The Science of Hysteresis : 3-volume set

Example Consider the system

L (~)x y(t) rl(t)

-

fit, x,y(t)),

=

(I)(r/(t)),

=

(F[rl(to)]x)(t),

(2.83)

with

L(p) = p~ + alp ~-1 + . . . q- a~, where x(t) and y(t) are inputs and outputs of the Preisach nonlinearity and the functionf(t, x, y) is supposed to be continuous and satisfiesflt, x, y) _-fit + T, x, y) for some T > 0. We split the set of all T-periodic solutions (x(t), r/(t)) = (x(t + T), tl(t + T)) (t>~to) of system (2.83) into disjoint equivalence classes, assigning two periodic solutions to the same class if they have the same first component and their second components ~lj(t) satisfy (I)(t/1 (t)) = q)0/2(t)), i.e. the variable states tlj(t) define the same periodic output of the Preisach nonlinearity. We denote the equivalence class of periodic solutions (x(t), rl(t)) by Ix(t), y(t)] where y(t) = (I)(r/(t)) for all its representatives. Classes [x(t), y(t)] are natural for several reasons. Firstly, the differential equation in system (2.83) contains inputs and outputs rather than

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184

variable states r/(t) of the hysteresis nonlinearity. This situation is typical since states of the hysteresis nonlinearity are basically considered as values that one can neither observe nor control. Secondly, the second components of all periodic solutions (x(t), q(t)) that belong to the same class Ix(t), y(t)] are related by very simple explicit formulas, which we discuss below. Let for all t, x e R

[fit, x, y)l<<.qlxl + b,

y e [0, 1],

(2.84)

(here [0, 1] is the range of output values of the Preisach nonlinearity). Suppose that all the zeros of the polynomial L(p) are different from the numbers nwoi (n 9 7/)where w0 = 21t/T. Set

(

k - max IL(nwoi)1-1,

kl - T -1/2 IL(O)1-2 + 2

neZ

~ )1/2 IL(nwoi)1-2 . n--1 (2.85)

In fact, these formulas define the norms k = IIH IIC2----~C2 and kl = IIH IIL2-,C of the solution operator H = H(T) sending a function u = u(t) to the solution

x

Hu of the T-periodic problem for the linear Eqn. L ( d ) x = u(t) in the

spaces L2 = L2[0, T] and C - C[0, T]. To keep things simple, we shall always assume in this subsection that the measure It of the Preisach nonlinearity has a bounded density h = h(~, t) with respect to the Lebesgue measure. Define the scalar function ~(r) = It({(~,//)'~ < - r , r
r>~0,

(2.86)

and denote by Ru the least nonnegative solution of the Eqn. ~(r) = 0 if any exists. If ~(r) > 0 for all r>~0 then we set Ru = c~. The value Ru plays an important role in all the statements below. PROPOSITION2.3.16. Let estimate (2.84) hold and

k l b V/-T 1 - qk < Ra.

qk < 1,

(2.87)

Then there exists a one-parameter continuum of classes [x,~(t), y2(t)] (0~<2~1) of T-periodic solutions to system (2.83), whose classes are different for different 2. Assume additionally that the density of the measure It satisfies h(~, fl)<.v(fl- ~),

~ < fl,

(2.88)

v(s) ds < cxz.

(2.89)

where v is integrable on the positive semiaxis, i.e. v~-

f

OO

0

185

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

Moreover, assume that the function fit, x, y) satisfies the Lipschitz condition Lf(t, Xl, yl) - f i t , x2, y2)l~qlxl - x21 if- qllyl - y2l

(2.90)

for all t, xj ~ JR, O<~yj<~l and 1 - qk ql < 2v~kl ~/T"

(2.91)

Then the set of all T-periodic solutions of system (2.83) is the join of the classes [x,~(t), y,~(t)] (0~<2~<1) above and there are numbers cj such that - 221,

IIx2~ (.) - x~ 2 ( . ) I I c ~ c l l , ~ l

IIY21(') - Y22(') IIc~c2121 - 221,

2j e [0, 1]. (2.92)

For example, consider Eqn. (2.9) of Section 2.1, i.e. x" + x = sin(~/2t) + p(2y(t) - 1),

(2.93)

where x and y are the input and output of the Preisach nonlinearity, p > 0. Here T - ~/2~z, estimate (2.84) holds with q = 0, b - 1+ p and the quantities (2.85) equal

k-

1 ~ v/2~z+ sin(V~n) ~. 0.829.

1,

kl - ~

1 - cos(~/2~z)

Therefore the first of the estimates (2.87) holds. Assume that the density h = h(c~, fl) of the measure/~ is defined by h(~, fl) = 1/(2r 2)

if

- r~<~ < fl<<.r;

h(~, fl) = 0

otherwise.

Then R~ = r and the second of the estimates (2.87) has the form r

> ~2~z 2 + ~/2~z sin(~/27r)

1+ p

4(1 - cos(~/~z))

1.747.

(2.94)

By Proposition 2.3.16 this estimate implies that Eqn. (2.93) has a oneparameter continuum of classes [x,~(t), y2 (t)] of periodic solutions with the period ~/2~z. Furthermore, the Lipschitz estimate (2.90) holds with ql = 2p and from

h(c~, fi)<~l/(2r 2)

if

0 < f l - c~<~2r,

h(c~, fi) = 0

otherwise

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186

it follows that Voo = 1/r, and therefore estimate (2.91) takes the form 4/t/2~ 2 q- ~/-2~sin (w/2~) P > V 4(1 -- c--0-os(---~~)) ~ 6.9876.

r

(2.95)

The second part of Proposition 2.3.16 implies that if estimates (2.94) and (2.95) are valid, then the set of all v/2~-periodic solutions of Eqn. (2.93) is the join of the classes [x2(t), y,l(t)] (0~<2~<1). In fact, these classes are related here by the simple formulas

x,~(t) = xo(t) + 2p6(2),

y~(t) = yo(t) + 6(2),

2 e [0, 1],

where the continuous function 6 = 6(2) strictly increases and 6(0) = 0. Note that estimate (2.94) implies r>1.747. One can use operator norms different from (2.85) to conclude that all Tperiodic solutions of system (2.83) constitute a one-parameter set of classes [x~(t), y,l(t)]. For example, that conclusion and estimates (2.92) are valid if the relation k2(q + 2vooql) < 1 holds with k2 - IIHIIc--.c. This relation and relation (2.91) of Proposition 2.3.16 do not imply each other and give different estimates for the coefficients q and ql of the Lipschitz condition (2.90). We remark that the norm k IIHIIL2__+L2 equals the spectral radius of the operator H = H(T) and that if qk < 1 (i.e., the first relation of (2.87) holds) then estimate (2.84) alone implies that system (2.83) has at least one T-periodic solution, which may be eventually unique. =

Structure of the Set of Periodic Variable States and Outputs The following proposition, describing the set of all possible periodic variable states and outputs of the Preisach nonlinearity for a given periodic input, explains why the one-parameter continua of equivalence classes of periodic solutions appear and play the main role in the proofs of Proposition 2.3.16 above and further statements of this subsection on such continua. Consider a continuous input, which is periodic on the semiaxis t>~to, i.e. x(t) - x(t + T) (t>~to). For some admissible initial states r/0 = r/0(~, fl), the variable state r/(t) = (F[r/0]x)(t) and the output y(t) = O(r/(t)) of the Preisach nonlinearity are also periodic on the semiaxis t>,to. To describe the class of all such variable states and outputs for a given input x(t), let us define the numbers Xm = min x(t), XM = max x(t) and the sets

G(xm, XM) = {(0r fl) : 0r < Xm~XM < fl}, Gc(xm, XM) = 1-I\G(xm, XM) on the half-plane FI = {(a, fl) : a < fl}. From the relay definition it follows that if (a, ]/) e Gc(xm, XM) then the relay output (R~,/~[t0, .]x)(t) is periodic

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

187

on the semiaxis t~to for exactly one of the two initial relay states 0 a n d 1, for the other initial state the o u t p u t is either not periodic or not defined. A u n i q u e (for a given x(t)) periodic relay o u t p u t is d e n o t e d by r/per(t, (x, fl) a n d we set

s s (J,~x)(t) - I ] r/per(t, (X, fl)d#(a:,//) + ,~#(G(Xm, XM)), d JGc(Xm,XM)

t~to,

(2.96) w h e r e 0 ~ 2 ~ 1. For each ,l, this operator is defined on the class of all continuous periodic functions x : [to, co) ~ R, acts in this class and sends a function x = x(t) with a period T to the function J , tx with the same period. Denote by (9 the class of all functions g : ~+ --> ~ such that Ig(yl) g(y2)l~lYl - y2[ (yl, y2~0) and define the function @(e) =

sup # {(~, fl) ~ 1-I : I~ + fl - g(fl - ~)lKe}, g=g(y)~q)

e~0,

(2.97)

w h i c h is nondecreasing, nonnegative a n d satisfies ~(e) ---> 0 as e ---> 0, since we a s s u m e that the m e a s u r e # has a b o u n d e d density. PROPOSITION2.3.17. The following statements hold:

(i) For any periodic input x(t) =_ x(t + T) (t~to) the formula y(t) = (J2x)(t) with 2 ranging over the segment 0 ~ 2 ~ 1 defines the class of all outputs of the Preisach nonlinearity that are periodic on t ~ to. (ii) For any periodic input x(t) =_ x(t + T) (t~to) the formula r/(t, a, fl) = {

r/,r/per(t fl)(X''(fl) (x, ififeithero~< Xm~XM fl > O~Xm<

[~,~ XM~fl

> O~or both,

(2.98) where r/~(~, fl) is an arbitrary measurable (with respect to the measure #) function with the values O, 1, defines the class of all periodic variable states r/(t) = r/(t, ~, fl) of the Preisach nonlinearity. Their periods equal the period of the input , i.e. r/( t) = r/( t + T) ( t ~ to ) , and the periodic output y(t) = ~(r/(t)) for the variable state (2.98) is defined by y(t) = (J2x)(t) with

2#(G(xm, XM)) -- #({(~, fl) 6 G(xm, XM): r/~(~, fl) -- 1}).

(2.99)

(iii) Each variable state r/(t) = (F[.]x)(t) (t~to) equals one of the variable states (2.98) on the semiaxis t~to + T. (iv) For every pair of periodic inputs xj = xj(t) (t~to) with a period T and every ,l ~ [0, 1] II,J2Xl - J2x211C[to,to+T]~26(llXl - x211C[to,to+T]).

(2.100)

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CHAPTER 2 Differential Equations with Hysteresis

From conclusion (iv) it follows that operator (2.96) is uniformly continuous in the space C[to, to + T] for each 2 (we identify periodic inputs and outputs with their restrictions to the period). If relations (2.88) and (2.89) hold, then I~,(E)I~
IIJ x X 1

- - ,.,']'2X2

]]C[to,to+T]<~2v~ ]Ix1 -

x2

[[C[to,to+T].

(2.101)

By Proposition 2.3.17 for every class [x(t), y(t)] of T-periodic solutions of system (2.83) the relation y(t) = (J,~x)(t) with some 2 E [0, 1] is valid. Under the conditions of Proposition 2.3.16 such classes [x(t), (J,~x)(t)] exist for all 0K2~<1 and are pairwise disjoint. In fact, the dependence of the periodic problem on the parameter 2 rather than any special behavior of the function fit, x, y) is the reason for the existence of a one-parameter continuum of classes of periodic solutions. Simple assumptions like estimates of the measure # density and Lipschitz continuity of the nonlinearity fit, x, y) used in Proposition 2.3.16 may guarantee that the set of all classes [x(t), y(t)] of T-periodic solutions is the image of the segment 0~2K 1 under a one-to-one continuous mapping. The easiest way to construct examples of periodic solution sets with any number of connected components is to consider open-loop systems consisting of the differential Eqn. L(d/dt)x --- fit, x) without y, and the relations r/(t) = (F[rl(to)]x)(t), y(t) = @(x(t)) that define variable states and outputs of the hysteresis nonlinearity. Here every periodic solution xj(t) of the differential equation (if any exists) defines a one-parameter set of classes [xj(t), (J,~xj)(t)] (0K2~1) of periodic solutions for the system.

Classes of Periodic S o l u t i o n s Now we consider individual T-periodic solutions (x(t), rl(t)) of the system (2.83) from any class [x(t), y(t)]. Simple relations between their second components r/(t) follow from Proposition 2.3.17. Recall that, by definition, (x(t), rl(t)) belongs to the class [x(t), y(t)] if and only if r/(t) = (F[tl(to)]x)(t), y(t) = (I)0/(t)) and r/(t) = r/(t + T) for all t>~to. Statements (i) and (ii) of Proposition 2.3.17 imply that for every periodic solution (x(t), rl(t)) E Ix(t), y(t)] the component r/(t) = r/(t, a, t) coincides on the domain (~, t) E Gc(xm, XM) for all t~to with the function ~/per(t, ~x, t) uniquely defined by the input x(t). Thus, the components ~/(t) = r/(t, a, t) of the representatives (x(t), rl(t)) of the class Ix(t), y(t)] differ from each other on the domain G(xm, XM) only, where these components do not change with time. That means that r/(t, :r fi) - ~/. (a, t) on G(xm, XM) for each individual solution (x(t), r/(t)). If #(G(xm, XM)) > 0, then one obtains all representatives of the class [x(t), y(t)] inserting here characteristic functions r/. (~, t) of all measurable subsets of G(xm, XM) that

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

189

satisfy a unique restriction, namely (2.99) should hold with the 2 ~ [0, 1] defined by the equality

y(t) - f f r/per(t, a, fl)d#(a, fl)+ 2#(G(xm, XM)), d JGc(Xm,XM) which is nothing other than as y - J, tx. Since there is a continuum of such functions r/. = r/~(a, ]/), each class Ix(t), y(t)] consists of a continuum of periodic solutions (x(t), r/(t)) if l~(G(xm, XM)) > 0. For example, under the conditions of Proposition 2.3.16, system (2.83) has a one-parameter continuum of classes [x,t(t), Y,t(t)] of T-periodic solutions and each of those classes is a continuum parameterized by the functional parameter r/~ Formula (2.96) implies that J2x - J0x for all 2 and the Preisach nonlinearity assigns a unique periodic output to a given periodic input x - x(t) if and only if #(G(xm, XM)) -- 0. In contrast to the situation considered above, it means that the a class [x(t), y(t)] of periodic solutions is not included in the one-parameter set of such classes [x,~(t), y2(t)] if #(G(xm, XM)) = O. Moreover, formula (2.99) implies that the class [x(t), y(t)] contains a unique element (x(t), r/per(t)) in this case, since we consider functions r / = r/(~, fl) that are, with respect to the measure #, equal almost everywhere as the same state of the Preisach nonlinearity. Thus, periodic solutions (x(t), r/(t)) such that #(G(xm, XM)) -- 0 are basically isolated (if the nonlinearity f itself is not so complicated that it 'produces' nonisolated solutions). For example, the relation #(G(xm, X M ) ) - - 0 holds if Xm < - R ~ and XM > Ra.

Guiding Functions It is instructive to modify the statements of Subsection 2.3.3 that guarantee the existence of at least one T-periodic solution for systems with the Preisach nonlinearity in order to formulate sufficient conditions for the existence of the continuum of such solutions. Under the hypotheses of each of the Propositions 2.3.5, 2.3.6, 2.3.7 the periodic solutions admit an a priori estimate llz(t)licKp of the z-component. Such an estimate implies the estimate llx(t)llc~P0 for the corresponding periodic inputs x = (c, z) of the Preisach nonlinearity, for example with P0 - plcl. If we assume that P0 < R~, in addition to the conditions of any of above propositions, then the set of periodic solutions will include the one-parameter continu u m of disjoint classes [z2(t), y,t(t)], 0K2K1. Here and henceforth we denote by [z(t), y(t)] the class of all periodic solutions (z(t), r/(t)) satisfying y(t) = (I)(r/(t)) for systems (2.104) below, similar to the notation Ix(t), y(t)] used for system (2.83). We consider in more detail how this approach works in another application based on the method of guiding functions.

