Abstract
This paper deals with the rational function approximation of the irrational transfer function \(G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]}\) of the fundamental linear fractional order differential equation \((\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t)\), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.
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References
Oustaloup, A.: Systèmes Asservis Linéaires d’Ordre Fractionnaire: Théorie et Pratique. Masson, Paris (1983)
Petras, I., Podlubny, I., O’Leary, P., Dorcak, L., Vinagre, B.M.: Analogue realization of fractional order controllers. Fakulta Berg, TU Kosice (2002)
Hilfer, R. (ed.): Applications of Calculus in Physics. World Scientific, Singapore (2000)
Sabatier, J., et al. (eds.): Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering. Springer, Dordrecht (2007)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlinear Dyn. 38, 117–131 (2004)
Charef, A.: Modeling and analog realization of the fundamental linear fractional order differential equation. Nonlinear Dyn. 46, 195–210 (2006)
Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16, 231–253 (1997)
Hartley, T.T., Lorenzo, C.F.: A solution of the fundamental linear fractional order differential equation. NASA TP-1998-208693, December 1998
Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)
Momani, S., Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162, 1351–1365 (2005)
Kumar, P., Agrawal, O.P.: An approximate method for numerical solution of fractional differential equations. Signal Process. 86, 2602–2610 (2006)
Momani, S., Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31, 1248–1255 (2007)
Arikoglu, A., Ozkol, I.: Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals 34, 1473–1481 (2007)
Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)
Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics, alternation current characterization. J. Chem. Phys. 9 (1941)
MacDonald, J.R.: Impedance Spectroscopy. Wiley, New York (1987)
Matignon, D.: Stability results for fractional differential equation with application to control processing. In: Proceedings of the Symposium on Control, Optimization and Supervision, CESA’96, Lille, France, pp. 963–968 (1996)
Malti, R., Moreau, X., Khemane, F.: Resonance of fractional transfer functions of the second kind. In: Proceedings of IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 5–7 November 2008 (2008)
Kuo, B.C.: Automatic Control Systems. Prentice-Hall, Englewood Cliffs (1981)
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Charef, A., Nezzari, H. On the fundamental linear fractional order differential equation. Nonlinear Dyn 65, 335–348 (2011). https://doi.org/10.1007/s11071-010-9895-z
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DOI: https://doi.org/10.1007/s11071-010-9895-z