Stochastic model of the hysteresis converter with a domain structure

S. V. Borzunov ; M. E. Semenov ; N. I. Sel'vesyuk ; P. A. Meleshenko ; A. M. Solovyov

Matematičeskoe modelirovanie, Tome 33 (2021) no. 9, pp. 60-86.

Résumé

The work is devoted to generalization of one of the most "popular" constructive models of hysteresis – the Preisach converter (a continuous system of non-ideal relays connected in parallel). In many applied problems related to modeling hysteresis effects, where the domain structure of carriers of hysteresis properties is a priori assumed (multiple magnetization reversal of ferromagnetic components of electromagnetic systems, polarization of a ferroelectric depending on the electric field strength, dependence of employment on price in mono-commodity markets, etc.), one has to face with the need to take into account the uncertainties in the reactions of individual domains to external influences. In this paper, we propose an instrumental method that makes it possible to take into account such uncertainties by means of stochastic models of elementary carriers of hysteresis properties. The basic properties of discrete and continuous relay systems are considered, the parameters of which are treated as random variables, while the output of the relay system is represented as a random process. The correctness of the definition is investigated, in particular, the independence of the output (of a random process) from the sampling method within the limit transition from a discrete to a continual system of nonideal relays is established, and the controllability and monotonicity (within the framework of the corresponding definition) of the stochastic analogue of the Preisach converter are established.

Détail

Keywords: stochastic process, hysteresis, non-ideal relay, Preisach operator.

Bibliographie

[1] J. Dho, C. W. Leung, M. G. Blamire, “Universal Time Relaxation Behavior of the Exchange Bias in Ferromagnetic/antiferromagnetic Bilayers”, J. of Applied Physics, 99 (2006), 033910-1–033910-5 | DOI

[2] J. Ortiz-Lopez, F. Luty, “Optical Studies of Thermal Cycling and Hysteresis Effects in Elastic Order-disorder Phase Transformations. I. Pure Alkali-metal Cyanide Crystals”, Phys. Rev. B, 37:10 (1988), 5452–5460 | DOI

[3] K. Zhang, T. Zhao, H. Fujiwara, “Training Effect of Exchange Biased Iron-oxide/ferromagnet Systems”, Journal of Applied Physics, 89 (2001), 6910–6912 | DOI

[4] R. Cross, Unemployment: Natural Rate Epicycles or Hysteresis?, European Journal of Economics and Economic Policies: Intervention, 11:2 (2014), 136–148 | DOI

[5] Baker Hughes US Rig Count by Basin, https://rigcount.bakerhughes.com/na-rig-count/

[6] Crude Oil Future Contract (Dollars per Barrel), https://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=RCLC1&f=D

[7] B. Carboni, W. Lacarbonara, “Nonlinear dynamic characterization of a new hysteretic device: experiments and computations”, Nonlinear Dynamics, 83 (2016), 23–39 | DOI

[8] R. Cross, H. McNamara, A. Pokrovskii, D. Rachinskii, “A new paradigm for modelling hysteresis in macroeconomic flows”, Physica B: Condensed Matter, 403:2-3 (2008), 231–236 | DOI

[9] O. Klein, P. Krejčí, “Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations”, Nonlinear Analysis: Real World Applications, 4:5 (2003), 755–785 | DOI | MR | Zbl

[10] M. A. Janaideh, R. Naldi, L. Marconi, P. Krejčí, “A hybrid model for the play hysteresis operator”, Physica B, 430 (2013), 95–98 | DOI

[11] W. Lacarbonara, F. Vestroni, “Nonclassical Responses of Oscillators with Hysteresis”, Nonlinear Dynamics, 32 (2003), 235–258 | DOI | Zbl

[12] S. F. Masri, R. Ghanem, F. Arrate, J. P. Caffrey, “Stochastic Nonparametric Models of Uncertain Hysteretic Oscillators”, AIAA Journal, 44:10 (2006), 2319–2330 | DOI

[13] A. L. Medvedskii, P. A. Meleshenko, V. A. Nesterov, O. O. Reshetova, M. E. Semenov, A. M. Solovyov, “Unstable Oscillating Systems with Hysteresis: Problems of Stabilization and Control”, J. of Computer and Systems Sciences International, 59:4 (2020), 533–556 | DOI | Zbl

