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Only a very simple modification of the non-linearity  "hysteron"  is described below. See [[#References|[a1]]] for the general definition and an identification theorem, that is, qualitative conditions under which a  "black box"  is a hysteron. Consider in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104301.png" />-plane the graphs of two continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104303.png" /> satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104305.png" />. Suppose that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104306.png" /> is partitioned into the disjoint union of the continuous family of graphs of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104308.png" /> is a parameter. Each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h1104309.png" /> is defined on its interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043013.png" />, that is, the end-points of the graphs of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043014.png" /> belong to the union of the graphs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043016.png" /> (see Fig.a2.).
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Only a very simple modification of the non-linearity  "hysteron"  is described below. See [[#References|[a1]]] for the general definition and an identification theorem, that is, qualitative conditions under which a  "black box"  is a hysteron. Consider in the $  ( x,g ) $-
 +
plane the graphs of two continuous functions $  H _ {1} ( x ) $,  
 +
$  H _ {2} ( x ) $
 +
satisfying the inequality $  H _ {1} ( x ) < H _ {2} ( x ) $,  
 +
$  x \in \mathbf R $.  
 +
Suppose that the set $  \Omega = \{ {\{ x,g \} } : {x \in \mathbf R,  H _ {1} ( x ) \leq  g \leq  H _ {2} ( x ) } \} $
 +
is partitioned into the disjoint union of the continuous family of graphs of continuous functions $  g _  \alpha  ( x ) $,  
 +
where $  \alpha $
 +
is a parameter. Each function $  g _  \alpha  ( x ) $
 +
is defined on its interval $  [ \eta _  \alpha  ^ {1} , \eta _  \alpha  ^ {2} ] $,  
 +
$  \eta _  \alpha  ^ {1} < \eta _  \alpha  ^ {2} $,  
 +
and $  g _  \alpha  ( \eta _  \alpha  ^ {1} ) = H _ {1} ( \eta _  \alpha  ^ {1} ) $,  
 +
$  g _  \alpha  ( \eta _  \alpha  ^ {2} ) = H _ {2} ( \eta _  \alpha  ^ {2} ) $,  
 +
that is, the end-points of the graphs of the functions $  g _  \alpha  ( x ) $
 +
belong to the union of the graphs of $  H _ {1} ( x ) $
 +
and $  H _ {2} ( x ) $(
 +
see Fig.a2.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110430a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110430a.gif" />
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Hysteron: Prandtl non-linearity
 
Hysteron: Prandtl non-linearity
  
A hysteron is a transducer with internal states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043017.png" /> from the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043018.png" /> and with the following input–output operators. The variable output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043020.png" />) is defined by the formula
+
A hysteron is a transducer with internal states $  \xi $
 +
from the segment $  [ 0,1 ] $
 +
and with the following input–output operators. The variable output $  {\mathcal H} ( \xi _ {0} ) x ( t ) \equiv {\mathcal H} ( \xi _ {0} ,t _ {0} ) x ( t ) $(
 +
$  t \geq  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/h/h110/h110430/h11043021.png" /></td> </tr></table>
+
$$
 +
{\mathcal H} ( \xi _ {0} ) x ( t ) = \left \{
 +
\begin{array}{l}
 +
{g _  \alpha  ( x ( t ) ) , \  \eta _  \alpha  ^ {1} \leq  x ( t ) \leq  \eta _  \alpha  ^ {2} , } \\
 +
{H _ {1} ( x ( t ) ) , \  x ( t ) \leq  \eta _  \alpha  ^ {1} , } \\
 +
{H _ {2} ( x ( t ) ) , \  \eta _  \alpha  ^ {2} \leq  x ( t ) , }
 +
\end{array}
 +
\right .
 +
$$
  
for monotone inputs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043023.png" />. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043024.png" /> is determined by the initial state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043025.png" /> to satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043026.png" /> and the corresponding variable internal state is defined by
+
for monotone inputs $  x ( t ) $,  
 +
$  t \geq  t _ {0} $.  
 +
The value of $  \alpha $
 +
is determined by the initial state $  \xi $
 +
to satisfy $  g _  \alpha  ( x ( t _ {0} ) ) = \xi _ {0} H _ {1} ( x ( t _ {0} ) ) + ( 1 - \xi _ {0} ) H _ {2} ( x ( t _ {0} ) ) $
 +
and the corresponding variable internal state is defined by
  
<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/h/h110/h110430/h11043027.png" /></td> </tr></table>
+
$$
 +
\Xi ( \xi _ {0} ) x ( t ) = {
 +
\frac{ {\mathcal H} ( \xi _ {0} ) x ( t ) - H _ {1} ( x ( t ) ) }{H _ {2} ( x ( t ) ) - H _ {1} ( x ( t ) ) }
 +
} .
 +
$$
  
For piecewise-monotone continuous inputs the output is constructed by the [[Semi-group|semi-group]] identity. The input–output operators can then be extended to the totality of all continuous inputs by continuity (see [[#References|[a1]]]). The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043029.png" /> are defined for each continuous input and each initial state. They are continuous as operators in the space of continuous functions with the uniform metric (cf. also [[Metric|Metric]]).
+
For piecewise-monotone continuous inputs the output is constructed by the [[Semi-group|semi-group]] identity. The input–output operators can then be extended to the totality of all continuous inputs by continuity (see [[#References|[a1]]]). The operators $  {\mathcal H} ( \xi _ {0} ) x ( t ) $,  
 +
$  \Xi ( \xi _ {0} ) x ( t ) $
 +
are defined for each continuous input and each initial state. They are continuous as operators in the space of continuous functions with the uniform metric (cf. also [[Metric|Metric]]).
  