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190

A scalar-valued, continuously differentiable function V(z) is called a guiding function for the system of ODEs z' = F(t, z) with z ~ Ra if the gradient VV(z) of this function satisfies for some p > 0 (VV(z), F(t, z)) < 0

if

Izl > p, t ~ E.

(2.102)

Let the continuous function F(t, z) be T-periodic with respect to t. The simplest fact illustrating the role of guiding functions in the study periodic problems is that estimate (2.102) and the relation IV(z)I --> oe as IzI --* c~ imply the existence of at least one T-periodic solution of system z' = F(t, z). An important idea that leads to less restrictive existence conditions is to use several guiding functions. From (2.102) it follows that outside some ball in Ra the vector fields VV(z) and -F(to, z) are linearly homotopic and do not have zeros for every to. Therefore if Vj(z) (j = 1 . . . . . k) are guiding functions for the system z' = F(t, z), then the index at infinity is defined for each gradient vector field VVj(z) and all these indices coincide. The common value of these indices is called the topological index of the finite set of guiding functions 111(z) . . . . . Vk (z). Such a set is called complete if IWl(Z)i-}-..--}-IWk(z)l--+

oo

as

[zl--*

oo.

(2.103)

There is a classical theorem which states that if there exists a complete finite set of guiding functions with a nonzero topological index, then the system has at least one T-periodic solution. If one uses a straightforward modification of the definition of a guiding function, then the same statement holds for systems with hysteresis nonlinearities. So, we consider the system

dz/dt y(t) ~l(t) x(t)

= = = =

F(t, z(t), y(t)), (I)0/(t)), (F[rl(to)]x)(t), (c, z(t))

Z E [~d,

(2.104)

with the Preisach nonlinearity, where F(t, z, y) =- F(t + T, z, y) is a continuous function. Assume that for j = 1 . . . . . k

(VVj(z), F(t, z, y)) < 0

if

Izl > pj, 0KyK1, t ~ ~,

(2.105)

the relation (2.103) holds and the common value of the index at infinity of the gradient vector fields VVj(z) is nonzero. Then the system (2.104) has at least one T-periodic solution (see, e.g. [61]). Combining this statement with Proposition 2.3.17 in accordance with the basic approach of this subsection, we arrive at the following conditions

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

191

for the existence of a continuum of periodic solutions. We use the notations mj and Mj for the smallest and the largest values of the guiding function Vj(z) on the ball Izl~Pj and the notation c j - {z ~ ~d " m j < . V j ( z ) < . M j } .

PROPOSITION2.3.18. Let scalar-valued continuously differentiable functions V] (z), . . . . Vk(z) satisfy (2.103) and (2.105) and the index at infinity of the gradient vector fields VVj(z) in ~cl be nonzero. Let a ball {z E ~ 9 Izl~p.} contain the set G1 D ... D Gk and let R~ > p.Ici. Then the set of T-periodic solutions of system (2.104) includes a one-parameter continuum of disjoint classes [z2 (t), y2 (t)] (0~2~1). We remark that relation (2.103) implies that the set G1D... NGk is bounded. The conditions of Proposition 2.3.18 imply the inclusion z(t) E G1Vi... DGk, t>~to for the z-component of each periodic solution of system (2.104) and therefore the a priori estimate Ilz(t)Ilc~p. for all such solutions. If IV(z)] --~ cx~ as ]z] --> cx~ (i.e., k = 1) then the index at infinity of the vector field VV(z) is either I or -1. There are many other simple conditions which guarantee that the topological index of the complete set of guiding functions Vl(Z) . . . . . Vk(z) is different from zero. For example, if one of these guiding functions is even, then the topological index is odd. The proofs of Propositions 2.3.16 and 2.3.18 are based on the same idea. Using the operator (2.96), one constructs an operator equation depending on 2, whose solutions define the classes Ix(t), y(t)] of T-periodic solutions of the system (2.83) (the classes [z(t), y(t)] for system (2.104), respectively), then proves the existence of a solution for each 2 E [0, 1] and shows that the solutions are different for different 2. For example, one can use the operator equation

~(t) =fit, (H~)(t), (,.72H~)(t))

(2.106)

in the space L2[0, T] to prove Proposition 2.3.16. From the definition of the linear operator H = H(T) and Proposition 2.3.17 it follows that each solution ~ = ~(t) of (2.106) defines the class Ix(t), y(t)] of T-periodic solutions of the system (2.83) with x(t) -- (H~)(t), y(t) = (J,~x)(t). The existence of a solution ~,~ of (2.106) for each 2 E [0, 1] follows from (2.84) and the first of the estimates (2.87) by the Schauder principle. The second of estimates (2.87) and the a priori estimate ]lxllcKklb~/T/(1 - qk) of all T-periodic solutions of (2.83) imply that all the classes [x,l(t), y,~(t)] of such solutions defined by the solutions ~,~ of (2.106) are disjoint. If the additional assumptions of the second part of Proposition 2.3.16 hold, then Eqn. (2.106) satisfies the conditions of the contraction mapping principle and therefore has a unique

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CHAPTER 2 Differential Equations with Hysteresis

solution r for each ;~ ~ [0, 1]; moreover, the map 2 --, r is Lipschitz continuous. For the proof of Proposition 2.3.18, one can use the operator equation

z(t) = z(O) +

~0t F(s, z(s), (J2x)(s))

ds

with

x(t) = (c, z(t))

in the space C T ( [ 0 , T]; •a) of continuous functions z - z(t) satisfying z(0) = z(T) with the uniform norm. The proof of the existence of a solution z2 of this operator equation for each ;~ ~ [0, 1] follows standard lines of the guiding functions method. For details of the proofs of Propositions 2.3.16 and 2.3.18 and for the proof of Proposition 2.3.17 we refer to [162].

A u t o n o m o u s Systems Our approach also works in periodic problems for autonomous systems, and here it leads to statements on the existence of continua of cycles. For the same reasons as in problems on forced periodic oscillations, the continua of cycles have a similar type of one-parameter structure as described above if the a priori estimate lixllc < R~ holds for the cycles. We give one particular statement. Consider the system

L (-~)x y(t) ~l(t)

= = =

f(x, y(t)), (I)(r/(t)), (F[~l(to)]x)(t)

(2.107)

with the Preisach nonlinearity. Its solutions of the form (x(t), ~l(t)) =(x0, r/0) are called stationary, since for such solutions y(t) = Yo with Y0 = 9 (r/0). We are interested in conditions that guarantee the existence of nonstationary periodic solutions (x(t), rl(t)) -- (x(t + T), ~l(t + T)) of any period T > 0. Since system (2.107) is autonomous, those periods are u n k n o w n a priori. Furthermore, one should take into account that the set of nonstationary periodic solutions contains, together with any solution (x(t), r/(t)), all its time shifts (x~(t), rl~(t)) = (x(t + ~), r/(t + ~)) with 0 < ~ < T. Passing to the equivalence classes [x(t), y(t)] = {(x(t), ~/(t)) : y(t) = (I)(r/(t))} of periodic solutions of the system (2.107), we see that the time shifts of solutions from a class Ix(t), y(t) ] by the time r form the class [x(t + z), y(t + ~)]. In order to simplify the description of shift invariant structures of periodic solution sets, we introduce closed curves 7 = {(x(t), y(t)) ~ [~2 . 0Kt K T} on the i n p u t output plane of the Preisach nonlinearity and say that the curve 7 represents the class of all periodic solutions from the join Uo<.~
2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

193

The curve 7 is called degenerate if (x(t), y(t)) =_ (xo, Yo) and nondegenerate otherwise. We consider stationary solutions in more detail. For 0<~A~
-

#({(a,/~)

9 ~ < fl < ,o}) + ,~ ~ ( { ( ~ , / ~ )

9 a < ,o < fl}).

(2.108)

This function is continuous with respect to the set of arguments A, p. As we know, the Preisach nonlinearity assigns stationary outputs to stationary inputs for any admissible initial states. Moreover, from conclusion (i) of Proposition 2.3.17 it follows that the values of all the outputs y(t) -- Yo corresponding to the input x(t) - xo form the segment {Y0 = ~b,t(x0) : 0<~A~
(2.109)

If L(0) ~- 0 and the continuous functionf(x, y) is bounded, then Eqn. (2.109) has at least one solution p - p*(2) for each 2 e [0, 1] with Ip*(2)l~pl = sup ~(x, y)llL(0)1-1, and therefore the system (2.107) has stationary solutions. In particular, if there is a unique solution p = p* (A) of (2.109) for each 2 and #({(a,//) 9 ~ < p < fl}) > 0 whenever p e [ - P l , Pl], then the points representing all the classes of stationary solutions of system (2.107) in the plane (x, y) form the continuous curve F -- {(x0, Y0) - (p*(2), q~,t(p*(A))) 9 0~<~.~<1} without self-intersections. N o w we are ready to formulate a statement on the existence of nonstationary solutions. Let fix, y) be a bounded continuously differentiable function satisfying the global Lipschitz condition

~(Xl, Yl) --f(x2, y2)l<~qlxl -- x21-+- qllyl -- Y21,

xj, yj 6 ~.

(2.110)

We assume that the relations (2.88), (2.89) are satisfied for the Preisach nonlinearity. It is easy to see that if there are no zeros of the polynomial L(p) on the imaginary axis and the coefficients q, ql in (2.110) are small enough, then the system (2.107) does not have nonstationary periodic solutions. Let us assume that the polynomial L(p) has a pair of the conjugate imaginary zeros 4-woi, and the so-called nonresonance condition L(nwoi) g= 0

for

n - 0, 4-2, 4-3 . . . .

(2.111)

holds. Define on the positive semiaxis r > 0, the function d(r) -

/0

sin t fir sin t, 1) dt -

/0

sin t f ( - r sin t, O) dt.

(2.112)

194

CHAPTER 2 Differential Equations with Hysteresis

PROPOSITION2.3.19. Let the polynomials L(wi) and ~mL(wi) of the variable w have a pair of common zeros • (wo > O) of the same odd multiplicity. Let the nonresonance condition (2.111) hold. Suppose that estimates (2.88) and (2.89) are valid, that the bounded continuously differentiable function fix, y) satisfies the Lipschitz condition (2.110) and function (2.112) satisfies Id(r)l~c > Ofor r~r,. Then there are numbers q0 > 0, Wl e (w0/2, w0) and w2 > w0, such that if q + vooql < q0, the following statements are valid:

(i) The set of all stationary solutions of the system (2.107) is represented on the plane (x, y) by a continuous curve F, which may degenerate to a unique point. (ii) If there is a point (x*, y*) o f f such that

ef

d(r.) ~xx(X*, y*) < 0,

(2.113)

then the system (2.107) has at least one class of nonstationary periodic solutions represented by a nondegenerate closed curve ~; = {(x, y) e R 2 9 x = x(t), y = y(t), t e [0, T)} with a period satisfying 2~/w2~T~2rc/w]. (iii) If estimate (2.113) is valid for all points of the curve F, then there exist numbers r], ro > 0 such that the x-components of all periodic solutions of the system (2.107) with periods 2~/w2<~TK2rc/wl satisfy IIx(t) - YClIL2[O,T]~rl,

]Ix(t)IIc[o,T]~ro,

(2.114)

where Ycdenotes the average value of the function x(t) over its period. If in addition Ra > ro, then the set of all nonstationary periodic solutions of the system (2.107) includes a one-parameter continuum of disjoint classes represented at the plane (x, y) by nondegenerate closed curves ?,~ where 2 ranges over the segment 0K2~l. These curves are different for different 2 and the periods of all periodic solutions represented by them satisfy 2rc/w2 ~ T K2rC/Wl. By conclusion (iii), the x-components of all periodic solutions represented by the curves 7,l satisfy the estimates (2.114) and consequently are separated from the one-dimensional subspace of constant functions and from infinity in L2, C and other functional spaces, which implies, in particular, that such periodic solutions are separated from stationary ones. Additional assumptions about the functionf and the measure # of the Preisach hysteresis nonlinearity guarantee that the one-parameter set of curves ?,t represents all periodic solutions of the system (2.107) and these curves depend continuously on 2. Formulas for the numbers q0, w], w2 (they are defined by the polynomial L(p)) and estimates (2.114) can be given explicitly. We do not present

195

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

these cumbersome explicit expressions here, 3 but consider, as an example, the equation x'" + x" + x' -t- x =fix, y(t)). (2.115) The polynomial L(p) = p3 + p2 + p + I has the pair of simple roots +i,

L(wi) = (wi + 1)(1

3mL(wi) = w(1

- w2),

- w2),

w0-1

and 1/2 < Wl < 1 < w2. For example, if the Lipschitz estimate (2.110) holds with q = 0.305 and v~q1~<0.295, the Eqn. p = tip, qbx(p)) has a unique solution p = p~ for some 2 ~ [0, 1] and for all sufficiently large r

d(r) -~x

<

o,

[d(r) l~c > O,

then Eqn. (2.115) with the Preisach nonlinearity has nonstationary T-periodic solutions with 5.575~
L ( d ) x - g(x) + qly(t). Then

d(r) = 2ql + 4

f rcl2 J0

sin t godd (r sin t) dt,

where godd (x) -- (g(x) -- g(--x))/2 is the odd part of g. If godd (x) ~ go as x --, +oe, then d(r) ~ do = 2ql + 4g0 as r --~ +cx~. If do ~ 0, then the condition Id(r)l>~c > 0 of Proposition 2.3.19 is satisfied for all sufficiently large r.

2.3.6 AVERAGING PRINCIPLE General Description The averaging principle, see [146] and bibliography therein, is a powerful tool in nonlinear dynamics. Here we apply averaging theory to differential equations with hysteresis operators on the right-hand side. We remind the reader, that the standard averaging principle is applicable to the ordinary differential equation of the form z ' = ef(t, z) 3Further details, statements and proofs can be found in [163].

(2.116)

196

CHAPTER 2 Differential Equations with Hysteresis

where the function f is periodic or almost periodic in t. The idea is to introduce the function

F(z)-

lim Z1 IJoT f(t, z) dt

T~oo

(2.117)

and then to consider the equation

z' = eF(z)

(2.118)

as an approximation to (2.116). It has been proved by N. Bogoljubov, that the averaged Eqn. (2.118) is a good approximation to (2.116) over intervals of the size O(1/e), which is the characteristic time for Eqn. (2.116). Now we consider the system with hysteresis given by

~(t) = el(t, z, y(t)), y(t) = (W[rlo]x)(t),

z ~ ~N,

(2.119)

x(t) = g(t, z(t)).

(2.120)

Here y(t) is the output of the hysteresis operator W with the initial state t/0, and e > 0 is a small parameter. The functions f and g are almost periodic in t. We note that the system (2.119), (2.120) is not a particular case of (2.13). We explain how these Eqns. (2.119), (2.120) 'come to life' in the analysis of Eqn. (2.13) below.

