Modeling of Duhem hysteresis with Riemann-Liouville fractional derivative

Beyza Billur İskender Eroğlu


Due to the memory effect of hysteresis, this work considers the fractional order
modeling of hysteresis with singular fractional derivative. For this purpose, the paper focuses on Duhem hysteresis since it is a model defined by a differential equation despite the most of hysteresis models. The model is adapted to a fractional order differential equation in the sense of Riemann-Liouville derivative
and solved by using Grünwald Letnikov approximation. Then, fractional order Duhem model is demonstrated for different order of the derivative by figures drawn using MATLAB. It is observed from the figures that the fractional model exhibits hysteresis behavior for fractional orders smaller than 1. 

Tam Metin:



Krasnosel'skii, M.A., and Pokrovskii, A.V., Systems with Hysteresis, Springer Verlag, (1989).

Mayergoyz, I.D., Mathematical Models of Hysteresis, Springer Verlag, Berlin, (1991).

Macki, J.W., Nistri, P., and Zecca, P., Mathematical Models for Hysteresis, Siam Review, 35, 1, 9-123, (1993).

Visintin, A., Differential Models of Hysteresis, Springer, Berlin. (1994).

Duhem, P., Die dauernden aenderungen und die thermodynamik, 0Zeitschrift fur Physikalische Chemie, 22, 543-589, (1897).

Bagley, R.L., and Torvik, P.J., On the Fractional Calculus Model of Viscoelastic Behavior, Journal of Rheology, 30, 1, 133—155, (1986).

Padovan, J., and Sawicki, J. T., Diophantine Type Fractional Derivative Representation of Structural Hysteresis, Computational Mechanics, 19, 335-340, (1997).

Machado, J.A., Analysis and design of fractional order digital control systems, Systems Analysis Modelling Simulation, 27, 107-122, 1997.

Darwish, M.A., and El-Bary, A.A., Existence of Fractional Integral Equation with Hysteresis, Applied Mathematics and Computation, 176, 684-687, (2006).

Schafer, I., and Kruger, K., Modeling of coils using fractional derivatives, Journal of Magnetism and Magnetic Materials, 307, 91-98, (2006).

Duarte, F. and Machado, J.A., Fractional dynamics in the describing function analysis of nonlinear friction, Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, 39, 11, 218-223, (2006).

Deng, W. and Lü, J., Generating Multi-Directional Multi-Scroll Chaotic Attractors via a Fractional Differential Hysteresis system, Physics Letters A, 369, 438-443, (2007).

Duarte, F. and Machado, J.A., Describing function of two masses with backlash, Nonlinear Dynamics, 56, 409-413, (2009).

Özdemir N. and İskender, B.B., Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis, Journal of Computational and Nonlinear Dynamics, ASME, 5, 2, 021002 (6 pages), (2010).

İskender, B. B., Özdemir, N. and N., Karaoglan, A.D., Parameter Optimization of Fractional Order PID Controller Using Response Surface Methodology, Discontinuity and Complexity in Nonlinear Physical Systems, Series: Nonlinear Systems and Complexity, Vol. 6, Machado, J. A. T, Baleanu, D., Luo, A.C.J. (Eds.), Chapter 5, ISBN 978-3-319-01411-1, (2014).

Spanos, P.D., Di Matteo, A. and Pirotta, A., Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements, Nonlinear Dynamics98,4,3113-3124, (2019).

Guyomar, D., Ducharne, B. and Sebald, G., Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives, Journal of Physics D: Applied Physics, 40, 6048-6054, (2007).

Zhu, Z. and Zhou, X., A novel fractional order model for dynamic hysteresis of piezoeletrically actuated fast tool servo, Materials, 5, 2465-2485. (2012).

Zhu, Z., To, S., Li, Y., Zhu, W-L. and Bian, L., External force estimation of a piezo-actuated compliant mechanism based on a fractional order hysteresis model, Mechanical Systems and Signal Processing, 110, 296-306, (2018).

Ding, C., Cao, J. and Chen, Y.Q., Fractional-order model and experimental verification for broadband hysteresis in piezoelectric actuators, Nonlinear Dynamics, 98,3143-3153, (2019).

Caputo, M., and Carcione, J.M., Hysteresis cycles and fatigue criteria using anelastic models based on fractional derivatives, Rheologica Acta, 50, 107-115, (2011).

Caputo, M. and Fabrizio, M., On the notion of fractional derivatvie and applications to the hysteresis phenomena, Meccanica, 52, 3043-3052, (2017).

Naser, M.F.M. and Ikhouane, F., Consistency of the Duhem model with hysteresis, Mathematical Problems in Engineering, 2013, 586130, (16 pages), (2013).

Colemann, B.D., and Hodgdon, M.L., A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials, International Journal of Engineering Science, 24, 897-919, (1986).

Colemann, B.D. and Hodgdon, M.L., On a class of constitutive relations for ferromagnetic hysteresis, Archive for Rational Mechanics and Analysis, 99, 375-396, (1987).

Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, (1974).

Miller, K.S. and 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).

Hodgdon, M.L., Applications of a theory of ferromagnetic hysteresis, IEEE Transactions on Magnetics, 24, 218-221, (1988).

Su, C.Y., Stepanenko, Y., Svoboda, J. and Leung, T.P., Robust and adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis, IEEE Transactions on Automatic Control, 45, 2427-2432, (2000).


  • Şu halde refbacks yoktur.

Telif Hakkı (c) 2020 Beyza Billur İskender Eroğlu

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.