View source for Non-ideal relay

The [[Hysteresis|hysteresis]] non-linearity denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100601.png" />, with thresholds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100603.png" />, and defined for a continuous input <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100605.png" />, and an initial state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100606.png" /> by the formulas (see Fig.a1.)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100607.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100608.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n1100609.png" /> denotes the last switching moment. The input–output operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006010.png" /> are discontinuous in the usual function spaces. These operators are monotone in a natural sense, which allows one to use the powerful methods of the theory of semi-ordered spaces in the analysis of systems with non-ideal relays.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n110060a.gif" />

Figure: n110060a

Non-ideal relay

The Preisach–Giltay model of ferromagnetic hysteresis is described as the spectral decomposition in a continual system of non-ideal relays in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006011.png" /> be a finite [[Borel measure|Borel measure]] in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006012.png" />. The input–output operators of the Preisach–Giltay hysteresis non-linearity at a given continuous input <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006014.png" />, and initial state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006015.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006016.png" /></td> </tr></table>

where the measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006017.png" /> describes the internal state of the non-linearity at the initial moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006018.png" />. In contrast to the individual non-ideal relay, the operators of a Preisach–Giltay non-linearity are continuous in the space of continuous functions, provided that the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110060/n11006019.png" /> is absolutely continuous with respect to the [[Lebesgue measure|Lebesgue measure]] (cf. [[Absolute continuity|Absolute continuity]]). For detailed properties of Preisach–Giltay hysteresis and further generalizations see [[#References|[a1]]], [[#References|[a2]]] and the references therein.

See also [[Hysteresis|Hysteresis]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.A. Krasnosel'skii,   A.V. Pokrovskii,   "Systems with hysteresis" , Springer  (1989)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.D. Mayergoyz,   "Mathematical models of hysteresis" , Springer  (1991)</TD></TR></table>