Averaged Hysteresis Operators First we need some analysis of hysteresis nonlinearities for inputs that are almost periodic functions depending on a slowly changing parameter s ~ ~+. Formally, the function u(t, s) is given in such a way that for each fixed s ~ R+ it is almost periodic as a function of t ~ ~ and the map s --+ u(.; s) is uniformly continuous on any bounded interval [0, T]. For each s ~ ~+ we define the sequence of inputs

VSk(t) = u(t, min{t/k, s}),

t~>0,

to the operator W[~/0] and denote z~ (t) = W[r/0]v~(t). Since v~(.) converges to an almost periodic function, the same is true for the output z~ (t); by y~ (t) we denote the almost periodic limit of z~ (t). Note that y~ (t) is defined for all t ~ R. If, for each s ~ ~+, the sequence y~ (t) converges to a function ys(t) as k --+ oo uniformly in t ~ 0~, then ys(t) is almost periodic and we say that the map y(s) = ys(') from ~+ to the space of almost periodic functions is the output of the averaged hysteresis operator W'[r/0]. We return to the system (2.119), (2.120) The solutions are traced on a finite interval of the 'slow' time ~ = et. The functions fit, x, z) and g(t, x)

197

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

are assumed to be almost periodic in t and uniformly continuous in all the variables. We denote 1 T + ~ -T /0 fit, z(s), W*[~lo]X s (t)) dt, (I)z(s) = T ~lim

x s(t) = g(t, z(s)).

The averaging m e t h o d replaces (2.119) by the equation

~(t) = ~(bz(t),

(2.121)

whose solutions converge to the corresponding solutions of (2.119) as e --+ +0, if the averaged operator W* exists. Existence of the averaged operator W* has been proven for m a n y classical models, including the Preisach nonlinearity and hysterons. We note especially Vladimirov's results concerning the multidimensional hysteresis operator [111] and their applications to IT problems.

Averaging of Preisach Nonlinearity Let us give a simple example of an explicit description of the limit nonlinearity. Let P be a Preisach nonlinearity and let function u(t, s) be T(s)periodic. We suppose also that the set f~ coincides with the half plane It is sufficient to consider the case w h e n the initial state ~/0 is the characteristic function of a set S, which looks like the shaded set in Fig. 2.35(a). That is, S is located to the left of the line a - ] / a n d below a continuous linear curve L = Ls, with a non-positive slope. We denote the set of states ~/describing the sets S as above by (~.. Additionally, we assume that the coordinates of the intersection of L with the diagonal coincide with the current value of the input x. We will discuss this type of initial condition in more detail in Section 2.4.1. N o w we will give a correspondence between a given function u(t, s) and a varying set S(s). For each s>~0 introduce u_ (s) = min u(t; s) t

and

u+(s) = max u(t; s).

(2.122)

t

For simplicity we consider only the case w h e n both these functions are piecewise monotone. We define S(0) as the part of S which is located to the left from the vertical line :~ = u_ (0), that is,

s(0)see Fig. 2.35(b).

< u_(0)}.

CHAPTER 2 Differential Equations with Hysteresis

198

FIGURE 2.35 The sets S, S(0). Suppose that S(cr) is defined for 0~cr~s0 and that the functions (2.122) are monotone at the interval so Ks. Let us consider four cases: Case Case Case Case

1. 2. 3. 4.

u_(s)~u_(so), u_(s)~u_(so), u_(s)Ku_(so), u_(s)Ku_(so),

and and and and

u+(s)Ku_(so). u+(s)~u_(so). (u_(s), u+(s)) r S(so). (u_(s), u+(s)) ~ S(so).

Figure 2.36 provides the 'picture definitions' for the corresponding set S(s), which is the union of the hatched and shaded regions. The bold line is given by the parametric Eqn. ~ = u_ (rT),//-- u+ (rr). For instance in Case 1, S(s) is the union of the set S(so) and the set {(a, ]/) "u_(so)~o~ < u_(s), and a < / / < u+(a)}. The limit Preisach nonlinearity P*[~/0] can be described by

(P*[~lo]xS)(t) = (P[rl(s)]xS)(t), where x s (t) = u(s, t) and r/(s) is the characteristic function of the set S(s).

Numerical Example Here we demonstrate how the techniques described above work in a particular example. Consider our standard scalar differential-operator equation of the form x"+x = ey(t), y(t) -- (I)0/(t)) 2, (2.123) r/(t) = (F[r/0])x(t).

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

FIGURE 2.36

199

Set S(s).

w h e r e O(r/(t)) is a Preisach n o n l i n e a r i t y w i t h a m e a s u r e #. We s u p p o s e that p is a L e b e s g u e m e a s u r e on the triangle A ( - 1 , 1) = {-1K0cKflK + 1} a n d zero a n y w h e r e else. U s i n g the s u b s t i t u t i o n x = r cos(0 + t),

x' = - r sin(0 + t)

the Eqn. (2.123) can be r e w r i t t e n in the f o r m of the s y s t e m

dr/dt dO/dt y(t)

= =

r/(t)

=

-

- e s i n ( t 4- O)y(t), ecos(t 4- O)y(t)/r, O(r/(t)) 2, (F[~/0])(r(t) cos(0(t) + t)).

C o n s i d e r the initial c o n d i t i o n s r(O) = ro e (0, 1],

0(0) = O,

rt(O) = ~o.

(2.124)

CHAPTER 2 Differential Equations with Hysteresis

200

FIGURE 2.37 Set S(s) for (2.124) is hatched; all possible sets described by characteristic function r/belong to the union of the filled areas.

For these initial conditions, the system (2.124) has a unique solution. The initial state of the Preisach nonlinearity ~/0 is chosen to be u n i o n of the s h a d e d and hatched areas on Fig. 2.37. Using the averaging m e t h o d from the previous subsection and the properties of the Lebesgue measure, we can write a differential equation for the average ~ of r

f2~+o

1 d~/dt= _ m2~ Jo

e Sin(S -}- 0)(I)((1P[r/0])(~ cos(0 if- S))) 2 ds

1 fo ~ e sin(s)(c q- -~~.2 (cos(s) - 1)2) 2 ds = - 2---~ ~2 1 f2~ 2re e sin(s)(c + 2~z - ~ ( 1 - cos(s))2) 2 ds

4~2(C if_ ~2) 3~ Here c > 0 and ~ is decreasing with time, that is the case of the evolution of a Preisach nonlinearity s h o w n in Fig. 2.35(a). The solution x - r sin(t + 0) has symmetric m i n i m a and m a x i m a which means that c is the area in the hatched p o l y g o n in Fig. 2.37. Thus, c - 1 - ~2 and the equation above can

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

201

be written as d~ dt

4~ 2 =

3rt

~(0) - r0

and has the solution 3~zr0 ?(t) = 3rr + 4erot"

Similarly, one can write d o w n and solve the equation for the average of 0. From the averaging principle d0 1 f020 ~ cos(s + 0)(I)((F[r/0])(r cos(0 + s))) 2 ds d--~ = 2~z~ = 2 1~ fo ~ e cos(s)(c q- -~~2 (cos(s) -- 1)2) 2 ds

1 ~2~r 2~z~

~2 e cos(s)(c + 2r 2 - ~-(1

- cos(s))2) 2 ds

= e?(c + 72). N o w the equation for dO dt

= ~

3~ro 3~z + 4roet '

o(o) - o

has the solution 0(t) - -3/4(zt log(3r0 - zr log(3rc + 4er0t)). In Fig. 2.38 we c o m p a r e the results of the averaging m e t h o d with straightforward numerical integration. We consider two values of p a r a m e t e r e: E = 0.01 a n d e = 0.001. Initial conditions are the same in both cases, r0 = 1 a n d we c o m p a r e the solutions on the time interval [0, 1000].

A p p l i c a t i o n to Q u a l i t a t i v e A n a l y s i s of S y s t e m s with Hysteresis We will d e m o n s t r a t e h o w it could w o r k by a particular example. We consider the scalar differential-operator equation of the form x" + x -

qb(t) + ey(t),

y(t) - (I)0/(t)),

(2.125)

r/(t) - (F[r/0])x(t),

(2.126)

CHAPTER 2

202

Differential E q u a t i o n s w i t h H y s t e r e s i s

in ! ! ....

0.2

0.5 0.1

o

!

C 0.1

-0.5

0.2

-10

|

--

4'0

i

6'0

i

i

900

(a) c = 0.01, t r [0, 100] 1

i

0.5

f

nir,f

|

1

10(

920 940 960 980 (C) c=O.O01, t E [900, 1000]

1000

-

0.5 :

i

0

2

-0.5 -1

920 940 960 980 (b) ~ -- 0.01, t E [900, 1000]

i 0

i

20

I

40

60

-0.5

I i

i~

.

,

80

-1 90O

100

(C) c=O.O01, t E [0, 100]

FIGURE 2.38 Dashed oscillating-numerical solution; solid oscillating-solution of averaging equations. w i t h the Preisach nonlinearity. We will s u p p o s e that the c o r r e s p o n d i n g m e a s u r e # is finite, positive, absolutely continuous w i t h b o u n d e d density, a n d defined in the half plane a < ft. Suppose that the external force ~b(t) is a s u m of periodic functions qbi(t), i = 1 . . . . . n, w i t h periods Ti, for w h i c h n

2~

Z ki-~ii + kn+l ~ 0

(2.127)

i=1

for a n y integers kin, m = 1 . . . . . n + 1. As w e know, this equation for a n y initial conditions x(0) = x0, x'(0) = x 0' has a u n i q u e solution x(t , x(O) , x'(O) , e). We denote b y ui(t), i = 1 . . . . . n, the periodic solution (it is unique) of the equation u" + u = qbi(t ), and we set n

u(t) = Z i=1

ui(t).

(2.128)

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

203

By m e a n s of the usual substitution x = u(t) + r cos(0 + t),

x' = u' (t) - r sin(0 + t)

the Eqn. (2.125) can be rewritten in the form of the system dr d--t = - ~ sin(t + O)y(t), dO d---t-= e cos(t + O)y(t)/r, y(t) = ~(r/(t)),

~/(t) = (F[~/0])x(t), x(t) = u(t) + r(t)cos(0(t) + t).

For any initial conditions r(0)=r0>0,

0(0)=00,

~/(0)=~/0

this system has a u n i q u e solution (r(t; ro, 0o, tl o, e), O(t; ro, 0o, ~1o, e), r/(t; r0, 00, ~/0, e)). We denote by f~(r), r > 0 the following d o m a i n in the half plane ct < fl: f~(r) =

~ <

m a x ui(t) + r or e > i=1

m i n ui(t) - r

.

i=1

For r > O, (e, fl) ~ f~(r) and 0 < Oi < 2re, i = 1 . . . . . n + 1, we set F(r; ~, fl, 01 . . . . On+l) =

lim

to---+ - - ~

(R[z(e, fl); ~, fl)x) (0)

n ( x(t) = ~

ui

t -- to -t- - ~ T i

-t- r cos(t - to -t- On+l).

i=1

The limit of the right-hand side always exists. For fixed values of r > 0 a n d (e, fl) e f~(r) it is convenient to consider this function to be defined on the (n + 1)-dimensional torus H n+l. For r > 0 we set P(r) = ~

(r)

d/.t(0c,fl)fH

n+l

F(r; ct, [3, 6)) sin On+l d |

Since in any d o m a i n 0 < r < R, P(r) satisfies a Lipschitz condition, each Cauchy problem d? m

= -ePff),

dt has a unique solution ?(t; ?0, e).

r(-0) = r0

204

CHAPTER 2 Differential Equations with Hysteresis

THEOREM2.3.4. The function P(r) is strictly positive for r > O. COROLLARY2.3.10. Let R > O. Then to each ~ > 0 there correspond numbers l eo > 0 and L > 0 such that for 0 < e < e0, and Ix0[ + Ix0] < R Ix(t; x0, x~) - u(t)] + Ix'(t; x0, x~) - u'(t)] < 6,

t > L/e.

2.3.7 BIFURCATIONANALYSIS In Section 2.2 we observed a rich bifurcation behavior of Eqn. (2.9) depending on the parameter b. Here we survey some statements about bifurcation points. We restrict ourselves to one type of bifurcations, namely Hopf bifurcations in autonomous systems, discussing mainly sufficient conditions for some parameter value to be a bifurcation point and the structures of solution sets, to some extent, and omitting the proofs.

H o p f Bifurcation Let some autonomous system depending on a scalar parameter b have an equilibrium z.(b) for each parameter value from some interval (b_, b+) and let those equilibria form a smooth curve in the phase space Z of the system. We say that Hopf bifurcation occurs and a parameter value b0 is a Hopf bifurcation point if for any sufficiently small r > 0 one can find a parameter value br E (bo - r, bo + r) such that the system with b -- br has a cycle which lies in the ball of radius r centered at the equilibrium z.(br) (this type of weak definition of Hopf bifurcation was introduced in [164], where nonsmooth problems were studied for the first time). Freely speaking, we consider bifurcation of a small cycle from an equilibrium point of the autonomous system. The simplest example is the equation x" + bx' + x = fix, x', b)

(2.129)

with f satisfying flZl, Z2, b) = o(]Zll q-IZ21) as Zl, Z2 --+ 0 uniformly with respect to b. Here the origin is an equilibrium of the equivalent planar firstorder system for every b; the value b = 0 is a unique Hopf bifurcation point. Figure 2.39 presents a typical structure of the set of cycles in the product of the phase space ~2 of the equivalent system and the axis of parameter values b in the case of a sufficiently smooth f. The cycles form a smooth two-dimensional surface ~ (a 'cup') passing through the origin (note that the picture is local both in b and z = (Zl, z2), i.e. we consider cycles lying in a small vicinity of the zero equilibrium and b small in absolute value only). Appropriate nondegeneracy conditions guarantee that ~ is tangent

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

205

I Z2 s ss

E1 %

FIGURE 2.39 Hopf bifurcation.

to the plane b = 0 and lies on one side of it. In Fig. 2.39, s lies to the right of this plane, which means that the system has no small cycles for b < 0 and has such cycles for small b > 0; for b = b' the cycle is a cross section of with the plane b = b'. Under the conditions of the classical Hopf Theorem (see e.g. [165]) a similar picture is observed for a u t o n o m o u s systems

d z / d t = F(z, b),

z c ~a,

(2.130)

of ODE of any dimension in some vicinity of the equilibrium z,(b). Let us recall more precisely the algorithm to determine bifurcation points by linearization of (2.130). Since the change of variables z ~ z + z, (b) places the equilibrium at the origin, let us assume without loss of generality that z, (b) ~ 0, i.e. system (2.130) has zero equilibrium for all b. In this situation, we call b0 a Hopf bifurcation point for system (2.130) with a limit period To > 0 if for any sufficiently small r > 0 the system has a periodic solution zr(t) of a minimal period Tr > 0 for some b = br and max Izr(t)l ~ O,

br ~ bo,

Tr ~ TO

as

r ~ 0.

t~

We remark that periods Tr are not k n o w n a priori and that we use an auxiliary parameter r. The reason w h y the parameter b is not good for the parameterization of the set of small cycles becomes clear if one considers linear equations, for example x" + bx' + x = 0, i.e. Eqn. (2.129) w i t h f ~ 0.

206

CHAPTER 2 Differential Equations with Hysteresis

Here all the cycles exist for one parameter value b0 = 0, which is a unique Hopf bifurcation point (the surface ~ in Fig. 2.39 becomes a piece of the plane b = 0). Suppose F is differentiable with respect to z at the point z = 0 uniformly with respect to the parameter b, i.e. F(z, b) = A(b)z + g(z, b) where suPb ]g(z, b)l/JzJ ~ 0 as izi --* 0, and let the matrix-valued function A(b) and the vector-valued function g(z, b) be continuous. Assume that the matrix A(bo) has a pair of simple imaginary eigenvalues 4 +iwo (w0 > 0) for some b0, which implies that A(b) has a unique pair of simple eigenvalues rr(b) 4- iw(b) close to 4-iwo for b close to b0. Let the derivative rr'(b0) of the real part of these eigenvalues at the point b = b0 exist and the so-called transversality condition cr'(b0) ~= 0 (2.131) and nonresonance condition

iwok ~ Sp(A(b0))

for all

k ~ 77, k ~= 4-1

(2.132)

hold, where Sp(A) is the spectrum of A. Then b0 is a Hopf bifurcation point for system (2.130) with the limit period 2rc/wo (for example, this follows from the results of [164]). If in addition F is sufficiently smooth, then according to the Hopf Theorem the small cycles form a two-dimensional smooth surface in the product of the phase space ~c/of (2.130) and the axis b. Moreover, information about the Taylor expansion of F at the origin allows us to study further important problems including stability of the cycles, but we do not consider those problems here.

