Application of Newton's Method to Analog and Digital Realization of Fractional-order Controllers

• Autorzy: Tepljakov A. , Petlenkov E. , Belikov J.
• Języki publikacji: EN

Abstrakty

EN

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 a modification of 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 analog and digital implementations of fractional-order systems and controllers with relevant examples and comments.

Słowa kluczowe

EN

fractional calculus; Newton's method; Carlson's method; MATLAB; implicit fractional transfer function; fractional power zero-pole

PL

pochodna ułamkowa; metoda Newtona; metoda Carlsona; MATLAB; transmitancja operatorowa

• Wydawca: Lodz University of Technology. Department of Microelectronics and Computer Science
• Czasopismo: International Journal of Microelectronics and Computer Science
• Rocznik: 2012
• Tom: Vol. 3, nr 2
• Strony: 45--52
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Tepljakov A.

autor

Petlenkov E.

autor

Belikov J.

• Department of Computer Control, Tallinn University of Technology Ehitajate tee 5, 19086, Tallinn, Estonia,

Bibliografia

• [1] I. Podlubny, Fractional differential equations , ser. Mathematics in science and engineering. Academic Press, 1999.
• [2] I. Podlubny, L. Dor cák, and I. Kostial, “On fractional derivatives, fractional-order dynamic systems and PI ? D ?-controllers,” in Proc. 36th IEEE Conf. Decision and Control , vol. 5, 1997, pp. 4985–4990.
• [3] I. Podlubny, “Fractional-order systems and PI ? D ?-controllers,” IEEE Trans. Autom. Control , vol. 44, no. 1, pp. 208–214, 1999.
• [4] A. Oustaloup, P. Melchior, P. Lanusse, O. Cois, and F. Dancla, “The CRONE toolbox for Matlab,” in Proc. IEEE Int. Symp. Computer-Aided Control System Design CACSD 2000 , 2000, pp. 190–195.
• [5] B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis , vol. 3, pp. 945–950, 2000.
• [6] A. Tepljakov, E. Petlenkov, and J. Belikov, “Application of the Newton method to first-order implicit fractional transfer function approximation,” in Proc. 19th Int Mixed Design of Integrated Circuits and Systems (MIXDES) Conference , A. Napieralski, Ed., 2012, pp. 473–477.
• [7] F. Hildebrand, Introduction to numerical analysis , ser. International series in pure and applied mathematics. McGraw-Hill, 1956.
• [8] H. S. Wall, “A modification of Newton’s method,” The American Mathematical Monthly , vol. 55, no. 2, pp. 90–94, 1948.
• [9] J. F. Traub, “Comparison of iterative methods for the calculation of n th roots,” Commun. ACM , vol. 4, pp. 143–145, March 1961.
• [10] G. Carlson and C. Halijak, “Approximation of fractional capacitors (1 /s ) 1 /n by a regular Newton process,” IEEE Trans. Circuit Theory , vol. 11, no. 2, pp. 210–213, 1964.
• [11] I. Podlubny, I. Petráš, B. M. Vinagre, P. O’Leary, and L. Dorčák, “Analogue realizations of fractional-order controllers,” Nonlinear Dynamics , vol. 29, pp. 281–296, 2002.
• [12] D. Valério. (2005) Toolbox ninteger for MatLab, v. 2.3. [Online]. Available: http://web.ist.utl.pt/duarte.valerio/ninteger/ninteger.htm
• [13] H. Peitgen, H. Jürgens, and D. Saupe, Chaos and fractals: new frontiers of science . Springer-Verlag, 1992.
• [14] A. Charef, H. H. Sun, Y. Y. Tsao, and B. Onaral, “Fractal system as represented by singularity function,” IEEE Trans. Autom. Control , vol. 37, no. 9, pp. 1465–1470, 1992.
• [15] A. Oustaloup, J. Sabatier, and P. Lanusse, “From fractal robustness to the CRONE control,” Fractionnal Calculus and Applied Analysis , vol. 2, no. 1, pp. 1–30, 1999.
• [16] H. Bensoudane, C. Gentil, and M. Neveu, “The local fractional derivative of fractal curves,” in Proceedings of the 2008 IEEE International Conference on Signal Image Technology and Internet Based Systems , ser. SITIS’08. Washington, DC, USA: IEEE Computer Society, 2008, pp. 422–429.
• [17] A. Householder, The numerical treatment of a single nonlinear equation , ser. International series in pure and applied mathematics. McGraw-Hill, 1970.
• [18] C. A. Monje, B. M. Vinagre, A. J. Calderon, V. Feliu, and Y. Q. Chen, “Auto-tuning of fractional lead-lag compensators,” in Proceedings of the 16th IFAC World Congress , 2005.
• [19] C. Monje, B. Vinagre, V. Feliu, and Y. Chen, “Tuning and auto-tuning of fractional order controllers for industry applications,” Control Engineering Practice , vol. 16, no. 7, pp. 798–812, 2008.
• [20] C. A. Monje, Y. Chen, B. Vinagre, D. Xue, and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications , ser. Advances in Industrial Control. Springer Verlag, 2010.
• [21] A. Varga, “Balancing free square-root algorithm for computing singular perturbation approximations,” in Proc. 30th IEEE Conf. Decision and Control , 1991, pp. 1062–1065.
• [22] A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequency-band complex noninteger differentiator: characterization and synthesis,” IEEE Trans. Circuits Syst. I , vol. 47, no. 1, pp. 25–39, 2000.
• [23] A. Tepljakov, E. Petlenkov, and J. Belikov, “FOMCON: Fractional-order modeling and control toolbox for MATLAB,” in Proc. 18th Int Mixed Design of Integrated Circuits and Systems (MIXDES) Conference , A. Napieralski, Ed., 2011, pp. 684–689.
• [24] I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis , vol. 5, no. 4, pp. 367–386, 2002.
• [25] S. Roy, “On the realization of a constant-argument immittance or fractional operator,” Circuit Theory, IEEE Transactions on , vol. 14, no. 3, pp. 264–274, september 1967.
• [26] B. Maundy, A. Elwakil, and S. Gift, “On a multivibrator that employs a fractional capacitor,” Analog Integr. Circuits Signal Process. , vol. 62, no. 1, pp. 99–103, 2010.
• [27] Y. Q. Chen, I. Petráš, and D. Xue, “Fractional order control - a tutorial,” in Proc. ACC ’09. American Control Conference , 2009, pp. 1397–1411.
• [28] A. Tepljakov, E. Petlenkov, and J. Belikov, “Implementation and real-time simulation of a fractional-order controller using a MATLAB based prototyping platform,” in Proc. 13th Biennial Baltic Electronics Conference , 2012, pp. 145–148.