Non-ideal relay

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)