Bifurcation in Systems with Preisach Hysteresis Consider the autonomous system

dz/dt y(t) rl(t) x(t)

= = = =

F(z(t),y(t),b), (I)(r/(t)), (F[rl(to)]x)(t), (c, z(t))

z ~ ~a (2.133)

with the Preisach nonlinearity, where the first equation depends on the scalar parameter b; the function F is supposed to be continuous. We use notation similar to that of Subsection 2.3.5. Solutions of the form (z(t), r/(t)) = (z0, r/0) are called stationary, the point (z0, ~/0) in the product ~a x (~. is called an equilibrium of the system. By a cycle we mean the trajectory of any nonstationary periodic solution (z(t), rl(t)) =_ (z(t + T), rl(t + T)) 4The existence of imaginary eigenvalues of A(bO) is a necessary condition for b0 to be a Hopf bifurcation point.

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

207

in ~a x (~., with any T > 0; actually, the cycle is a c o m m o n trajectory for all the time shifted periodic solutions (z~(t), ~/~(t)) = (z(t + ~), rl(t + ~)), 0~ (c, z0). Since the Eqn. y0 = ~0/0) with a given Y0 has a solution r/0 in this class of admissible states if and only if #({(~, f l ) : ~ < fl < (c, z o ) } ) ~ y o ~ # ( { ( ~ , fl): ~
(2.134)

we conclude that system (2.133) has an equilibrium if and only if the Eqn. F(zo, Y0, b) = 0 has a solution (z0, Y0) satisfying (2.134). Similarly to the definition for ODEs above, we call b0 a H o p f bifurcation point for system (2.133) with a limit period To if for any sufficiently small r > 0 the system has a nonstationary periodic solution (zr(t), rlr(t)) with a period Tr > 0 for some b = br and max I z r ( t ) - z o l ~ 0, max/9 l(r/r(t ), r/0) ~ 0, br -+ bo, Tr ~ TO t t

as

r --, 0,

where (z0, ~/0) is an equilibrium for b = b0 and/91 (., .) is the metric (2.20) in the state space @. of the Preisach nonlinearity. To formulate the simplest statement on H o p f bifurcation for the system (2.133), suppose that there are z0, Y0 satisfying (2.134) such that F(z0, y0, b) - 0

for all b.

Therefore system (2.133) has an equilibrium at the same point (z0, r/0) ~a x @. for all b. A s s u m e that the function F can be linearized at the point (z0, Y0), more precisely F(z, y, b) - A ( b ) ( z - zo) + a(b)(y - Yo) 4- g(z, y, b),

(2.135)

with the matrix A = A ( b ) and the vector a - a(b) d e p e n d i n g continuously on b and lim maxlb-b01~R Ig(z, y, b)l - 0 (2.136) z--,zo, y-*yo Iz - z01 + lY - Y01 for some R > 0. In this situation, H o p f bifurcation points for system (2.133) are d e t e r m i n e d by the matrix A (b) in the same w a y as for ODE. PROPOSITION 2.3.20. Let relations (2.135), (2.136) be valid and strict estimates #({(a, fl) 9 a < fl < (c, z0)}) < Y0
CHAPTER 2 Differential Equations with Hysteresis

208

hold in place of (2.134). Let the measure # have a bounded density with respect to the Lebesgue measure. Suppose that the matrix A(bo) has a pair of simple eigenvalues • that the transversality condition (2.131)and the nonresonance condition (2.132) hold. Then bo is a Hopf bifurcation point for the system (2.133) with a limit period 2~/wo. As an example, consider the equation x" + bx' + x = 2y(t) - 1 where x(t) and y(t) are the input and output of the Preisach nonlinearity. If #({(a, f l ) : a < f l < 0 } ) < l / 2 < # ( { ( a , fl):a 0 for some b = br and max [zr(t)- Zr(Z)[--+ cx~, br ~ bo, Tr ~ TO as t,'~

r ~ cx~.

(2.138)

In other words, the system has cycles with arbitrarily large amplitudes of the z-component and periods close to To for some parameter values arbitrarily close to b0. Suppose that the function F can be represented as

F(z, y, b) = A(b)z + g(z, y, b),

(2.139)

lim max Ig(z, y, b)] = 0 Izl-~c~ Ib-bol<~R,lYI~R~ ]Z[

(2.140)

where

for some R > 0 and every R1 > 0. Then the derivative A(b) of F with respect to z at infinity plays the same role in problems on Hopf bifurcation of large cycles as the derivative at the equilibrium point in local problems above.

209

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

PROPOSITION2.3.21. Suppose that relations (2.139), (2.140) hold, the matrix A (bo) has a pair of simple eigenvalues +iwo, the transversality condition (2.131) and the nonresonance condition (2.132) are valid. Then bo is an asymptotic Hop]: bifurcation point for system (2.133) with a limit period 2n/wo. This proposition is a consequence of a more general statement of [166]. Proposition 2.3.20 and some further statements on Hopf bifurcation from an equilibrium can be found in [167].

Bifurcation with Respect to Internal Parameter As in Subsection 2.3.5, let [z(t), y(t)] denote the class of periodic solutions (z(t), rl(t)) = (z(t + T), rl(t + T)) (t~to) such that y(t) = (I)(r/(t)) and let us say that the closed curve 7 - {(z(t), y(t)) 9 0~t < T} in the space Ea+l of pairs (z, y) represents the periodic solutions of the join Uo<~
dz/dt y(t) rl(t) x(t)

= = = =

F(z(t), y(t)), (I)(r/(t)), (F[ri(to)]x)(t), (c, z(t))

ZE[~ d, (2.141)

that does not depend explicitly on parameters. Nevertheless, as we already know from considerations in Subsection 2.3.5, problems on cycles of this system contain the parameter 2, which we call 'internal'. This results in the fact that each closed curve 7 c ~a+l representing a class of cycles (if any exists) such that the amplitude of their z-component is smaller than the characteristic size of the Preisach nonlinearity is generically included in the one-parameter continual set of such curves 7,~; a typical example is presented by the conclusion (iii) of Proposition 2.3.19. It turns out that the curves 7 representing small cycles of the system (2.141) can accumulate at its equilibrium and form the structure similar to that observed for Hopf bifurcation problems for ODEs with a scalar parameter. Moreover, sufficient conditions for the existence of those structures for system (2.141) with hysteresis are close to the classical assumptions of the Hopf Theorem. Let us consider one example. Let the Eqn. F(z, 2) = 0 define a continuous branch z = ~p(2) of the implicit function on some open interval A*. Denote by F the curve (z, y) = (~p(2), 2), 2 E A* in [~d+l and assume that for each point (z0, Y0) ~ F relations (2.137) hold, which implies that this point represents the class [z0, Y0] of equilibria of system (2.141). We shall suggest sufficient conditions that

CHAPTER 2 Differential Equations with Hysteresis

210

guarantee the existence of a one-parameter set of closed curves ~ with vanishing diameters that accumulate at one of the points Ado = (z0, Y0) 6 F. The density of the m e a s u r e / t of the Preisach nonlinearity will be everywhere assumed to be bounded. Let us suppose that the function F(z, y) is continuously differentiable with respect to z and Lipschitz continuous with respect to y in some neighborhood of the curve F. As usual, set A(2) = 8F/8z(q)(2), 2),

2 e A*.

Suppose that at some point .Ado = {q~(20), 2o} e F the spectrum Sp(A(20)) of the d x d matrix A(20) = 0F/8z(q~(20), 2o) contains a pair of simple eigenvalues +iwo (w0 > 0) and the nonresonance condition iwok r Sp(A(20)) holds for all integer k 4= +1. Consider the pair of the simple eigenvalues cr(2) + iw(2) of A(2) satisfying cr(20) 4- iw(20) = +iwo, where the real continuous functions a(2), w(2) are defined in some neighborhood of 2o. PROPOSITION2.3.22. Let the function r~(2) take values of both sign in each neighborhood of 2 = 20. Then for every small r > 0 system (2.141) has a class [Zr(t), yr(t) ] of nonstationary periodic solutions of a period Tr, all the closed curves ~ r = { ( Z , y) 6 [ ~ d + l . z - - Zr(t), y -- yr(t), t ~ [0, Tr)} are different for different r and max ]zr(t)- q~(,~0)] ~ 0, t

max lyr(t)- ,~0] ~ 0, t

Tr --* TO = 2~/wo

(2.142) as r --+ 0, i.e. the curves ~r shrink to the point .Ado and periods approach To. Particularly, the conditions of this proposition are satisfied if the function r~(2) is differentiable at the point 2 = 20 and the transversality condition r~'(20) ~= 0 holds, which one can consider as the main case. The simplest example of system (2.141) satisfying the conditions of Proposition 2.3.22 is x" + (y(t) - K)x' + x = g(x, y), where x and y are the input and output of the Preisach nonlinearity and g(x, y) = o(]xl + lYl) as x,y --+ 0. If u ( { ( ~ , / ~ ) : a < / ~ < 0}) < ~: < u ( { ( a , / ~ ) : a < / ~ , a < 0})

then this second-order equation has the continuum of disjoint classes [Xr(t), yr(t)] (0 < r < r0) of nonstationary periodic solutions with periods Tr such that maxt Ixr(t)l --+ 0, maxt lYr(t) - tel ~ 0 and Tr --+ 2~z as r --+ 0. If the function F(z, y) is sufficiently smooth in a neighborhood of the point M0, then the functions r~(2), w(2) are continuously differentiable and F(q~(2) + ~, 2 + v) = A(2)~ + a(2)v + g(2, ~, v)

(2.143)

2.3 EXAMPLESOF RIGOROUS CONCLUSIONS

211

for 2 close to 20 and ( ~ ~a, v ~ R small in absolute value. Let the matrixvalued function A(2) and the vector-valued function a(2) be Lipschitz continuous and let the residual term g satisfy uniformly in 2 ]g(2, ~1, Vl)]~ o(p),

(2.144)

]g(21, ~1,121) - g(22, ~2,122)]~ o(P)]~,l-22]nus(P) (]~1-~2] if- ]121-122])

(2.145)

whenever ]~jl, ]vj]~fi with e(p) = o(1), p ~ 0. In this situation, Proposition 2.3.22 is supplemented by the following statement. PROPOSITION2.3.23. Let relations (2.143)-(2.145) hold and r~'(20) ~= 0. Let the density of the measure t2 of the Preisach nonlinearity be Lipschitz continuous with respect to both the arguments ~, ft. Then there is an rl > 0 such that the curves ~/r -- {(Z, y) E ~ d + l zr(t), y -- yr(t), t ~ [0, TF)} representing cycles of (2.141) in Ra+l and satisfying (2.142) as well as their periods Tr depend Lipschitz continuously on r for 0 < r < rl. Moreover, there is a neighborhood U c Ra+l of the point ;~do ~ F such that each closed curve ~ lying in U and representing a class of cycles of (2.141) with a period sufficiently close to To belongs to the set of curves ?r, 0 < r < rl, i.e. there are no other such curves in U. .

Z

--

We stress the difference between the conclusion of this proposition and that of the statements in the begining of this section on Hopf bifurcations in systems without hysteresis. System (2.141) with the Preisach nonlinearity has a continuum of small cycles in the phase space (this continuum is the join of the classes of cycles represented by the curves 7r with 0 < r < rl), while system (2.130) has typically at most one small cycle in the phase space for each value of the parameter, the surface like that shown in Fig. 2.39 in the product ~a+l of the phase space and the parameter axis consists of cycles existing for different parameter values. Another remarkable difference is seen if we consider the asymptotics of the cycles. Small cycles of system (2.130) form a smooth surface 5 in the space [~d+l of pairs (z, b). The curves ~/r representing cycles of system (2.141) in the space R~a+l of pairs (z, y) form a structure close to a cone, as is shown in Fig. 2.40. More precisely, the curves ?r are asymptotically close to the ellipses gr = Ado + r(z,, y,) + r(C, 0) for small r > 0, where Ado + r(z,, y,) is a ray tangent to the curve F at the point ,Ad0 and C is an appropriate cycle of the linear system z' = A(20)z (the cycles rC form a plane in the phase space ~d of z' = A(20)z). The Hausdorf distance between 7r and gr is o(r) as r ~ 0; Fig. 2.40 shows the curves ~r and the 5We consider sufficiently smooth nonlinearities F(., .).

212

CHAPTER 2 Differential Equations with Hysteresis

~Z2

/ z1

FIGURE2.40 Curves 7r. two-dimensional cone surface formed by the ellipses gr with the vertex at the point .Ado. Some further details can be found in [117].

D i m e n s i o n of the Set of Cycles of System with a Parameter Let us consider again small cycles of the system (2.133) in the case of Hopf bifurcation from an equilibrium (z0, ~/0) at the Hopf bifurcation point b = b0. Here the Problem includes the 'external' parameter b and the 'internal' parameter 2. So, one might expect that the set of closed curves 7 = {(z(t), y(t), b) ~ [ ~ d + 2 . t E •} representing small cycles of (2.133) in the space R a+2 of triples (z, y, b) in a vicinity of the point.M0 - (z0, Y0, b0) with Y0 = (I)0/0) is generically two-parametric and that these curves accumulate at the points of some curve $ passing through the point.M0 and representing equilibria of (2.133). Namely, if F(zo, Y0, b0) = 0 with Y0 = cI)(r/0) satisfying (2.137), the matrix A(.,., .) = cgF/cgz(., .~ .) has a pair of simple eigenvalues • iwo (w0 > 0) at the point AF0 and proper smoothness and nondegeneracy conditions are satisfied, then the equality F(z, y, b) = 0 defines a two-dimensional surface | in some vicinity of the point d~0 in ~a+2, each point (z~, y~ b~) ~ | defines the stationary solution (z(t), y(t)) = (z~, y~) of (2.133) for b = b~ and the curve 8 c | is defined by the relations Sp(A(z, y, b)) ~ iw, Iw - w0J < ~, w > 0 with a sufficiently small ~. Yet, we do not know exact statements of this type. In problems on Hopf bifurcation from infinity for system (2.133) one should distinguish between the cases R~ = cx~ and Ra < cx~. For example, suppose that the conditions of Proposition 2.3.21 are satisfied. If R~ -- cx~

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

213

then the set of closed curves 7 - {(z(t), y(t), b) e ~d+2. t e R~}representing large cycles of (2.133) for b close to the asymptotic Hopf bifurcation point b0 is generically two-parametric; if R~ < cx~ then this set is generically (for 'nonexotic' nonlinearities F) one-parametric. One can see it more precisely from operator equations equivalent to the problem. We outline the approach roughly, not presenting any particular operator equation, which can be constructed in different ways in the form W r (z, y, b, T) = 0. Here r is an auxiliary parameter; the u n k n o w n s are the functions z = z(t), y = y(t), the parameter value b for which the class Uo<.~ r0 and every 0~<2~<1, the relations m a x I Z r , 2 ( t ) - Zr,2('c)l---> cx~, br, 2 ---> bo, Tr,2 ~ t,'c

TO = 27r/wo

as

r ~ cx~

similar to (2.138) hold uniformly with respect to 2 e [0, 1], and Zr, 2 -Zr,,2', Yr,2 -- Yr',2' if and only if r = r', 2 = 2 I. This implies that there is the two-parameter set of c u r v e s 7r,2 - {(Zr,2(t), Yr,2(t), br,2) 6 [~d+2 . t e [0, Tr,,~)} (r > r0, 0K2~<1) which are different for different (r, ;~) and that the diameter of 7r,,~ goes to infinity as r increases. Moreover, it turns out that the m a x i m u m max{d(b, 2, b,,V) " 0~2, 2' <~1} of the Hausdorf distance d(., .) between the c u r v e s 7r,2 and 7r,2' with the same r vanishes as r increases, i.e. the set U0~<,~<~l7r,;~ shrinks to the individual curve 7r,0 as r ~ cx~. If the vector c does not belong to the invariant plane of the matrix A(bo) corresponding to the pair of its simple eigenvalues +iwo, then the relations mint (c, z(t) ) ~ - c x t maxt (c, z(t) ) ~ ooasb ~ bo hold a priori for the cycles bifurcating from infinity. In case R~ < cx~ this implies (J,~x)(t) = (Jox)(t) with x(t) = (c, z(t)) for all 0~2~<1 and all the large cycles. Therefore in place of (2.146) one arrives at the system Wr(z, y, b, T) = 0, y = J0x, x = (c, z) with only the parameter r. One proves that for all r > r0 this system has solutions (Zr, yr, br, TF) satisfying (2.138). In this case, large cycles of system (2.133) form the one-parameter set of classes represented by the curves 7r (r > r0) with the diameters going to infinity as r increases. The auxiliary parameter r can be chosen in different ways. For example, a ' g o o d ' choice (which allows one to construct topologically nondegenerate systems (2.146)) is r -- Ilzllc. With this choice, the above statements

CHAPTER 2 Differential Equations with Hysteresis

214

guarantee the existence of cycles for all sufficiently large amplitudes of the z-component. Statements of this type on Hopf bifurcations from infinity for systems with the Prandtl-Ishlinskii hysteresis nonlinearities (a class of nonlinearities close to a particular class of the Preisach models) in case Ra - ~ are presented in [116]. Case Ra < cx~ for systems with the stop hysteresis nonlinearity was studied in [168], including stability analysis of large cycles; a similar approach can be applied here.

