tsirimokou G Psychalinos C, Elwakil A S 2015 Analog. Integr. Circ. Sig. Process. 85 413
[18] Pu Y F, Yuan X, Liao K, Zhou J L 2006 J. Sichuan Univ. (Eng. Sci. Ed.) 38 128 (in Chinese) [蒲亦非, 袁晓, 廖科, 周激流 2006 四川大学学报(工程科学版) 38 128]
[19] Ortigueira M D Batista A G 2008 Phys. Lett. A 372 958
[20] Ortigueira M D 2008 IEEE Circuits Syst. Mag. 38 19
[21] Magin R, Ortigueira M D, Podlubny I, Trujillo J 2011 Signal Process. 91 350
[22] Sheng H, Chen Y Q, Qiu T S 2012 Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications (Springer London, Dordrecht, Heidelberg, New York: Signals and Communication Technology) pp31-39
[23] Elwakil A S 2010 IEEE Circuits Syst. Mag. 4 40
[24] Podlubny I 1999 Fractional Differential Equations (San Diego(USA): Academic Press) pp252-259
[25] Machado J A T, Silva M F, Barbosa R S, Jesus I S, Reis C M, Marcos M G, Galhano A F 2010 Math. Probl. Eng. 2010 639801
[26] Hu K X, Zhu K Q 2009 Chin. Phys. Lett. 26 108301
[27] Ni J K, Liu C X, Liu K, Liu L 2014 Chin. Phys. B 23 100504
[28] Pan G, Wei J 2015 Acta Phys. Sin. 64 040505 (in Chinese) [潘光, 魏静 2015 64 040505]
[29] Huang Y, Liu Y F Peng Z M, Ding Y J 2015 Acta Phys. Sin. 64 030505 (in Chinese) [黄宇, 刘玉峰, 彭志敏, 丁艳军 2015 64 030505]
[30] Yuan X 2015 Mathematical Principles of Fractance Approximation Circuits (Beijing: Science Press) pp218-236 (in Chinese) [袁晓 2015 分抗逼近电路之数学原理(北京:科学出版社) 第218-236页]
[31] Valkenburg V M E (translated by Yang X J, Zheng J L, Yang W L) 1982 Network Synthesis (Beijing: Science Press) pp222-225 (in Chinese) [〔美〕Valkenburg V M E 著 (杨行峻, 郑君里, 杨为理 译) 1982 网络分析(北京: 科学出版社)第222 -225页]
[32] Yi Z, Yuan X, Tao L, Liu P P 2015 J. Sichuan Univ. (Nat. Sci. Ed.) 6 1255 (in Chinese) [易舟, 袁晓, 陶磊, 刘盼盼 2015 四川大学学报 (自然科学版) 6 1255]
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[1] Steiglitz K 1964 IEEE Trans. Circuit Theory 11 160
[2] Halijak C A 1964 IEEE Trans. Circuit Theory 11 494
[3] Ren Y, Yuan X 2008 J. Sichuan Univ. (Nat. Sci. Ed.) 45 1100 (in Chinese) [任毅, 袁晓 2008 四川大学学报(自然科学版) 45 1100]
[4] Dutta R S C 1967 IEEE Trans. Circuit Theory 14 264
[5] Krishna B T, Reddy K V V S 2008 Act. Passive Electron.Compon. 2008 369421
[6] Krishna B T 2011 Signal Process. 91 386
[7] Liu Y, Pu Y F, Shen X D, Zhou J L 2012 J. Sichuan Univ. (Eng. Sci. Ed.) 44 153 (in Chinese) [刘 彦, 蒲亦非, 沈晓东, 周激流 2012 四川大学学报(工程科学版) 44 153]
[8] Sun H H, Abdelwahab A A, Onaral B 1984 IEEE Trans. Autom Control 29 441
[9] Zou D, Yuan X 2013 J. Sichuan Univ. (Nat. Sci. Ed.) 50 293 (in Chinese) [邹道, 袁晓 2013 四川大学学报(自然科学版) 50 293]
[10] Carlson G E 1960 M. S. Thesis (Manhattan: Kansas State University)
