Stochastic Preisach operator: definition within the design approach
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ORIGINAL PAPER
Stochastic Preisach operator: definition within the design approach Mikhail E. Semenov Peter A. Meleshenko
· Sergei V. Borzunov ·
Received: 12 February 2020 / Accepted: 19 August 2020 © Springer Nature B.V. 2020
Abstract In this paper, we generalize the Preisach model to the case when parameters of elementary relays are random variables. We show that an output state of the stochastic Preisach operator can be treated as a random process. For this random process the first and second statistical moments are obtained in an explicit form. We prove also the correctness of the definition of
such an operator. Illustrative numerical examples are also presented and discussed. Keywords Hysteresis · Non-ideal relay · Preisach operator · Random process
1 Introduction This work was supported by the RFBR (Grants 18-08-00053-a and 19-08-00158). The contributions by M. E. Semenov and P. A. Meleshenko (System of stochastic non-ideal relays and stochastic Preisach operator: definition and properties) were supported by the RSF Grant No. 19-11-00197. M. E. Semenov (B)· S. V. Borzunov · P. A. Meleshenko Digital Technologies Department, Voronezh State University, Universitetskaya sq.1, Voronezh, Russia 394018 e-mail: [email protected] M. E. Semenov Geophysical Survey of Russia Academy of Science, Lenina av. 189, Obninsk, Russia 249035 M. E. Semenov Applied Mathematics and Mechanics Department, Voronezh State Technical University, XX-letiya Oktyabrya st. 84, Voronezh, Russia 394006 M. E. Semenov Metheorology Department, Zhukovsky-Gagarin Air Force Academy, Starykh Bolshevikov st. 54 “A”, Voronezh, Russia 394006 P. A. Meleshenko Target Search Lab of Groundbreaking Radio Communication Technologies of Advanced Research Foundation, Plekhanovskaya st. 14, Voronezh, Russia 394018
Investigation and modelling of real-life complex technical systems involves formalization of the unconventional nonlinearities, including nonlinearities of a hysteretic nature. As is well known, a wide class of physical, physico-chemical, biological and economic systems exhibit hysteretic behaviour, which is caused either by their internal structure or by dynamical features of the processes occurring in these systems (for details see, e.g. [15,20,26,30,36,40,48] and related references). To date there are two main approaches to description of the hysteresis phenomena: design models, e.g. a backlash (or play-operator), a non-ideal relay, the Preisach model, the Ishlinskii–Prandtl model (see [10,13,22,26,32,34,52]), and phenomenological models, such as the Bouc–Wen model [19], the Duhem model [42], the Iwan model and others [21,33,39]. Note that parameters of all these approaches are supposed to be deterministic. In works of Krasnoselskii [25,26] a special mathematical apparatus was developed for the analysis of systems with hysteresis. This apparatus is based on the
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idea of elementary carriers of hysteretic properties (the so-called hysterones). Hysterones are converters with state spaces defined by the “input–output” and “
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