2.3.8 CHAOTIC BEHAVIOR Definition of Chaos Letf : S ~ S be a continuous mapping in a subset S of an Euclidean space Re. Important attributes of chaotic behavior include sensitive dependence on initial conditions, an abundance of periodic trajectories and an irregular mixing effect describable informally by the existence of a finite number of separated subsets U0. . . . . Urn-1 of Re which can be visited by trajectories of some fixed iteratef k off in any prescribed order. Let/d = {U0 . . . . . Urn-l}, m > I be a family of disjoint subsets of S and let us denote the set of onesided sequences co = o90, o91. . . . " ogi 9 {0, 1 . . . . . m - 1} by f~R m. Sequences in f~R m will be used to prescribe the order in which sets Ui are to be visited. m-1 For x 9 Ui=0 Ui we denote by I(x) the number i satisfying x 9 Ui. A mapping f is called (/d, k)-chaotic (k is a positive integer) if there exists a compact f-invariant set Z C ~Ji Ui with the following properties: (p l) for any o9 9 f2R there exists x 9 Z with fik (x) = (fk')i 9 Uo~i for i~1; (p2) for any p-periodic sequence o9 9 f~R m, there exists a pk-periodic point x 9 Z with fik(x) 9 U~i; (p3) for each ~/> 0 there exists an uncountable subset Z(r/) of S, such that the simultaneous relationships lim supi__,~

I(fik(xl)) -- l(fik(x2))] ~1,

lim infi~ c~

~ik(xl) -- fik(x2)l < rI

h o l d for all Xl, x2 6 I ( r ] ) , Xl # x2.

The above-defined properties of chaotic behavior are similar to those in the Smale transverse homoclinic trajectory theorem with an important difference being that we do not require the existence of an invariant Cantor

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS TABLE 2.2 i

V~1) (i)

V~2) (i)

0 1 2 3 4 5 6 7 8

2.595 2.597 2.606 2.641 2.808 3.045 2.677 2.622 2.602

1.286 1.278 1.258 1.189 0.9016 1.150 1.599 1.366 1.304

Va(1) (i) 0.001328 0.001194 0.001099 0.001086 0.001535 -0.002766 0.001789 0.001615 0.001465

215

Parallelograms Va(2) (i) -0.002690 -0.002419 -0.002190 -0.002059 -0.002426 0.004905 -0.002620 -0.003255 -0.002989

v~1) (i) 0.003545 0.004168 0.004869 0.005637 0.007217 0.0003166 0.001140 0.002219 0.002922

v~2) (i) 0.009351 0.01097 0.01284 0.01472 0.01352 -0.009191 0.006788 0.007069 0.008053

set. Instead, the definition includes property (p2), which is usually a corollary of uniqueness, and (p3), which is a form of sensitivity and irregular mixing as in the Li-Yorke definition of chaos [169], with the subset/701) corresponding to the Li-Yorke scrambled subset So. A similar definition was used previously in [170]. Note also that the one-sided left shift cr+ is a factor of the restrictionfkls (See [171], p. 68).

Chaos in an ODE with Preisach Nonlinearity Consider Eqn. (2.9) with b = I and an appropriate Poincare map F1 : FI 1-I generated by it. Let v = (v (1), v (2)) denote the element of FI. A part of the co-limit set of F that was constructed with help of the computer is given in Fig. 2.21, (p). Introduce the parallelogram sets R(i) ~ l-I, i = 0 . . . . . 8 given by the center points vc(i), and the halves of the side vectors va(i) and vb(i):

R(i) -- {vc(i) 4- ~va(i) 4- flvb(i) : --1 < ~, fl < 1}.

(2.147)

Values of vc(i), va(i) and vb(i) are given in Table 2.2. Let/d denote the family of connected components of the set ~8=0 R(i). 2.3.6. The family bl has nine components, i.e. R(i) N R(j) = O, i ~ j, i , j = O . . . . . 8.

LEMMA

PROOF. This is clear from the definition of set R(i). See also Fig. 2.41.

THEOREM2.3.5. Mapping F1 is (ld, 16)-chaotic. Proof of this theorem is given in Section 2.4.4.

[]

CHAPTER 2

216

Differential Equations with Hysteresis

I

I

'

I

'

I

'

1.35 1.8 1.3 1.6 1.25 1.4

-

,

I

,

2.59

I

,

2.6

I

,

2.61

I

,

2.62

I

2.63

1.2 -

1-

0.8 2.5

2.6

F I G U R E 2.41

2.7

2.8

2.9

3

Sets R(i), i = 0 . . . . . 8 in c o o r d i n a t e s t m o d V ~ r , z (2).

2.5 -

~,,~

-

1.4

r :Y~'G

1.5

v-,,,, #?.. , : , . ~ , - 4 d ~ ".

:V,

..

~" \'~

2.6

2.7

2.8

2.9

0.5

1 F I G U R E 2.42

2

3

4

Sets R(i), i = 0 . . . . . 8 a n d a c h a o t i c a t t r a c t o r .

5

,

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

217

Note also the following. PROPOSITION2.3.24. The map F1 has a fixed point v, ~ R(0) and for any positive

integer p greater than eight there is a periodic point v ~ R(O) which has the minimal period p, and which is not asymptotically stable. Moreover, Eqn. (2.9) has periodic solutions of minimal periods v/2rc 9 (3kl + 29k2), where kl, k2~>0, kl + k2 > 0. In particular there exist solutions of all periods x/-2rcn, with n > 55. Proof of the proposition closely follows ideas in [172,173].

Interpretation In this subsection we discuss another interpretation of our main result from the previous section. It is convenient to begin with some definitions in terms of Eqn. (2.13). We will always suppose that the functionf(t, z, u) is periodic in t with minimal period T > 0. Thus any periodic solution of that equation must have a period which is a multiple of T. Let no be a positive integer and zo(t), - ~ < t < ~ , (2.148) be a function with the minimal period noT. This function will play the role of an approximate n0T-periodic solution of Eqn. (2.13). We will always deal with the situation when the function zo(t) is piecewise continuous, and the limits z(t - 0) = lim z(t)

rSt

are well defined at the points of discontinuity. In our setting the wording 'the function zo(t) can be considered as an approximate n0T-periodic solution of Eqn. (2.13)' means that, firstly, the jump Iz0(t) - zo(t - 0)l at any discontinuity point t is rather small, and, secondly, the function almost satisfies the equation between successive points of discontinuity (the quantitative bounds for the corresponding discrepancies will be specified below). Suppose that for a positive integer nl < no, the number Iz0(0) - z0(nlT - 0)1 is small. Then we can treat the niT-periodic function Zl(t), - o e < t < oe, which is defined by zl(t) = zo(t), 0 <~t < niT, as an approximate niTperiodic solution of Eqn. (2.13). Since no is the least period of zo(t) and

CHAPTER 2 Differential Equations with Hysteresis

218

z

Zl.

$_;

,..G/ Zo

x

t

!

FIGURE 2.43 Functions z0 (solid) and Zl (dotted). Here z0 has period 2T, z I has period T; thus z0, Zl coincide on 2nT<.t < (2n + 1)T, n = 0, 4-1, 4-2. . . . in this case. nl < no, w e define the distance r b e t w e e n the c o r r e s p o n d i n g functions

zo(t), zl (t) b y r = sup Izo(t) - zl(t)l > 0.

(2.149)

t

Figure 2.43 illustrates the situation for the case no = 2, nl - 1: the functions z0 a n d Zl are s h o w n by a solid a n d a dotted line, respectively. We can use as a p p r o x i m a t e patterns for possible b o u n d e d o u t p u t profiles, not only the functions zo(t) a n d zl(t) themselves, b u t also all possible 'shuffiings of these functions'. To explain w h a t this m e a n s w e need additional notation a n d definitions. Let f~ be the totality of all bi-infinite oo b i n a r y sequences o9 = {o9i}i=-oo w i t h ogi E {0, 1} for i = 0, 4-1, 4-2 . . . . . To each sequence co e f2 there corresponds the sequence of n u m b e r s ti to be defined b y to = 0, i-1

ti = T E no~] j=0 for i = 1, 2, 3 . . . . . a n d b y i

ti--T

E

no~j

j=-I

for i = - 1 , - 2 , - 3 . . . . . . That is, the real line R 1 is partitioned into the intervals Ii = [ti, ti+l) w h e r e the length of the interval Ii equals no~iT. Figure 2.44 illustrates this idea: here 09= . . . . 0 , 1 , 0 , 1 , 1 , 0 . . . . . a n d the element coo is in bold.

(2.150)

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

-T

0

T

219

2T

3T

4T

FIGURE 2.44 Partition of R 1 corresponding to o~ . . . . . 0, 1, 0, 1, 1, 0 . . . . ; w e show the intervals corresponding to coi = 0 by the solid lines and intervals I i, corresponding to coi = 1, by the dotted line, with co0 = 0 in bold.

Z l ZI.

9

M

-.~.. .~"

Z1"

Z1.

ir

".~

If ~

l

,

'0

i

r

CT

l

~3T

i

/

/

if

,

,

I

~4T

t

I

!

z0 FIGURE 2.45

Function zo~ for o~ . . . . .

0, 1, 0, 1, 1, 0 . . . .

C o r r e s p o n d i n g to a n y o9 ~ f~ t h e f u n c t i o n zoo(t) is d e f i n e d b y

zoo(t) - zo~i(t - ti),

for

t i ~ t < ti+l.

F i g u r e 2.45 s h o w s a f r a g m e n t of t h e f u n c t i o n zo~ for t h e case w h e n co satisfies (2.150). We will u s e t h e n o t a t i o n Z ~ = {zm(t) 9 o9 ~ f~}. PROPOSITION 2.3.25. A n y two functions Zm~l), zo)(2) 09(2) . Moreover, sup

IZco(1)--Z(o(2) I -- r(09 (1), to(2))~rmin > 0,

~_

Z ~ are different if 09(1)

09(1) 09(2) E ~'-~, ~(1) ~: 09(2).

-c~
inf r(09 (1), 09(2))/~rmin > O, co(l) ~_co(2)

09(1), 09(2) E ~

is strictly positive. We call r ( Z n ) t h e s e p a r a t i o n t h r e s h o l d for t h e set Z n .

(2.152)

CHAPTER 2 Differential Equations with Hysteresis

220

Z

zo)(t) + e / I ~" ~ " ~ ,

II l ~

,/

"

~ '",

,'

'" .... ',,

"

,' ,'

"',

'" ....

"~, 9 ,

'

"'"

r

, i

,

2~T.... "'/

',,

!

I

.

I

~

9 ~ aL

~" ~

'

,' I 9

~

IL

""

9 ',.: v

:'T--'" " "

t ,"

",

'

zto(t)- e

FIGURE 2.46 A function z(t) (solid dark curve) satisfying (2.155), with o~ . . . . . O, 1, O, 1, 1, 0 .... If the sequence o9 is p-periodic in the sense that ogi = 0, 4-1, +2 . . . . . then the function z~(t) is T~-periodic where

ogi+p

for i =

p-1

To~ - T Z

n~i"

(2.153)

i=0

The number (2.153) is the minimal period of the function zoo, provided that p is the minimal period of the sequence co. Thus, among the elements of the set Zfl there are periodic functions with arbitrary large minimal periods. However, the main bulk of the set Zfl consists of functions which do not follow any particular pattern. Let e be a positive number satisfying e < r(Zn)/2.

(2.154)

We will say that Eqn. (2.13) is (zo, Zl, e)-compatible if the following two conditions hold: (C1) For any function zoo(t), o9 ~ f2 there exists a solution z(t) of Eqn. (2.13) satisfying the uniform estimate max

-cx~
Iz(t) - z~(t)l < e.

(2.155)

(C2) Additionally, if the sequence o9 is p-periodic, then there exists a T~periodic solution z(t) of Eqn. (2.13) satisfying the uniform estimate (2.155), where To~ is given by (2.153). Figure 2.46 graphs a function z(t) that satisfies (2.155), where o9 satisfies (2.150).

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

'

i

i

!

'

i '

2

"

221

i

'

i

'

'

1'0

'

i

'

_

0-1 -2

,...

,

0

|

20

,

i

40

,

!

60

,

i~ , 80 100 120 140

-20

;

1;

'

20

FIGURE 2.47 A function zo(t) on the left and a function z l(t) on the right. In the scalar function z for small e, functions z(t) satisfying (2.155) follow closely the 'guidance' of a c o r r e s p o n d i n g function zoo(t). In the term i n o l o g y of hyperbolicity theory, a function z(.) is an e-shadow of zoo(-). Since e is strictly less than half of the separation threshold r(Zr any solution z(t) can be considered as an e-shadow of not more than for one guide zoo. Thus, the m e a n i n g of (z0, Zl, e)-compatibility is that for any pattern given b y a function zo~ ~ Zr we can find a b o u n d e d solution profile which is an e-shadow of zoo. Consider Eqn. (2.9) with b = 1. As zo(t) = (xo, x'o) we choose the numerical solution of (2.9) c o r r e s p o n d i n g to the pseudo-trajectory of F1 of the length nine, which is given as the u n i o n of nine numerical trajectories with initial conditions x(ti) = 0.5, x'(ti) v~2) (i) on the intervals [ti, ti+l), i - 0 . . . . . 8; V~1) , V~2) are given in Table 2.2 and ti is given b y -

ti =

-

v~1) (i) + 3 i - v ~ n , v~1) (i) + (3i + i - 4 ) . ~/2n, v~1) (i) + (3i + 2). ~/2n,

i - 0 . . . . . 4, i - 5 . . . . . 6, i = 7 . . . . . 8,

vf 1) (0) + 29. v ~ n ,

i - 9.