[14] M. E. Semenov, A. M. Solovyov, P. A. Meleshenko, “Stabilization of coupled inverted pendula: From discrete to continuous case”, J. of Vibration and Control, 27:1–2 (2021), 43–56 | DOI | MR

[15] M. E. Semenov, A. M. Solovyov, P. A. Meleshenko, O. O. Reshetova, “Efficiency of hysteretic damper in oscillating systems”, Math. Modell. of Natur. Phenomena, 15 (2020), 43-1–43-14 | DOI | MR

[16] M. A. Krasnosel'skii, A. V. Pokrovskii, Systems with hysteresis, Springer-Verlag, Berlin–Heidelberg, 1989, 410 pp. | MR | MR | Zbl

[17] F. Ikhouane, J. Rodellar, “On the Hysteretic Bouc-Wen Model”, Nonlinear Dynamics, 42 (2005), 63–78 | DOI | MR | Zbl

[18] W. D. Iwan, “A distributed-element model for hysteresis and its steady-state dynamic response”, Journal of Applied Mechanics, Transactions ASME, 33:4 (1966), 893–900 | DOI

[19] C.-J. Lin, P. T. Lin, “Tracking control of a biaxial piezo-actuated positioning stage using generalized Duhem model”, Comp. and Mathem. with Applications, 64 (2012), 766–787 | DOI

[20] M. F.M. Naser, F. Ikhouane, “Consistency of the Duhem Model with Hysteresis”, Math. Problems in Engineering, 2013, 586130, 16 pp. | DOI | MR | Zbl

[21] M. Belhaq, A. Bichri, J. Der Hogapian, J. Mahfoud, “Effect of electromagnetic actuations on the dynamics of a harmonically excited cantilever beam”, International Journal of Non-Linear Mechanics, 46:6 (2011), 828–833 | DOI

[22] M. Brokate, A. Pokrovskii, D. Rachinskii, O. Rasskazov, “Differential Equations with Hysteresis via a Canonical Example”, The Science of Hysteresis, 3-volume set, v. 1, eds. G. Bertotti, I.D. Mayergoyz, 2006, 125–291 | DOI | MR | Zbl

[23] H. J. Khasawneh, Z. S. Abo-Hammour, M. I. Al Saaideh, S. M. Momani, “Identification of hysteresis models using real-coded genetic algorithms”, Eur. Phys. J. Plus, 134 (2019), 507-1–507-17 | DOI

[24] C. Kuehn, C. Münch, “Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems”, SIAM J. Applied Dynamical Systems, 16:3 (2017), 1650–1685 | DOI | MR | Zbl

[25] W. Lacarbonara, M. Cetraro, “Flutter Control of a Lifting Surface via Visco-Hysteretic Vibration Absorbers”, Intern. J. of Aeronautical Space Science, 12:4 (2011), 331–345 | DOI

[26] D. Rachinskii, “Realization of Arbitrary Hysteresis by a Low-dimensional Gradient Flow”, Discrete and Continuous Dynamical Systems B, 21:1 (2016), 227–243 | DOI | MR | Zbl

[27] P. D. Spanos, A. Matteo, A. Di Pirrotta, “Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements”, Nonlinear Dynamics, 98 (2019), 3113–3124 | DOI

[28] M. Eleuteri, E. Ipocoana, J. Kopfová, P. Krejčí, “Periodic solutions of a hysteresis model for breathing”, ESAIM: Mathematical Modelling and Numerical Analysis, 54:1 (2020), 255–271 | DOI | MR

[29] G. Friedman, P. Gurevich, S. McCarthy, D. Rachinskii, “Switching behaviour of two-phenotype bacteria in varying environment”, J. of Physics: Conference Series, 585:1 (2015), 012012-1–012012-12 | DOI

[30] A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. O'Callaghan, A. Pokrovskii, “Systems with Hysteresis in Mathematical Biology via a Canonical Example”, Mathematical Modeling, Clustering Algorithms and Applications, ed. C. L. Wilson, Nova Science Publishers, Inc., 2010, 34 pp.

