Difference between revisions of "Non-ideal relay"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | n1100601.png | ||
| + | $#A+1 = 19 n = 0 | ||
| + | $#C+1 = 19 : ~/encyclopedia/old_files/data/N110/N.1100060 Non\AAhideal relay | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| − | + | {{TEX|auto}} | |
| + | {{TEX|done}} | ||
| − | where | + | The [[Hysteresis|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.) | ||
| + | |||
| + | $$ | ||
| + | {\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{ | ||
| + | |||
| + | where $ \tau = \sup \{ s : {s \leq t, u ( s ) = \beta \textrm{ or } u ( s ) = \alpha } \} $, | ||
| + | that is, $ \tau $ | ||
| + | denotes the last switching moment. The input–output operators $ {\mathcal R} ( r _ {0} ; \alpha, \beta ) $ | ||
| + | are discontinuous in the usual function spaces. These operators are monotone in a natural sense, which allows one to use the powerful methods of the theory of semi-ordered spaces in the analysis of systems with non-ideal relays. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n110060a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n110060a.gif" /> | ||
| Line 11: | Line 33: | ||
Non-ideal relay | Non-ideal relay | ||
| − | The Preisach–Giltay model of ferromagnetic hysteresis is described as the spectral decomposition in a continual system of non-ideal relays in the following way. Let | + | The Preisach–Giltay model of ferromagnetic hysteresis is described as the spectral decomposition in a continual system of non-ideal relays in the following way. Let $ \mu ( \alpha, \beta ) $ |
| + | be a finite [[Borel measure|Borel measure]] in the half-plane $ \Pi = \{ {( \alpha, \beta ) } : {\alpha > \beta } \} $. | ||
| + | The input–output operators of the Preisach–Giltay hysteresis non-linearity at a given continuous input $ u ( t ) $, | ||
| + | $ t \geq t _ {0} $, | ||
| + | and initial state $ S ( t _ {0} ) $ | ||
| + | is defined by the formula | ||
| − | + | $$ | |
| + | x ( t ) = \int\limits { {\mathcal R} ( r _ {0} ( \alpha, \beta ) ; \alpha, \beta ) u ( t ) } {d \mu ( \alpha, \beta ) } , | ||
| + | $$ | ||
| − | where the measurable function | + | where the measurable function $ r _ {0} ( \alpha, \beta ) $ |
| + | describes the internal state of the non-linearity at the initial moment $ t = t _ {0} $. | ||
| + | In contrast to the individual non-ideal relay, the operators of a Preisach–Giltay non-linearity are continuous in the space of continuous functions, provided that the measure $ \mu ( \alpha, \beta ) $ | ||
| + | is absolutely continuous with respect to the [[Lebesgue measure|Lebesgue measure]] (cf. [[Absolute continuity|Absolute continuity]]). For detailed properties of Preisach–Giltay hysteresis and further generalizations see [[#References|[a1]]], [[#References|[a2]]] and the references therein. | ||
See also [[Hysteresis|Hysteresis]]. | See also [[Hysteresis|Hysteresis]]. | ||
Revision as of 08:03, 6 June 2020
The hysteresis non-linearity denoted by $ {\mathcal R} ( \alpha, \beta ) $,
with thresholds $ \alpha $
and $ \beta $,
and defined for a continuous input $ u ( t ) $,
$ t \geq t _ {0} $,
and an initial state $ r _ {0} \in \{ 0,1 \} $
by the formulas (see Fig.a1.)
$$ {\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{ 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"/> 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|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) |
Non-ideal relay. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-ideal_relay&oldid=47988