Thus no = 29. Let us choose z] (t) as zo(t) on the interval [to, to + 3- ~/2n), in which case n l -- 3. The pictures of the w a v e g u i d e s are given in Fig. 2.47. LEMMA2.3.7. The following relation holds: r(Zn) > 0.1. By this l e m m a the (z0, Zl, e)-compatibility is well defined w h e n e v e r e<0.05. The principal result of the previous section, T h e o r e m 2.3.5, m a y be rewritten n o w as follows.

222

CHAPTER 2 Differential Equations with Hysteresis

THEOREM2.3.6. The equation is (zo, Z l , O.O1)-compatible.

Why It is Interesting and Important The (z0, zl, s)-compatibility implies, in particular, that the totality of bounded solutions is spatially chaotic in a topological sense. We recall, as an aside, that the three main aspects of chaotic behavior are equally important: the measure theory aspect such as the existence of the Sinai-Ruelle-Bowen invariant measures [174]; the 'sensitivity' aspect, such as positivity of Lyapunov exponents [171]; and the topological aspect describing irregular mixing properties in terms of symbolic dynamics [171]. At an informal level the topological aspect of the spatially chaotic behavior can be described as follows. Suppose that we register the forms of the solutions within a certain precision ~, where ~ is greater than s, but significantly less than the separation threshold (see the relevant numerical information in the next section). Then a 'shadow solution' z(t), satisfying (2.155), is indistinguishable from its guide zo~(t), although different guides are quite distinguishable. Thus any chaotic shuffling zo~ of two basic patterns z0, Zl provides a possible steady profile (within the accuracy ~). The next level of formalization requires ideas of symbolic dynamics. Denote by n the least common multiple of the numbers no, nl. Also let e [0, noT) be a fixed number such that Iz0(z) - Zl(Z)l > 2s (such a ~ exists by (2.154)). Firstly, we characterize a function z(t), -cx~ < t < cx~,by the bi-infinite sequence of its values Zi - -

z(inT + z),

i = O, 4-1, 4-2 . . . . .

(2.156)

That is, we measure the values of z(t) 'stroboscopically' with the uniform interval nT between the measurements. Further, encode the function z(t) by a binary sequence o9 e f~, which is constructed as follows. Let

Io = {z:lz- z00:)l~e},

I1 -- {z:lz- Zl('01- ~s}.

The sets I0 and 11 do not intersect since Iz0(z) - Zl('01 > 2S. If for any i the value zi belongs either to I0 or to I1, then we define the code sequence o9 - co(z(.)) by setting o g i = 0 in the first instance, and ogi = 1 in the second. In other words, we indicate the measurement positions inT, i = 0, 4-1, 4-2 . . . . where the function z(t) goes through a window I0 or I1 (see Fig. 2.48 for one-dimensional interpretation). If the value zi does not belong to one of these two windows for some i, then our coding procedure fails, and we are not interested in such 'unencodable' functions z(t). Since I0 N I1 -- 0, not more than one code can correspond to a particular function z(t). N o w the compatibility condition (C1) implies that any random

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

223

Z

I I f

t

FIGURE 2.48 A function z(t) with the code .... 0, 1, 0, 1, 1, 0 . . . . . sequence o9 a p p e a r s as a code for some b o u n d e d profile. (Indeed, let us introduce a sequence o9* b y substituting n/no ones (n/nl zeros) instead of a n y one (any zero) entry of the sequence o9. Then, the function z~. (t) ~ Z~ has the required properties.) For a given function z(t) w e denote b y ;:(-) = anTZ(.) its (left) nT-shift to be defined b y ;:(t) = z(t + nT). Let us denote also b y a the s t a n d a r d (left) shift on f2 given b y a(og) = o91 -- ( . . . . . o9I 1' 090' r 091t . . . . ) !

w h e r e ogi = o9i+1. For instance, if o9 - . . . , 0, 1, 0, 1, 1, 0 . . . . , then ogl

...,0,1,0,1,1,0

.....

PROPOSITION 2.3.26. Suppose that Eqn. (2.13) is (zo, Zl, e)-compatible. Then to

each o9 ~ ~ there is a solution z~(x) of Eqn. (2.13), such that z~(.) has the code co, and this correspondence og~-+zo~is shift invariant: O'nTZo~(') = Zao~(') for og~f~. Proposition 2.3.26 can be p r o v e d in a s t a n d a r d w a y using the Z o r n L e m m a . In technical terms, the above proposition m e a n s that the restriction O'nT of the shift an0T to the set {z~ : o9 ~ f2} is semi-conjugate to the shift a in f2. Such semi-conjugacy is w i d e l y recognized as the m o s t i m p o r t a n t attribute of topologically chaotic behavior, see, for example, [171]. It is also i m p o r t a n t to note that (z0, Zl, e)-compatibility implies the existence of solutions w i t h a w i d e range of nontrivial regular patterns. For instance, one more basic characteristic of chaotic b e h a v i o r is an a b u n d a n c e of periodic solutions and, in particular, an exponential rate of increase of the m a x i m a l n u m b e r of pairwise e-separated 6 kT-periodic solutions w h e n k increases, see [ 171]. 6Two functions z(x), y(x) are E-separated if sup Iz(x) - y(x)l > e.

224

CHAPTER 2 Differential Equations with Hysteresis

PROPOSITION2.3.27. Suppose that Eqn. (2.13) is (z0, Zl, e)-compatible and denote by n the least common multiple of the numbers no, nl. Then it has at least 2 k periodic solutions of period kn which are pairwise e-separated. This assertion follows immediately from the definitions. Another example of functions with a regular pattern is given by the socalled recurrent functions [97], which exhibit fractal-type features. The recurrence property of a function z(t) with - c ~ < t < o0 means that for each pair of positive numbers (e, a) there exists A > 0 such that for each real r, any interval which is longer than A contains r such that the function t --, z(t + r) is e-close to the function t ~ z(t + a), t ~ [-T, T]. Recurrent nonperiodic functions are important today in the analysis of desynchronized systems, various new models in biology, etc. The notion of a recurrent sequence co is defined analogously. Recall, in particular, the classical Morse example of a recurrent sequence. We begin from the one-element sequence Sl = 0 and then substitute at each step the two-element sequence 0, 1 instead of 0 and two-element sequence 1, 0 instead of 1. Thus, Sl = 0 , 1 ; s2 = 0 , 1 , 1 , 0 ; $3 = 0 , 1 , 1 , 0 , 1 , 0 , 0 , 1 ; S4 = 0, 1, 1,0, 1,0,0, 1, 1,0,0, 1,0, 1, 1,0; etc.

After infinitely many iterations we obtain the famous forward Morse sequence M. Considering backward shifts of this sequence and their weak limits we arrive at the set A//of bi-infinite Morse sequences. PROPOSITION2.3.28. Suppose that Eqn. (2.13) is (zo, zl, e)-compatible and the sequence co is recurrent in the Morse sense (for instance, one of the set .M). Then there exists a recurrent, nonperiodic, solution satisfying (2.155). The proof follows the standard constructions [175]. 2.3.9 RECURRENT OSCILLATIONS

We will be interested in recurrent solutions of the general equations

x' =fit, x, y(t)), y(t) = (Fit0, y(to)]x) (t).

(2.157) (2.158)

Here x ~ ~a, and y belongs to a complete metric space 3; with the metric py. Also, for a fixed Y0, the operator F[t0, Y0] transforms functions

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

225

x 9 [to, oo) ~ ~a into functions y 9 [to, oo) ~ 3;. For a particular function x(.), the notation (Fit0, y0]x) (t) refers as usual to the value of the corresponding function y(.) at the m o m e n t t. The operators describing the hysteresis nonlinearities can be treated with a typically infinite dimensional set 3; of internal states y. We will use the definition of recurrent property introduced on the previous page. Periodic, quasiperiodic and almost-periodic functions are recurrent, however the class of recurrent functions is m u c h wider. The typical examples of recurrent, but not almost-periodic behavior are given by various modifications of the classical Morse construction [176]. Such functions are getting rather important today due to analysis of desynchronized systems [177], various new models in biology, etc. We will establish, u n d e r some natural technical restriction, that the system (2.157)-(2.158) has a recurrent solution, providing that the function f in the right-hand side of Eqn. (2.157) is recurrent in t, and that there exists at least one solution (x(t), y(t)) which is uniformly b o u n d e d at t --+ oo. Surprisingly, some of our results seem to be new even for Eqns. (2.157)(2.158) without hysteresis.

(W, C)- and s

Functions

Let us introduce some notation. 9 Let a function a of scalar a r g u m e n t be measurable at the interval [r, r~] and take values in a finite-dimensional Euclidean space. We denote II~ll&.~ =

I~(t)l dt.

(2.159)

9 If a function ~ is additionally absolutely continuous at [~, rr], denote

Ilallw~,~ = la(~)l + Ila'lls

(2.160)

9 For any continuous functions yl, Y2 : [z, rT] --+ 3;, we define

P:,cr(Yl, y2)

-- max P3;(yl(t), y2(t)). te[~,cr]

(2.161)

We denote by Wr,r aa the Banach space of absolutely continuous functions x 9 [r, cr] --+ R a with the n o r m (2.160). Analogously, denote by C~,~;3; the complete metric space of continuous functions y : [~, or] ~ 3; with the uniform metric (2.161).

CHAPTER 2 Differential Equations with Hysteresis

226

Denote the z-shift of a function y by 0(r)y:

(O(r)y) (t) = y(t + r). We say that a set ,5' c ~ is A-dense if any interval [t, t + A] contains at least one element from $; that is ,_q~[t, t + A] ~a 0, for all t ~ ( - ~ , ~ ) . Let a function x 9 R ~ R~a be absolutely continuous on any finite interval, whereas a function y : ~ ~ 3; is continuous. The pair (x, y) is said to be (W, C)-recurrent if for any positive T, ~ there exists A = A(T, e) such that the set of rT ~ ( - ~ , o0) satisfying

II0(~)x- O(~)xlIw_T,T + P-T,T(O(r)Y, O(cr)y) < e is A-dense for each z ~ (-cx~, cx~). Let a functionf be measurable in t and continuous in x, y (that is, it satisfies the Caratheodory condition). Let also f satisfy the following growing condition: TT Ixl,
T< ~

(2.162)

for any T, R > 0. The function f is s in t if for any positive numbers T, R, e there existsA = A(T, R, e) such that the set of cr ~ (-oo, o0) satisfying T

max Ll:(t + ,, x, y) - f i t + or, x, y)l dt < e T ]xI<<.R,Y~3;

(2.163)

is A-dense for each 9 ~ (-oo, oo).

(W, C)-Correct Hysteresis Nonlinearities We will use a standard approach to describe hysteresis nonlinearities, s e e [15] and the references therein. We suppose that the operator F[t0, Y0] can be understood as a description of an autonomous control dynamical system F. That is, the family of operators F[t0, Y0] must satisfy the following properties: 9 the Volterra property: X(S) =-- Xl(S), S E

[to, t],

implies

(F[t0, y0]x) (t) = (F[t0, y0]xl) (t);

9 the semigroup property: (F[tl, (F[t0, y0]x)(tl)]X)(t) - (F[t0, y0]x)(t), for to < tl < t;

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

227

9 the autonomy property: (Fit0, y0]x) (t) = (F[t0, y0]Xl) (t -- t o q- tl), where Xl(t) = x(t - to + tl) and to < tl < t. N o w we introduce some additional properties of the hysteresis operators F. We will suppose that the operators F[t0, Y0] are defined for all absolutely continuous functions x 9 [0, oe) --~ ~ct and that their outputs are continuous. Using the Volterra property, the operators Fit0, Y0] are also well defined at any finite interval [to, tl]. Recall, that a set $1 of measurable functions x 9 [0, t] ~ ~a is uniformly integrable if the relationship f lim / [x(t)[ dt = 0 2--,oo dlx(t)l>2 holds uniformly in x 6 51. A set $2 of Y-valued continuous functions is said to be equicontinuous if for any e > 0 there exists 6 > 0 such that p y ( y ( t l ) , y(t2)) < e,

It1 - t 2 ] < 6, y 6 $2,

holds. We say that a nonlinearity F is (W, C)-correct if for any uniformly bounded set $1 of absolutely continuous functions, such that the set of derivatives {x', x 6 $1} is uniformly integrable, the set $2 -- {l-'[0, yo]x : Yo ~ 3;, x

~

$1}

is equicontinuous, and if the relationships [IXn - X]lWto,t 4- P 3; (,YO(n) ' Yo) ---> 0

imply p y ((F[t0, y~n)]xn)(t), (F[t0, yo]x)(t)) ~ O.

for any - ~ < to < t < c~. As examples of (W, C)-correctness we mention the Prandtl, Besseling, Ishlinskii, Mizes, Tresca, etc. models of plasticity, Madelung, Preisach, Giltay, Mayergoys, etc. models in ferromagnetism and many, many others, see [7,14,15,157] and the bibliographies therein. Actually all hysteresis nonlinearities with continuous output signals known to us belong to the class of (W, C)-continuous nonlinearities.

228

CHAPTER 2 Differential Equations with Hysteresis

Main Result Let us come back to the system (2.157)-(2.158). The function f in the right-hand side of (2.157) will always satisfy the Caratheodory condition. We will understand the solutions of Eqns. (2.157)-(2.158) in the classical sense: solution is a pair (x, y) where x 9 [to, cx~) "---> RJ is a locally absolutely continuous function and y : [to, cx~) :--+ 32 is continuous; the relationships (2.157) should hold almost everywhere and the relationship (2.158) everywhere. (Recall [178] that the function x 9 [~, a] ~ ~d is absolutely continuous if x' exists almost everywhere, is integrable on [r, a], and x'(t)dt = x(tl) - x(t2) for r < tl < t2 < a.) The pair (x, y) where the functions x and y are defined over the whole interval (-cxz, oo) is said to be a solution of the system (2.157)-(2.158) if the restrictions of the functions x, y to the interval [to, ~ ) provide a solution of this system for each to 6 (-cx~, oo). The last definition is correct by the Volterra property.

ft~

THEOREM2.3.7. Let f be s

in t and let F be (W, C)-correct. Let (x,(t), y,(t)),

t>~O,

be a solution of the system (2.157)-(2.158) with the function x,(t) uniformly bounded. Then there exists a (W, C)-recurrent solution (x(t), y(t)) of the system (2.157)-(2.158). N o w we will give some general comments concerning the use of the preceding theorem. Under simple additional assumptions, that hold for all practically important hysteresis nonlinearities, any initial problem

x(to) = xo ~ Rd,

y(to) -- Yo ~ Y,

(2.164)

has at least one solution (x, y) defined at the maximal interval [to, T) where either T = ~ or Ix(t)l --* ~ as t -+ T. It is sufficient, for instance, to suppose that the hysteresis nonlinearity F is continuous from C to C (as in the case of Prandtl, Besseling, Tresca, Mises, Giltay, etc. nonlinearities) or is continuous from W to W (as is the case for the Madelung nonlinearity). The proof proceeds in the usual way using the Showder fixed-point theorem. We will say in this situation that the initial problems for Eqns. (2.157)-(2.158) have local solution. If the initial problems have local solutions, we say that the system (2.157)-(2.158) is dissipative if for any R > 0 there exists R1 > 0 such that lim sup Ix(t)l~R, t--, oo ix01~
2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

229

where x(t) is a solution of Eqn. (2.164); the sufficient conditions of dissipativity are easy to suggest in terms of Lyapunov functions, etc. COROLLARY2.3.12. Let f be s in t and let F be (W, C)-continuous. Let the system (2.157)-(2.158) be dissipative. Then there exists a (W, C)-recurrent solution (x(t), y(t)) of the system (2.157)-(2.158). Note that for analysis of stability of recurrent solutions, the method suggested in [83] is applicable without trouble.

