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$$  
 
$$  
 
{\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{
 
{\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 } \} $,  
 
where  $  \tau =  \sup  \{ s : {s \leq  t, u ( s ) = \beta  \textrm{ or  }  u ( s ) = \alpha } \} $,  

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=47988
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