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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.)
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<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>
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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.
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The [[Hysteresis|hysteresis]] non-linearity denoted by  $  {\mathcal R} ( \alpha, \beta ) $,
 +
with thresholds  $  \alpha $
 +
and  $  \beta $,
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and defined for a continuous input  $  u ( t ) $,
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$  t \geq  t _ {0} $,
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and an initial state  $  r _ {0} \in \{ 0,1 \} $
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by the formulas (see Fig.a1.)
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$$
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{\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{
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 +
\begin{array}{ll}
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r _ {0}  &\textrm{ if  }  \alpha < u ( s ) < \beta,  t _ {0} \leq  s \leq  t,  \\
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0  &\textrm{ if  either  }  u ( t ) \leq  \beta  \textrm{ or  }  \\
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{}  & u ( t ) \in ( \beta, \alpha )  \textrm{ and  }  u ( \tau ) = \beta,  \\
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1  &\textrm{ if  either  }  u ( t ) \geq  \alpha  \textrm{ or  }  \\
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{}  & u ( t ) \in ( \beta, \alpha )  \textrm{ and  }  u ( \tau ) = \alpha,  \\
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\end{array}
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\right .
 +
$$
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 +
where  $  \tau = \sup  \{ s : {s \leq  t, u ( s ) = \beta  \textrm{ or  }  u ( s ) = \alpha } \} $,  
 +
that is, $  \tau $
 +
denotes the last switching moment. The input–output operators $  {\mathcal R} ( r _ {0} ; \alpha, \beta ) $
 +
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n110060a.gif" />
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Non-ideal relay
 
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
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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 $  \mu ( \alpha, \beta ) $
 +
be a finite [[Borel measure|Borel measure]] in the half-plane $  \Pi = \{ {( \alpha, \beta ) } : {\alpha > \beta } \} $.  
 +
The input–output operators of the Preisach–Giltay hysteresis non-linearity at a given continuous input $  u ( t ) $,  
 +
$  t \geq  t _ {0} $,  
 +
and initial state $  S ( t _ {0} ) $
 +
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>
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$$
 +
x ( t ) = \int\limits { {\mathcal R} ( r _ {0} ( \alpha, \beta ) ; \alpha, \beta ) u ( t ) }  {d \mu ( \alpha, \beta ) } ,
 +
$$
  
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.
+
where the measurable function $  r _ {0} ( \alpha, \beta ) $
 +
describes the internal state of the non-linearity at the initial moment $  t = t _ {0} $.  
 +
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 $  \mu ( \alpha, \beta ) $
 +
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]].
 
See also [[Hysteresis|Hysteresis]].

Latest revision as of 14:54, 7 June 2020


The hysteresis non-linearity denoted by $ {\mathcal R} ( \alpha, \beta ) $, with thresholds $ \alpha $ and $ \beta $, and defined for a continuous input $ u ( t ) $, $ t \geq t _ {0} $, and an initial state $ r _ {0} \in \{ 0,1 \} $ by the formulas (see Fig.a1.)

$$ {\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{ \begin{array}{ll} r _ {0} &\textrm{ if } \alpha < u ( s ) < \beta, t _ {0} \leq s \leq t, \\ 0 &\textrm{ if either } u ( t ) \leq \beta \textrm{ or } \\ {} & u ( t ) \in ( \beta, \alpha ) \textrm{ and } u ( \tau ) = \beta, \\ 1 &\textrm{ if either } u ( t ) \geq \alpha \textrm{ or } \\ {} & u ( t ) \in ( \beta, \alpha ) \textrm{ and } u ( \tau ) = \alpha, \\ \end{array} \right . $$

where $ \tau = \sup \{ s : {s \leq t, u ( s ) = \beta \textrm{ or } u ( s ) = \alpha } \} $, that is, $ \tau $ denotes the last switching moment. The input–output operators $ {\mathcal R} ( r _ {0} ; \alpha, \beta ) $ 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.

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 $ \mu ( \alpha, \beta ) $ be a finite Borel measure in the half-plane $ \Pi = \{ {( \alpha, \beta ) } : {\alpha > \beta } \} $. The input–output operators of the Preisach–Giltay hysteresis non-linearity at a given continuous input $ u ( t ) $, $ t \geq t _ {0} $, and initial state $ S ( t _ {0} ) $ is defined by the formula

$$ x ( t ) = \int\limits { {\mathcal R} ( r _ {0} ( \alpha, \beta ) ; \alpha, \beta ) u ( t ) } {d \mu ( \alpha, \beta ) } , $$

where the measurable function $ r _ {0} ( \alpha, \beta ) $ describes the internal state of the non-linearity at the initial moment $ t = t _ {0} $. 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 $ \mu ( \alpha, \beta ) $ is absolutely continuous with respect to the Lebesgue measure (cf. Absolute continuity). For detailed properties of Preisach–Giltay hysteresis and further generalizations see [a1], [a2] and the references therein.

See also Hysteresis.

References

[a1] M.A. Krasnosel'skii, A.V. Pokrovskii, "Systems with hysteresis" , Springer (1989) (In Russian)
[a2] I.D. Mayergoyz, "Mathematical models of hysteresis" , Springer (1991)
How to Cite This Entry:
Non-ideal relay. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-ideal_relay&oldid=49331
This article was adapted from an original article by A.M. Krasnosel'skiiM.A. Krasnosel'skiiA.V. Pokrovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article