2.3.10 HYSTERESIS VERSUS TIME RELAXATION Generalities Time relaxation and hysteresis are common strongly nonlinear phenomena which occur in many industrial, physical and economic systems. Often these two types of phenomena manifest different stages of the same or similar processes. A number of fundamental hysteresis models can be considered as limit cases of time relaxation processes, or admit an approximation by a differential equation which is singular with respect to a particular parameter. However, the level of interaction between theories of systems with time relaxation and systems with hysteresis (and between the 'relaxation' and 'hysteresis' research communities) is quite low. The aim of this section is to discuss, at the very basic level, relationships between those two topics. An instructive singularly perturbed system is the following

x' =fit, x, y), ey' = g(x, y),

(2.165) (2.166)

where e > 0 is a small parameter. We concentrate first on the auxiliary equation

~y' = g(x(t), y), where x(t) is an arbitrary function. We denote the solution of this equation, with a given initial condition y(0) = Y0, by

y(t) = VE[t0, yo]x(t). Under suitable technical conditions it can be proved that for a generic function x(t) there exists a limit

y~(t) = V[to, yo]x(t) = lim V~[t0, yo]x(t). 6--+0

CHAPTER 2 Differential Equations with Hysteresis

230

g(x,y)
/s x,~ X

g(x,y)>O

FIGURE 2.49 Relay as a singularly perturbed ODE. The solutions of the original system are then approximated as e ~ 0 by the solutions of the differential-operator equation

x' =fit, x, y(t)),

y(t) = V[to, yo]x(t).

(2.167)

The following observation is straightforward: LEMMA2.3.8. The limit operator V is rate-independent. Thus, the global analysis of singularly perturbed equations becomes, in the first approximation, an investigation of equations with hysteresis nonlinearities. This is the first point of contact between equations with hysteresis and singularly perturbed equations. We will demonstrate how this observation works below. Let the function g be described by the picture in Fig. 2.49. Suppose x(t) does not have local minima equal to a or local maxima equal to ft. Then the operator V is well defined on x(t) and

V[to, yo]x(t) = R~,fl[to, yo]x(t) for all t > to such that

x(t) # a

and

x(t) # fl.

(See Section 2.1.2 for the definition of Ra,fl.) This example shows that singularly perturbed equations can approximate equations with hysteresis. This is the second point of contact between equations with hysteresis and singularly perturbed equations. This approach is especially important if there are some physical reasons to believe that the relay (or any other discontinuous hysteresis nonlinearity) is an idealization of a limit operator similar to V. Sometimes, however,

2.3 EXAMPLESOF RIGOROUS CONCLUSIONS

231

discontinuous hysteresis operators are idealizations of quite different continuous operators. The application of methods used in systems with singular perturbations to these new types of singular approximations is the third point of contact between equations with hysteresis and singularly perturbed equations. We note in conclusion that numerous mathematical models of physical, mechanical and other systems can be described by the equations of the form

x' =fit, x, y, z)~ y = Fx(t),

ez' = g(x, y, z) where F is a continuous hysteresis nonlinearity acting on function x(t) (for, instance, Preisach nonlinearity P a with an absolutely continuous measure #). The analysis of such systems will require the combination of ideas both in the area of systems with hysteresis and in the area of singularly perturbed systems. The main technical problem lies in the fact that hysteresis operators are not smooth in a standard sense: they only satisfy a Lipschitz condition as operators in function spaces. Thus, even the notions of stable and unstable manifolds must be adjusted accordingly.

Generalized Play Ferromagnetic monocrystals can be considered as an idealization of an interesting hysteresis nonlinearity: the generalized play [15]. This will be denoted by L~, and is illustrated in Fig. 2.50. Formally, this nonlinearity can be described as follows. We denote by f2 the set of pairs (x, y) satisfying

?_(x)~y~7+(x). For a continuous input x(t) and for initial state r/0 satisfying (x(to), rio) ~ f2, the output y(t) = L~[to, r/0]x(t) is the unique continuous function such that, firstly, y(to) = r/0, secondly,

(x(t), y(t)) ~ f2,

t~to,

and, thirdly, the inclusion

(x(t), y(~)) ~ f2,

~t~r~

implies y(t) =_ y(r), z~tKr~. Thus, the output behaves 'lazily': it prefers to be unchanged when the phase pair (x(t), y(t)) belongs to f2. Fig. 2.51 shows a typical phase trajectory.

232

CHAPTER 2 Differential Equations with Hysteresis

V_(x)

,

1

='? / 13+~

x

(x) FIGURE 2.50 Generalized play Le.

Generalized play L~ can be also represented by the Preisach m o d e l with an appropriate measure. LEMMA2.3.9. Suppose x(t) has no local minima equal to ~ or local maxima equal

to fl, and let Yo equal either 0 or 1. Then the limit y,(t) = lim L~[to, yo]x(t) e-+ O

is well defined, and y,(t) - 2R_Kl~[t0, yo]x(t) - 1 for all t > to such that x(t) # ~

and

x(t) # fl.

In other words, for generic input functions x(t) the discontinuous nonlinearity 2R_K~x(t) - I is a limit of continuous nonlinearities LE[to, Y0] as E ----~ 0.

233

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS y

ar

ill

v

X

x~

FIGURE 2.51 Phase portrait of a typical trajectory for the generalized play Le.

Hysteretic Canards We consider the equation x" + x - b sin(t) + y(t),

y(t) - 2 R _ K ~ x ( t ) - 1.

(2.168)

By L e m m a 2.3.9 this is to be considered as an idealization of the equation x" 4- x = b sin(t) 4- y(t)~

y(t) = L~x(t)

(2.169)

where Le is a generalized play from the previous subsection, and e > 0 is a small parameter. We will say that a pair (x,(t)~ y~(t)) is a limit periodic solution of Eqn. (2.169), if it can be approximated by some 2~-periodic solutions (x~ (t), ye (t))

234

CHAPTER 2 Differential Equations with Hysteresis

of the system (2.169) with arbitrary small e. That is, if for any a > 0 there exists e0 > 0 such that for e < e0 Eqn. (2.169) has a periodic solution xe(t), ye (t) satisfying max [xe(t)- x,(t)[ < a. THEOREM 2.3.8. Let 0 < b < 4/n, then Eqn. (2.169) has exactly three limit periodic solutions:

(xo(t), yo(t)),

(xl(t), yl(t)),

(x2(t), y2(t)).

(2.170)

We n o w give explicit formulas for the solutions (2.170). Introduce the functions b x0(t, b) = - f l cos t + ~ (sin t - t cos t)

nb +-v-(cos t - 1) sin t, 4

(

Xl t + n-~:.b 2 '

)

=-(l+fl)

+I+

cost+~

- n K t < n,

sin~sint-tcos

t+

2

nb (cos t - 1)(sin t + 1), ~ - 2nKt < ~, 4 sin(z/2)

x2(t) = - x l ( t - n). Here ~ = v(b), 0 < b < 4 / n is defined by the relationships 1-fl arccosl_Ffl
b=2

f l - 1 + (fl+ 1)cosz sin(r/2) (sin ~ + "c- 2n)

Introduce also 2n-periodic functions y0 (t), yl(t), y2(t) by the equalities nb ~ sint,

y0(t;b)=

Yl t +

n-r.b)= 2 '

1

-nKt
nb r --~-sin~(l+sint),

y2(t) = - y l ( t - n).

r-2n~
235

2.4 EXAMPLES OF TECHNICAL TOOLBOXES PROPOSITION2.3.29. The functions given above satisfy Theorem 2.3.8.

Theorem 2.3.8 and Proposition 2.3.29 are similar to those proved in [179]. Figures 2.52, 2.53 show x0(t), yo(t) w h e n / / = 1. Similarly, Figs. 2.54, 2.55 show the functions xl(t), Yl (t). We claim that these limit solutions are analogs of canard solutions, see [180-182]. In particular, (xo(t), yo(t)) is an analog of the symmetric canard and (xl (t), Yl (t)) is an analog of the asymmetric canard. 9 The first reason is that classical canards and these solutions are mathematical explanations of the same physical phenomena: the limit behavior of periodic solutions of a family of differential equations approximating a given equation with a discontinuous, relaytype nonlinearity. 9 Secondly, in both cases the limit solutions demonstrate strange behavior: in the classical canard situation for a time they follow unstable manifolds; in the 'hysteretic canard' situation, the solutions can stop in the course of a jump, and then hover for a while at an intermediate level. 9 Thirdly, similar methods could probably be applied to analyze both situations. For example, the topological degree approach used to prove the last theorem can be useful for the rough location of classical canards. On the other hand, the typical 'canard community' methods of investigation will be useful in a more subtle asymptotic analysis of hysteresis canards.

2.4 E X A M P L E S

OF TECHNICAL

TOOLBOXES

2.4.1 DESCRIPTIONS OF PREISACH NONLINEARITY Standard Representation States and Outputs Consider the set f2 on which the internal states ~/ : f2 ~ {0, 1} of the Preisach nonlinearity P are defined. Under natural technical restrictions this set can be chosen to be the two-dimensional half-plane FI = {(~, fl) : fl > ~} (see [15]). Moreover, the formula (2.3) is replaced by:

y(t) - y[to, r/0](t) - f~<~ Ra,~[to, r/0(co)]x(t) d#,

t~to.

(2.171)

Here # is a probability Borel measure in FI. Below, unless the opposite is stated explicitly, we suppose that this measure is absolutely continuous

236

CHAPTER 2

Differential Equations with Hysteresis

1

0.5

.

.

-3

FIGURE 2.52

.

.

.

-2

.

.

2

-1

3

Functions xo(t) (sine-like function) and yo(t) (step-like function).

l-

v

-1-

FIGURE 2.53

Parametric plot of (x0(t), yo(t)).

X

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

237

2

1.5 1

0.5

9

/-5

FIGURE 2.54

,

-4

-3

,

.

|

.

.

.

.

.

.

-2

.

.

.

1

Functions x l(t) (sine-like function) and yo(t) (step-like function).

1 I I I I I

-1

J

1

-1-

FIGURE 2.55

Parametric plot of (x l(t), yl(t)).

X

238

CHAPTER 2 Differential Equations with Hysteresis

with respect to the standard Lebesgue measure. That is, for S c II #(S) = L h(~x, fl)d~dfl, where h is nonnegative, and satisfies

/ii

h(~, fl) dadfl = 1.

It is sufficient to consider the case when the initial and varying states r/0, 1/(t) are characteristic functions of sets S, which look like the shaded set in Fig. 2.56. That is, S is located to the left of the line a - fl and below a continuous curve L = Ls, with a nonpositive slope. Additionally, in this standard case the coordinates of the intersection of L with the diagonal coincide with the current value of the input x. In other words such sets S admit the representation S = {(a,/~) :/~ + a ~ < ~ s ( / ~ - a)}

where 6s(U), 0~
- vl.

The curve Ls is connected with the function ~'s by

LS = {(a, t) : fl + ~ = ~/s(fl- a)}.

(2.172)

We denote by (~, the set of states r/of the Preisach nonlinearity the characteristic functions of which are described above (see Fig. 2.42).

Dynamics

The set ~, is convenient because the evolution of the every state r/e ~, admits a simple geometrical interpretation in terms of the area described by i/, see Figs. 2.57, 2.58. Here the input x(t), moves along the horizontal scroll-bar, and controls the point on the diagonal a = fl above itself. When moving toward the upper right corner, this point on the diagonal drags the horizontal line, and hatches the domain below this line and above the diagonal. (For instance, if, in Fig. 2.57, x increases from the value u to the value v, the filled area is increased by the hatched triangle.) When moving towards the bottom-left comer, the diagonal point drags the vertical line, and hatches everything to the right of this line and above the diagonal as in Fig. 2.58. The output y(t) is the area of both shaded and hatched domains with respect to the measure #.

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

239

FIGURE 2.56 The set S given by typical characteristic function r/E (~,. In particular the set @, is invariant with respect to piecewise continuous inputs. Clearly, after some time the 'tail' of the curve Ls(t) becomes a continuous piecewise linear curve L, whose links are parallel to one of the coordinate axes.

The Krej~i Theorem The 'picture definition' of the dynamics given above works formally only for piecewise monotone functions. There exists also a simple and very convenient general analytic definition, first discovered, probably, by Pavel Krej~i, which we will also use below. We recall that the play nonlinearity describes a mechanical device illustrated by Fig. 2.59. Here a bean B can be moved along the string S. We control a frame F which can be moved along a string S forward and backwards. The position x(t) of the handle is a varying input. The bean moves only if the left end of the frame pushes it forward, or if the right end of the frame pushes it backward. The position y(t) of the center of the bean is the output. This 'device' is the play operator with the width w = s - r/2 where s is the width of the frame, and r is the radius of the bean. We will illustrate the relationship between the play operator and the Preisach model first by Fig. 2.60 below.

240

CHAPTER 2 Differential Equations with Hysteresis

FIGURE 2.57 Evolution of the set S corresponding to the state r/of the Preisach nonlinearity; increasing input.

Here we have many (ideally infinitely many) strings all parallel to the diagonal fl = a. There are also (infinitely) small beans, one on each string. There is no gravity, so beans do not move without external influence. We control the position of the angle-shape frame F, whose origin O is always at the diagonal a - ft. The position of the origin with respect to the scale x is the input. Initially, let the beans be located along a certain line L = L0, which bounds the set described by the initial state 1/= r/(0) of the Preisach system. We suppose that the initial position of O coincides with the intersection of L with the diagonal. Since r/~ ~, the whole line L is within the initial position of the frame. Then, when time passes, the beans will be moved (to the north-east when the input increases, and to the south-west, when the input decreases). It is easy to see that at any timemoment the beans will be positioned along the line L~l(t). This sketches the picture proof of the following Krej~i theorem. THEOREM2.4.1. Let P[to, x(t0); u] denote a one parameter family of the play nonlinearities with the initial state x(to) E ~ at to and the parameter u~O. Let rl(t) ~ if,,, t~to be the varying state of a Preisach nonlinearity for an input x(t), t~to and let the set S(t), corresponding to r/(t), be defined by a continuous

241

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

FIGURE 2.58 Evolution of the set S corresponding to the state r/of the Preisach nonlinearity; decreasing input.

FIGURE 2.59 The play nonlinearity.

function

~S(t), see

~l(t)(U)

(2.172). Then

-- ~S(t)(U)

-- P[t0, ~S(to)~ u]y(t),

where x(t) is the input of the Preisach nonlinearity.

y(t) - 2x(t),

t~to.

242

CHAPTER 2 Differential Equations with Hysteresis

FIGURE 2.60 The Preisach nonlinearity as a family of play nonlinearities. Interestingly, with some minor restrictions a play nonlinearity m a y be considered also as a Preisach nonlinearity.

Algorithmic Description Restricted Set of States In this text we are mainly interested in an even more restricted set of states of the Preisach model. These are the states ~/ corresponding to the sets S ~ FI b o u n d e d by the lines L with an infinite horizontal link, and with a finite n u m b e r of vertical and horizontal links, see Fig. 2.61. We denote the height of the infinite horizontal link by w (1). Let w (2k), 1 < 2kKK, be the ascending sequence of the horizontal coordinates of the vertical links. That is, the vertical links belong to the straight lines L2k = (W(2k), fl), 1 < 2 k ~ K . Let also w(2k+l), 1 < 2k + I ~ K , be the descending

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

243

0

f

I

I

161

14

I21

i

w2

w4

w6

w5

i

w3

w1

FIGURE 2.61 State r/ ~ ~, that defines a set S bounded by the piecewise-linear curve L. sequence of the vertical coordinates of the finite horizontal links. By the definition numbers w (k), l <.k <<.K, satisfy the inequalities W (1) > . . .

