The Preisach Model
Delayed relay operators are introduced; by integrating them with respect to a Borel measure μ over the half plane ρ of admissible thresholds, the Preisach operator H μ , is then constructed. Memory is represented by means of an antimonotone graph in ρ, wh
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Outline Delayed relay operators are introduced; by integrating them with respect to a Borel measure f.l over the half plane P of admissible thresholds, the Preisach operator H,. is then constructed. Memory is represented by means of an antimonotone graph in p, which separates relays staying in different states. Relations between Preisach and Prandtl-Ishlinskiloperators of play-type are pointed out. The continuity properties of the Preisach operator in several function spaces are then studied. Conditions on the measure f.l are given such that H,. is either continuous, or uniformly continuous, or Lipschitz continuous in CO([O, T)), and such that H,. operates either in the Sobolev spaces W1,P(0, T) (1 ~ p ~ +00), or in the Holder spaces CO, II ([0, T)) (0 < v ~ 1), or in CO([O, T)) n BV(O, T). Other conditions on f.l guarantee the existence of the inverse operator H; I, and yield its continuity in the spaces above. Two vectorial extensions of this model are then outlined, and their properties are briefly discussed. Delayed relays and generalized plays are then characterized by a simple variational principle.
Prerequisites. Basic notions of measure theory and of functional analysis are applied. Definitions of some function spaces are recalled in Sects. Xll.I and Xll.2. Reduced memory sequences, introduced in Sect. rn.6, are used in Sects. IV.I, IV.2. Prandtl-IshlinskiY models, see Sect. rnA, are considered in the final part of Sect. IV.2.
IV.1 The Preisach Operator In this section we introduce the Preisach operator, and outline its most elementary properties.
Delayed Relays. We denote by BV(O, T) the Banach space of functions [0, T] ---t R having finite total variation, cf. Sect. Xll.7, and by C~([O, TD the linear space of A. Visintin, Differential Models of Hysteresis © Springer-Verlag Berlin Heidelberg 1994
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IV. The Preisach Model
functions which are continuous on the right in [0, T[. For any couple P := (PI, P2) E R2, with PI < P2, we introduce the delayed relay operator hp : Co([o, Tn x {-I, I} -+ BV(O, T)
For any u E CO([o, Tn and any {-1, I} is defined as follows:
z(O) :=
{
n C~([O, TD.
e=-lor 1, the function z =hp(u, e) : [0, T] -+ -I
if u(O) ::; PI,
;
if PI < u(O) < P2, if u(O) ~ P2;
(1.1)
for any t E ]0, T], setting X t := {r E ]0, t] : u(r) = PI or P2}, z(O) z(t):=
{
~l
if X t
=0,
=Ph if X t '" 0 and u(maxXt ) =P2. if X t '" 0 and u(maxXt )
(1.2)
Then z is uniquely defined in [0, T]. For instance, let u(O) < PI; then z(O) = -1, and z(t) = -1 as long as u(t) < P2; if at some instant u reaches P2, then z jumps up to 1, where it remains as long as u(t) > PI; if later u reaches PI, then z jumps down to -1; and so on, cf. Fig. 1. Note that for any function u E CO([o, Tn, the number of oscillations of u between PI and P2 is necessarily finite, because of the uniform continuity; hence z can have just a finite number of jumps between -1 and 1. Therefore z is piecewise constant and its total variation in [0, T], denoted by Var(z), is actually finite. It is straightforward to check that z is a
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