[31] A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. O'Callaghan, D. Rachinskii, “Memory and Adaptive Behavior in Population Dynamics: Anti-predator Behavior as a Case Study”, Journal of mathematical biology, 74:6 (2017), 1533–1559 | DOI | MR | Zbl

[32] H. R. Noori, Hysteresis Phenomena in Biology, Springer, Berlin–Heidelberg, 2014, 45 pp. | MR | Zbl

[33] P. Krejčí, J. P. O'Kane, A. Pokrovskii, D. Rachinskii, “Properties of solutions to a class of differential models incorporating Preisach hysteresis operator”, Physica D: Nonlinear Phenomena, 241:22 (2012), 2010–2028 | DOI | MR

[34] F. Baronti, N. Femia, R. Saletti, C. Visone, W. Zamboni, “Preisach modelling of lithium-iron-phosphate battery hysteresis”, Journal of Energy Storage, 4 (2015), 51–61 | DOI

[35] J. Carmeliet, K. E.A. Van Den Abeele, “Application of the Preisach-Mayergoyz space model to analyze moisture effects on the nonlinear elastic response of rock”, Geophysical Research Letters, 29:7 (2002), 48-1–48-4 | DOI

[36] V. Franzitta, A. Viola, M. Trapanese, “Description of hysteresis in Lithium battery by classical Preisach model Advanced Materials Research”, Trans. Tech. Publications, 622 (2013), 1099–1103

[37] P. Krejčí, G. A. Monteiro, “Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling”, Discrete Continuous Dynamical Systems B, 24:7 (2019), 3051–3066 | DOI | MR | Zbl

[38] K. Łygas, P. Wolszczak, G. Litak, P. Sta̧czek, “Complex response of an oscillating vertical cantilever with clearance”, Meccanica, 54 (2019), 1689–1702 | DOI | MR

[39] G. Friedman, S. McCarthy, D. Rachinskii, “Hysteresis can grant fitness in stochastically varying environment”, PLoS One, 9,:7 (2014), e103241-1–e103241-9 | DOI

[40] I. D. Mayergoyz, M. Dimian, “Analysis of spectral noise density of hysteretic systems driven by stochastic processes”, Journal of Applied Physics, 93:10 (2003), 6826–6828 | DOI | MR

[41] D. Rachinskii, M. Ruderman, “Convergence of direct recursive algorithm for identification of Preisach hysteresis model with stochastic input”, SIAM Journal on Applied Mathematics, 76:4 (2016), 1270–1295 | DOI | MR | Zbl

[42] S. V. Borzunov, M. E. Semenov, N. I. Sel'vesyuk, P. A. Meleshenko, “Hysteretic Converters with Stochastic Parameters”, Math. Mod. and Comp. Simul., 12:2 (2020), 164–175 | DOI | MR

[43] S. V. Borzunov, M. E. Semenov, N. I. Sel'vesyuk, P. A. Meleshenko, “Generalized Play-Operator Under Stochastic Perturbations: An Analytic Approach”, Journal of Vibration Engineering Technologies, 9 (2021), 355–365 | DOI

[44] M. E. Semenov, S. V. Borzunov, P. A. Meleshenko, “Stochastic Preisach operator: definition within the design approach”, Nonlinear Dynamics, 101 (2020), 2599–2614 | DOI

[45] M. E. Semenov, P. A. Meleshenko, I. N. Ishchuk, D. D. Dmitriev, S. V. Borzunov, N. N. Nekrasova, “Non-ideal Relay with Random Parameters”, Trends in Mathematics, 11, eds. A. Korobeinikov, M. Caubergh, T. Lázaro, J. Sardanyés, Springer, 2019, 253–258 | DOI | MR | Zbl

[46] D. R. Cox, H. D. Miller, The Theory of Stochastic Processes, Methuen's monographs on applied probability and statistics, CRC Press, 1977, 408 pp.

[47] A. D. Venttsel, Kurs teorii sluchainykh protsessov, 2-e izd., dop., Nauka. Fizmatlit, M., 1996, 399 pp. | MR

[48] A. N. Shiryaev, Probability, Second ed., Springer, 2016, 621 pp. | MR | Zbl

[49] M. M. Borovikova, V. G. Zadorozhnii, “Nakhozhdenie momentnykh funktsii resheniia dvumernogo uravneniia diffuzii so sluchainymi koeffitsientami”, Izvestiia Rossiiskoi akademii nauk. Seriia matematicheskaia, 74:6 (2010), 27–54 | MR | Zbl

[50] V. G. Zadorozhniy, G. A. Kurina, “Mean Periodic Solutions of a Linear Inhomogeneous First-Order Differential Equation with Random Coefficients”, Differential Equations, 50:6 (2014), 722–741 | DOI | MR | Zbl