A hysteron is called a Prandtl non-linearity if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043031.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043034.png" />. This non-linearity describes the Prandtl model of ideal plasticity with Young modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043035.png" /> and elastic limits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110430/h11043036.png" />. The parallel connections of a finite numbers of such elements describe the Besseling model of elasto-plasticity and the continual counterpart describe the Ishlinskii model.
+
A hysteron is called a Prandtl non-linearity if $  H _ {1} ( x ) \equiv - h $,  
 +
$  H _ {2} ( x ) \equiv h $;  
 +
$  g _  \alpha  = kx - \alpha $,
 +
$  \alpha - h \leq  x \leq  \alpha + h $,  
 +
$  \alpha \in \mathbf R $.  
 +
This non-linearity describes the Prandtl model of ideal plasticity with Young modulus $  k $
 +
and elastic limits h $.  
 +
The parallel connections of a finite numbers of such elements describe the Besseling model of elasto-plasticity and the continual counterpart describe the Ishlinskii model.
  
 
See also [[Hysteresis|Hysteresis]].
 
See also [[Hysteresis|Hysteresis]].

Latest revision as of 22:11, 5 June 2020


Only a very simple modification of the non-linearity "hysteron" is described below. See [a1] for the general definition and an identification theorem, that is, qualitative conditions under which a "black box" is a hysteron. Consider in the $ ( x,g ) $- plane the graphs of two continuous functions $ H _ {1} ( x ) $, $ H _ {2} ( x ) $ satisfying the inequality $ H _ {1} ( x ) < H _ {2} ( x ) $, $ x \in \mathbf R $. Suppose that the set $ \Omega = \{ {\{ x,g \} } : {x \in \mathbf R, H _ {1} ( x ) \leq g \leq H _ {2} ( x ) } \} $ is partitioned into the disjoint union of the continuous family of graphs of continuous functions $ g _ \alpha ( x ) $, where $ \alpha $ is a parameter. Each function $ g _ \alpha ( x ) $ is defined on its interval $ [ \eta _ \alpha ^ {1} , \eta _ \alpha ^ {2} ] $, $ \eta _ \alpha ^ {1} < \eta _ \alpha ^ {2} $, and $ g _ \alpha ( \eta _ \alpha ^ {1} ) = H _ {1} ( \eta _ \alpha ^ {1} ) $, $ g _ \alpha ( \eta _ \alpha ^ {2} ) = H _ {2} ( \eta _ \alpha ^ {2} ) $, that is, the end-points of the graphs of the functions $ g _ \alpha ( x ) $ belong to the union of the graphs of $ H _ {1} ( x ) $ and $ H _ {2} ( x ) $( see Fig.a2.).

Figure: h110430a

Figure: h110430b

Hysteron: Prandtl non-linearity

A hysteron is a transducer with internal states $ \xi $ from the segment $ [ 0,1 ] $ and with the following input–output operators. The variable output $ {\mathcal H} ( \xi _ {0} ) x ( t ) \equiv {\mathcal H} ( \xi _ {0} ,t _ {0} ) x ( t ) $( $ t \geq t _ {0} $) is defined by the formula

$$ {\mathcal H} ( \xi _ {0} ) x ( t ) = \left \{ \begin{array}{l} {g _ \alpha ( x ( t ) ) , \ \eta _ \alpha ^ {1} \leq x ( t ) \leq \eta _ \alpha ^ {2} , } \\ {H _ {1} ( x ( t ) ) , \ x ( t ) \leq \eta _ \alpha ^ {1} , } \\ {H _ {2} ( x ( t ) ) , \ \eta _ \alpha ^ {2} \leq x ( t ) , } \end{array} \right . $$

for monotone inputs $ x ( t ) $, $ t \geq t _ {0} $. The value of $ \alpha $ is determined by the initial state $ \xi $ to satisfy $ g _ \alpha ( x ( t _ {0} ) ) = \xi _ {0} H _ {1} ( x ( t _ {0} ) ) + ( 1 - \xi _ {0} ) H _ {2} ( x ( t _ {0} ) ) $ and the corresponding variable internal state is defined by

$$ \Xi ( \xi _ {0} ) x ( t ) = { \frac{ {\mathcal H} ( \xi _ {0} ) x ( t ) - H _ {1} ( x ( t ) ) }{H _ {2} ( x ( t ) ) - H _ {1} ( x ( t ) ) } } . $$

For piecewise-monotone continuous inputs the output is constructed by the semi-group identity. The input–output operators can then be extended to the totality of all continuous inputs by continuity (see [a1]). The operators $ {\mathcal H} ( \xi _ {0} ) x ( t ) $, $ \Xi ( \xi _ {0} ) x ( t ) $ are defined for each continuous input and each initial state. They are continuous as operators in the space of continuous functions with the uniform metric (cf. also Metric).

A hysteron is called a Prandtl non-linearity if $ H _ {1} ( x ) \equiv - h $, $ H _ {2} ( x ) \equiv h $; $ g _ \alpha = kx - \alpha $, $ \alpha - h \leq x \leq \alpha + h $, $ \alpha \in \mathbf R $. This non-linearity describes the Prandtl model of ideal plasticity with Young modulus $ k $ and elastic limits $ h $. The parallel connections of a finite numbers of such elements describe the Besseling model of elasto-plasticity and the continual counterpart describe the Ishlinskii model.

See also Hysteresis.

References

[a1] M.A. Krasnosel'skii, A.V. Pokrovskii, "Systems with hysteresis" , Springer (1989) (In Russian)
How to Cite This Entry:
Hysteron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hysteron&oldid=11383
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