Abstract
An approach to the design of analogue circuits, implementingfractional-order controllers, is presented. The suggestedapproach is based on the use of continued fraction expansions;in the case of negative coefficients in a continued fractionexpansion, the use of negative impedance converters is proposed.Several possible methods for obtaining suitable rational appromixationsand continued fraction expansions are discussed. An exampleof realization of a fractional-order Iλ controlleris presented and illustrated by obtained measurements.The suggested approach can be used for the control of veryfast processes, where the use of digital controllers isdifficult or impossible.
Similar content being viewed by others
References
Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.
Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
Samko, S. G., Kilbas, A. A., and Maritchev, O. I., Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk, 1987 [in Russian].
Podlubny, I., 'Fractional-order systems and PIλDμ-controllers', IEEE Transactions on Automatic Control 44, 1999, 208–214.
Podlubny, I., 'Fractional-order systems and fractional-order controllers', UEF-03–94, Slovak Academy of Sciences, Kosice, 1994.
Dorf, R. C. and Bishop, R. H., Modern Control Systems, Addison-Wesley, New York, 1990.
Oustaloup, A., Systèmes Asservis Linéaires d'Ordre Fractionnaire: Théorie et Pratique, Editions Masson, Paris, 1983.
Oustaloup, A., La Dérivation non Entière, Hermès, Paris, 1995.
Oustaloup, A., Levron, F., Mathieu, B., and Nanot, F. M., 'Frequency-band complex noninteger differentiator: Characterization and synthesis', IEEE Transactions on Circuit and Systems-I: Fundamental Theory and Application 47, 2000, 25–39.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical Recipes in C. The Art of Scientific Computing, 2nd edition, Cambridge University Press, Cambridge, 1992.
Dutta Roy, S. C., 'On the realization of a constant-argument immitance of fractional operator', IEEE Transactions on Circuit Theory 14, 1967, 264–374.
Vinagre, B. M., Podlubny, I., Hernandez, A., and Feliu, V., 'On realization of fractional-order controllers', in Proceedings of the Conference Internationale Francophone d'Automatique, Lille, Jule 5–8, P. Borne, J.-P. Richard, and Ph. Vanheeghe (eds.), 2000, pp. 945–950.
Carlson, G. E. and Halijak, C. A., 'Approximation of fractional capacitors (1/s)1/n by a regular Newton process', IEEE Transactions on Circuit Theory 11, 1964, 210–213.
Matsuda, K. and Fujii, H., 'H ∞-optimized wave-absorbing control: analytical and experimental results', Journal of Guidance, Control, and Dynamics 16, 1993, 1146–1153.
Heymans, N. and Bauwens, J.-C., 'Fractal rheological models and fractional differential equations for viscoelastic behavior', Rheologica Acta 33, 1994, 210–219.
Kvasil, J. and Čajka, J., An Introduction to Synthesis of Linear Circuits, SNTL/ALFA, Prague, 1981 [in Czech].
Khovanskii, A. N., The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, Noordhoff, Groningen, 1963.
Bode, H. W., Network Analysis and Feedback Amplifier Design, Tung Hwa Book Company, Shanghai, China, 1949.
Dostal, J., Operational Amplifiers, Butterworth-Heinemann, Boston, MA, 1993.
Nakagava, M. and Sorimachi, K., 'Basic characteristics of a fractance device', IEICE Transactions Fundamentals E75–A, 1992, 1814–1818.
Oldham, K. B. and Zoski, C. G., 'Analogue instrumentation for processing polarographic data', Journal of Electroanalytical Chemistry 157, 1983, 27–51.
Petráš, I. and Dorčák, L'., 'Some possibilities for realization of fractional order controller', Envirautom 4, 1999, 83–90 [in Slovak].
Wang, J. C., 'Realizations of generalized Warburg impedance with RC ladder networks and transmission lines', Journal of Electrochemical Society 134, 1987, 1915–1920.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Podlubny, I., Petráš, I., Vinagre, B.M. et al. Analogue Realizations of Fractional-Order Controllers. Nonlinear Dynamics 29, 281–296 (2002). https://doi.org/10.1023/A:1016556604320
Issue Date:
DOI: https://doi.org/10.1023/A:1016556604320