Editing Hysteresis (section)

===Rate-independent===

Systems with ''rate-independent hysteresis'' have a ''persistent'' memory of the past that remains after the transients have died out.<ref name=rate>The term is attributed to {{harvnb|Truesdell|Noll|1965}} by {{harvnb|Visintin|1994|loc=page 13}}.</ref> The future development of such a system depends on the history of states visited, but does not fade as the events recede into the past. If an input variable {{math|<var>X(t)</var>}} cycles from  {{math|<var>X</var><sub>0</sub>}}  to {{math|<var>X</var><sub>1</sub>}} and back again, the output {{math|<var>Y(t)</var>}}  may be  {{math|<var>Y</var><sub>0</sub>}} initially but a different value  {{math|<var>Y</var><sub>2</sub>}} upon return. The values of  {{math|<var>Y(t)</var>}} depend on the path of values that  {{math|<var>X(t)</var>}} passes through but not on the speed at which it traverses the path.<ref name="Bertotti1998ch2"/> Many authors restrict the term hysteresis to mean only rate-independent hysteresis.<ref name=Visintin1994p13>{{harvnb|Visintin|1994|loc=page 13}}</ref> Hysteresis effects can be characterized using the [[Preisach model]] and the generalized [[Prandtl−Ishlinskii model]].<ref>Mohammad Al Janaideh, Subhash Rakheja, Chun-Yi Su [https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5510155&tag=1 An analytical generalized Prandtl–Ishlinskii model inversion for hysteresis compensation in micropositioning control], IEEE/ASME Transactions on Mechatronics,  Volume:16 Issue:4, pp 734−744, 15 July 2010</ref>