Editing Hysteresis (section)

==Types==

===Rate-dependent===

One type of hysteresis is a [[Latency (engineering)|lag]] between input and output. An example is a [[sine wave|sinusoidal]] input {{math|<var>X(t)</var>}} that results in a sinusoidal output {{math|<var>Y(t)</var>}}, but with a phase lag {{math|<var>&phi;</var>}}:
:<math> \begin{align}
X(t) &= X_0 \sin \omega t \\ Y(t) &= Y_0 \sin\left(\omega t-\varphi\right).
\end{align}</math>
Such behavior can occur in linear systems, and a more general form of response is
:<math> Y(t) = \chi_\text{i} X(t) + \int_0^{\infty} \Phi_\text{d} (\tau) X(t-\tau) \, \mathrm{d}\tau, </math>
where <math>\chi_\text{i}</math> is the instantaneous response and <math>\Phi_d(\tau)</math> is the [[impulse response]] to an impulse that occurred <math>\tau</math> time units in the past.  In the [[frequency domain]], input and output are related by a complex ''generalized susceptibility'' that can be computed from <math>\Phi_d</math>; it is mathematically equivalent to a [[transfer function]] in linear filter theory and analogue signal processing.<ref name=Bertotti1998ch2>{{cite book |last1=Bertotti |first1=Giorgio |title=Hysteresis in magnetism: For physicists, materials scientists, and engineers |publisher=[[Academic Press]] |year=1998 |isbn=978-0-12-093270-2 |chapter=Ch. 2}}</ref>

This kind of hysteresis is often referred to as ''rate-dependent hysteresis''. If the input is reduced to zero, the output continues to respond for a finite time. This constitutes a memory of the past, but a limited one because it disappears as the output decays to zero. The phase lag depends on the frequency of the input, and goes to zero as the frequency decreases.<ref name="Bertotti1998ch2" />

When rate-dependent hysteresis is due to [[dissipative]] effects like [[friction]], it is associated with power loss.<ref name="Bertotti1998ch2" />

===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>