[11] Carlson G E, Halijak C A 1962 IRE Trans. Circuit Theory 9 302
[12] Carlson G E, Halijak C A 1964 IEEE Trans. Circuit Theory 11 210
[13] Zu Y X, Lu Y Q 2007 Network Analysis and Synthesis (Beijing: China Machine Press) pp111-120 (in Chinese) [俎云霄, 吕玉琴 2007 网络分析与综合(北京:机械工业出版社) 第111-120页]
[14] Liao K, Yuan X, Pu Y F, Zhou J L 2006 J. Sichuan Univ. (Eng. Sci. Ed.) 43 104 (in Chinese) [廖科, 袁晓, 蒲亦非, 周激流 2006 四川大学学报(自然科学版) 43 104]
[15] Pu Y F, Yuan X, Liao K, Zhou J L, Zhang N, Zeng Y 2005 Proceedings of IEEE International Conference on Mechatronics and Automation Niagara Falls, Canada, July 29-August 1, 2005 p1375
[16] Liao K, Yuan X, Pu Y F, Zhou J L 2005 J. Sichuan Univ. (Eng. Sci. Ed.) 37 150 (in Chinese) [廖科, 袁晓, 蒲亦非, 周激流 2005四川大学学报(工程科学版) 37 150]
[17] Tsirimokou G Psychalinos C, Elwakil A S 2015 Analog. Integr. Circ. Sig. Process. 85 413
[18] Pu Y F, Yuan X, Liao K, Zhou J L 2006 J. Sichuan Univ. (Eng. Sci. Ed.) 38 128 (in Chinese) [蒲亦非, 袁晓, 廖科, 周激流 2006 四川大学学报(工程科学版) 38 128]
[19] Ortigueira M D Batista A G 2008 Phys. Lett. A 372 958
[20] Ortigueira M D 2008 IEEE Circuits Syst. Mag. 38 19
[21] Magin R, Ortigueira M D, Podlubny I, Trujillo J 2011 Signal Process. 91 350
[22] Sheng H, Chen Y Q, Qiu T S 2012 Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications (Springer London, Dordrecht, Heidelberg, New York: Signals and Communication Technology) pp31-39
[23] Elwakil A S 2010 IEEE Circuits Syst. Mag. 4 40
[24] Podlubny I 1999 Fractional Differential Equations (San Diego(USA): Academic Press) pp252-259
[25] Machado J A T, Silva M F, Barbosa R S, Jesus I S, Reis C M, Marcos M G, Galhano A F 2010 Math. Probl. Eng. 2010 639801
[26] Hu K X, Zhu K Q 2009 Chin. Phys. Lett. 26 108301
[27] Ni J K, Liu C X, Liu K, Liu L 2014 Chin. Phys. B 23 100504
[28] Pan G, Wei J 2015 Acta Phys. Sin. 64 040505 (in Chinese) [潘光, 魏静 2015 64 040505]
[29] Huang Y, Liu Y F Peng Z M, Ding Y J 2015 Acta Phys. Sin. 64 030505 (in Chinese) [黄宇, 刘玉峰, 彭志敏, 丁艳军 2015 64 030505]
[30] Yuan X 2015 Mathematical Principles of Fractance Approximation Circuits (Beijing: Science Press) pp218-236 (in Chinese) [袁晓 2015 分抗逼近电路之数学原理(北京:科学出版社) 第218-236页]
[31] Valkenburg V M E (translated by Yang X J, Zheng J L, Yang W L) 1982 Network Synthesis (Beijing: Science Press) pp222-225 (in Chinese) [〔美〕Valkenburg V M E 著 (杨行峻, 郑君里, 杨为理 译) 1982 网络分析(北京: 科学出版社)第222 -225页]
[32] Yi Z, Yuan X, Tao L, Liu P P 2015 J. Sichuan Univ. (Nat. Sci. Ed.) 6 1255 (in Chinese) [易舟, 袁晓, 陶磊, 刘盼盼 2015 四川大学学报 (自然科学版) 6 1255]
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