Dept. of Appl.Math @

Systems with Hysteresis


What is `Hysteresis'

Preisach Model

Closed loop systems

Complex behavior

Other Hysterons
   Vector Preisach
   Three state relays
   2D play nonlinearity

Advanced modelling


2D play nonlinearity

Consider a polyhedral coil $\Gamma$ and a pivot $M$ which is placed in the interior of $\Gamma$. The pivot can move within the plane according to any arbitrary continuous rule and the coil remains fixed as long as the pivot moves only in its interior. If, however, the pivot touches the coil, the latter starts moving. We assume that only translations of the coil may occur, i.e., its sides remain parallel to themselves.

The relationships between variable position $x(t)$ of the pivot $M$, and the corresponding movement $y(t)$ of a reference point of the coil describes a hysteresis nonlinearity which is called two dimensional play operator with the characteristic $\Gamma$. This nonlinearity, its straightforward multidimensional analogs, and further modifications, are extremely important in various subject areas including mechanics and network analysis. For a rigorous definition of this nonlinearity and detailed studies of of its properties see, for instance,

  1. M. Krasnosel'skii and A. Pokrovskii, ``Systems with Hysteresis'', Springer, 1989,
  2. P. Krejci and A. Vladimirov. Lipschitz continuity of polyhedral Skorokhod maps. J. Analysis Appl., 20:817844, 2001.
  3. P. Krejci and A. Vladimirov. Polyhedral sweeping processes with oblique reflection in the space of regulated functions. Set-Valued Anal., 11:91110, 2003. I.
  4. Nedaiborshch, K. Nikolaev, and A. Vladimirov. Lipschitz continuity and unique solvability of fluid models of queueing networks. Information Processes, Electronic Scientific Journal, 3:138150, 2003.