> w(2k0 - 1 ) > w(2k0) > . . .

> W (2),

W (1) >

> w(2k0 +1) > w(2k0) > . . .

> W (2),

...

if K - 2k0; if K = 2k0 + 1.

(2.173)

The totality W of all states ~/, described by the finite n u m b e r of links as above, m a y be naturally identified with the family of all finite sequences w = {w k }lK

satisfying the inequalities (2.173). We emphasize that the lengths K - #w of different elements from W m a y be different. We introduce also the semiclosed intervals I0 = (cx~, w(1)], Ii=(-cx~, W(2)] and !k--[W (k-l), w(k+l)), 2Kk
CHAPTER 2 Differential Equations with Hysteresis

244

(TR) Let s be defined by x(a) ~ Ir Then assign K = K(a) = #w(a) := s + 1; afterwards assign w (K(~)) (a) := x(a). Note that the function K(t) = #w(t), zKtKa, is piecewise constant with the jump-points ti given by ti -- x - 1 ( w (ki) ('~)),

where

w (ki) ('~) E [x('~), x(o)].

Moreover, all components of w(t) except the last one are unchanged between the successive jumps. This rule (TR), together with the semigroup property, can be conveniently used as a base for an algorithmic description of the evolution of the state of the Preisach model, and in numerical algorithms for solving ODEs with the Preisach model.

Transformation Rule for Outputs

Let us also discuss a similar transformation rule to compute the varying outputs of the Preisach model together with its states. Let A(a, b) be the triangle with the vertices (a, a),(a, b),(b, b): A(a, b ) = {(a, fl) : aKa < fl~b}. Denote for a~b #(a, b) = #(A(a, b)) = f h(~, fl) d~ dfl, dA(a,b)

(2.174)

and define/z(b, a) = -#(a, b). In some cases there can be found an explicit formula for #(a, b). For a sequence w ~ W, with #w = K we correspond the array #(0) . . . . . #(K) by the equalities #(0) = 0, fl(1) : fl(--OO, W (1)) and the recurrent relationships fl(k) = ]z(k-1) if_ fl(w(k-1), w(k)). Clearly, (I)(w) = #(K). Since we are interested in outputs of the Preisach model for piecewise monotone inputs, then the Transformation Rule (TR), should be changed. Let the input x(t) be monotone at the interval r ~ t ~ a . Then w(a), K(a), and the output u(a) are obtained from the state w(r) via the following rule (TR*) below, see aslo Fig. 2.62: (TR*) Let ~ be defined by x(a) ~ I~. Then assign K = K(a) = #w(a) := ~ + 1; assign w (K)(a) := x(a); assign #(K)(a) "- fl(K-1)('0 -}- fi(w(K-1) , w(K)), with w (~ = -cx~; assign u(a) "- #(K) (a).

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

245

Let the function x(t) have all different local extrema whose values are different from the components of w(~). Then K(t) jumps up only at extremal points and points equal to w (i) (~). Note that if x(t) is increasing in the interval t ~ (r, a), i.e. x1(t) > 0, then K(t) is odd. Similarly, if x'(t) < 0 for t ~ (r, a), then K(t) is even. It is convenient to introduce the following notations K t = K~(t) - K t ( K ( t ) ) - K 4- 1 - ( K m o d 2 ) ; K S - K S (t) = K S (K(t)) = K 4- (K m o d 2).

Sometimes it is also convenient to describe the future dynamics of the output in terms of the current state w. LEMMA2.4.1. Let x(t) be increasing and x(t) ~ I~ for r K t < a. Then u(t) = y(K~-I)(r) + #(w(K~-I)(~), x(t)) u(t) -- # ( - ~ , x(t))

if if

K(r) > 1; K(r) - 1.

Let x(t) be decreasing and x(t) ~ I~ for r ~ t < or. Then u(t) - fl(K4-1)(1;) 4- fl(w (K4-1) (T,), x(t)).

Transformation Rules as Restrictions of Smooth Functional Dependencies Smooth Density It is important to treat the i n p u t - o u t p u t relationships as localizations of some smooth functions. Below we describe the convenient modifications of the transformation rule (TR*). We note initially that the function it(a, b), as defined for all pairs a, b in formula (2.174), is often n o n s m o o t h enough at the line a = b. We suppose that in the domain of its definition the function g(a, b), F ~ a ~ b ~ b coincides with some smooth function ~(a, b), -cx~ < aKb < cx~. Usually, this function/t can be defined as: /, ft(a, b) - / h(~, fl) d~ dfl; JA (a,b) where h(a, fl) - h(max{~, a}, min{b, fl}). Introduce the families of functions #+ (x), #w (x), w ~ W, x ~ ~1 as follows: g+(x)-

/ /t(-oc, x),

/

~(w(Kt -1),x),

if

K-1

if

K>I

246

CHAPTER 2

Differential Equations with Hysteresis

FIGURE 2.62 Evolution scenarios (TR*) for the state of the Preisach nonlinearity depending on parity of K=K(to) and direction of the change of the input x.

and

#w (X) = fl(X, w(K4-1)). These functions are s m o o t h in the x variable. LEMMA2.4.2. Let x(t) be increasing and x(t) E I~ for T~t < a. Then a smooth

function u(t) = fl (Kt-1) (-c) q- flw(T) + (x) coincides with the output u(t) of the Preisach nonlinearity for r ~ t < a. Let x(t) be decreasing and x(t) E I~ for TKt < a. Then a smooth function

~(t) = fi(K4-1)(.~) _ #w(T)(x). coincides with the output u(t) of the Preisach nonlinearity for r ~ t < r

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

247

N o n - S m o o t h Densities Sometimes the density h(a, fl) defining #(a, fl), see (2.174), is not smooth itself. In this case the description of the functions should be modified. Here we describe how it could be done. We restrict ourselves to the case when the density h(~, fl) coincides with some smooth function h 9 ~2 ~ [~2 inside a given triangle A(~, b), and is zero outside of this triangle. Consider the function

~(a, b) =/A(a,b)

h(~, fl) d~ dfl

which is smooth on {(~, fl) : -cx~ < aKb < cx~}. The function #(a, b) coincides with ~(a, b) on the triangle A(~, b). Introduce the families of functions #+(x), #w (x), w ~ W, x ~ [~1 as follows.

#+ (x) --

~(~, x)

if

[t(max{w (K1"-1), ~}, x)

if

0

otherwise

K-

1 and a ~ w (1) < b, ~Kw (K~ < b,

and

/ It(X, W(K4-1)) /~w(x)

/

0

if

~ w (K) < b,

otherwise

These functions are smooth in the x variable. LEMMA2.4.3. Let x(t) be increasing and x(t) E I~ for r ~ t < a. Then a smooth function

~(t) = fl (Kt - 1)(.~) if_ flw(r) +

(x)

coincides with the output u(t) of the Preisach nonlinearity for rKt < r Let x(t) be decreasing and x(t) E I~ for r ~ t < r Then a smooth function f~(t) = I~(K,-I) (T) -- l~w(r)(x). coincides with the output u(t) of the Preisach nonlinearity for r ~ t < a. In this sense i n p u t - o u t p u t relationships of the Preisach model are restrictions of some smooth functional dependencies.

248

CHAPTER 2

Differential Equations with Hysteresis

2.4.2 LIPSCHITZ CONSTANT OF THE SHIFT OPERATOR

Let us consider Eqn. (2.13) with the initial conditions (2.14); we repeat them below for the reader's convenience: dz/dt u(t) ~/(t) x(t)

= = = =

fit, z, u(t)), (I)(r/(t)), (F[r/0]x)(t), (c, z(t)) ;

(2.175)

and to, z(to) = z0, r/(t0) = ~/0 ~ (~..

(2.176)

Here t ~ R+, z e ~n, the scalar function u(t) is the output of the Preisach nonlinearity and the operator F describes the evolution of the state ~/of the nonlinearity depending on the scalar input x(t). We assume t h a t f is differentiable in all variables and satisfies [fit, z, u) - f i t , z, ~t)l<~2fu iu - ~tl.

(2.177)

Recall that every r/ e (~. is uniquely defined by a graph L given by the function 0r : N+ ~ N, see (2.172). Thus a metric between functions 0r(u), Oo(u) can be used as a metric between the states ~/, ~ ~ (~. of the Preisach nonlinearity. To estimate the Lipschitz constant of the shift operator we use the metric given by

P'(~' ~)

=

max

0
J6.(u)

-

O~(u)l.

(2.178)

Let (I)0/) satisfy the Lipschitz condition with the constant 20:

I~(~) - ~(~)l~2~p~(~, ~).

(2.179)

In this case the system (2.175), (2.176) has a unique solution. The state of the system (2.175) is uniquely defined by t, z, r/. We define a metric, r the space of all possible pairs (z, r/) as Q( (z, r/), (~, ~) ) = max{21cl. Iz - zl, Pn(~/, ~)}.

(2.180)

Lipschitz Constant of the Shift Operator The logarithmic norm A/'(Q) of a square matrix Q (with respect to a norm i l 9 l i) is defined by .M(Q) =

III + h Q i I - 1 h---~0, h>0 h lim

249

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

where I is the identity matrix and [[-[[ is the operator norm of the matrix. In the Euclidean metric the logarithmic norm A/'(Q) of the matrix Q coincides with the largest eigenvalue of the symmetric matrix 1 (Q, + Q) where Q' is a transposed Q (see [183], p. 59). Let E be a subset of Ea. PROPOSITION 2.4.1. Let (z(t), rl(t)), (~(t), O(t)) be two trajectories of(2.175) with

the initial conditions at time to being (zo, ~1o) and (zo, rio) respectively. Let the initial conditions be coherent, i.e., (F[r/0]z)(t0) = r/0

(F[O0]~)(t0) = ~0,

and let z(t), ;:(t) ~ E for all t ~ [to; tl]. Let A>~ sup~2~QAf((~) where (2 is a family of matrices given by

sup

Q - { f ~ - ( F / / j ) ' ~ / / j ~ [ [.zea,t,u inf -~z] 0j~ (t' z, u), z~E,t,u -&jzj(t' z, u)

]/

.

Then the following inequality holds O((z(t), rl(t)), (~(t), ~l(t)) )~e (A+22fu2~)[cl)(t-t~ O( (z0, t/0), (z0, ~0) )- (2.181) PROOF. Introduce the difference r(t) = z(t) - ~(t). To make the formulas below more compact we use z, w, u, x, r to denote z(t), r/(t), u(t), x(t), r(t), respectively, and x0, r0 to denote x(to), r(to). Similar notations are used for ~(t), ~(t), ~(t), ~:(t), ~(t). Let us estimate [r[':

1 (r, flt, z, u) - f i t [ r [ ' - (v/(r, r))' = {r,[r[r') = [r--[

~, ~))

_- 1 (r, f(t, z, u) - f i t , ~, u)) + ~[1 (r,f(t , ~, u) - f i t , ~, ~)) Irl

<.~1 (r,f(t , z, u) - f i t ,

~, u)) + [fit, ~, u) - f i t , ~ ~)l.

The distance between ri and ~ changes depending on the distance between inputs x, 2. From the definition of p~ it follows that

P~(r/' O)K max { p~(r/~ ~~ 2 maxs~[to,I(c,z(s)-~(s))l}-p(t). t]

(2.182)

CHAPTER 2 Differential Equations with Hysteresis

250

Using the Lipschitz properties off on u we can rewrite if(t, ~, u) -fit, ~, ~)[ as

If(t, ;:, u) -fit, ;:, ~)1 ~< 2Su lu - ~1 ~<XS,, I@(~) - @(4)1

'~fuX~Prl(rl, ["l)~Xfu 2@p(t). Let us rewritef(t, z, u) -fit, ;:, u) as

f(t, z, u) - f(t, ~, u) -

= Q(t, z, 5, u)r,

where Q(t, z, ;:, u) is a square matrix given by the integral. Using properties of the inner product we can write

Irl'

<~ ~1 (r, Q(t, z, ;:, u)r) + ,1.f~2@p(t) Af'(Q(t, z, Y:, u))lrl + 2fu 2@p(t).

(2.183)

Taking the m a x i m u m over the time interval [to, t] in (2.183) we get

d(maxse[to,t] Ir(s)l) ~ A max Ir(s)l + 2fu2<s)p(t). dt s~[to,t] Letus introduce the function Q(t) = max{21cl "maxse[to,t] Ir(s)l, p(t)}. Using the inequality p'(t) = (2 max I(c, r(s))I)'K21cl( max Ir(s)l)'

s~[to,t]

s~[to,t]

that follows from the definition of p(t), we can estimate Q(t)'r

~ max{21cl(maxse[to,t] Ir(s)l)', p(t)'}~21cl(maxse[to,t] Ir(s)l)' ~< 2AIcl maxs~[to,t] Ir(s)l + 22f~2@lclp(t) (A + 22f~2@1cl)- max{21cl maxse[to,t] Ir(s)l, p(t)} (A + 22f~2@1cl). r

By the Gronwall inequality we can write

r

(A+22f~2@lcl)(t-t~ Q(to).

To obtain the statement of the proposition it remains to note that

~( (z(t), ~l(t)), (;:(t), ~(t)) )Kr and that since the initial conditions are coherent,

p ( t o ) - max {p~(r/0, ~0), 21 (c,z(to)- ~(to))I} = P~(r/0, ~0) and r

- O((z(to), r/(t0)), (z(t0), ~(t0)) ). The proposition is proved.

[]

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

251

Lipschitz Constant Estimates for the Pendulum Example As an illustrative example of Proposition 2.4.1 we estimate the Lipschitz constant of the shift operator along the trajectories of (2.9). We estimate the constant L for the trajectories in z(t) ~ ~2 and for the trajectories {z 9 Iz(1)1~0.5} separately. The results will be used in the computer-assisted proof in Subsection 2.4.4. We begin with the estimate of A, which turns out to be the same for all z(t) ~ ~2. Consider the vector form of the Eqn. (2.9)

d z / d t =fit, z, (P (c, z))(t)), where z = (z (1), z (2)) and

fit, z, (P (c, z))(t))

-

(Z (2), --Z (1) q- sin(V~t) + 2b(Pz(]))(t) - b) .

From here

Q(t,z,~,u)-

-~z =

-1

0

"

Thus A = 0 as 0.5(Q + Q') is a zero matrix and has two eigenvalues equal to 0. For Eqn. (2.9) c = (1, 0), Icl = I and constant 2du is equal to 2b. Let (z(t), r/(t)) and (~(t), ~(t)) be the solutions of Eqn. (2.9) with the coherent initial conditions z(to) = z0, r/(t0) = r/0 and z(t0) = z0, ~(t0) = ~0Firstly, we consider the situation w h e n z(t), ~(t) ~ {z : Iz(1)1~0.5}. The output of the nonlinearity u = (I)(r/) is constant as long as Ix(t)[ = Iz(1) I~0.5. Thus 2~ = 0, L = 0 and using (2.181) we may write r

r/(t)), (~(t), ~(t)) )KQ( (z0, r/0), (~0, ~0) ).

(2.184)

Secondly, let z(t), ~(t) ~ ~2. In this case 2~ is estimated as the largest possible distance between two curves in triangle (2.8), a = {(a, fl) : -0.5~<~
Thus 2~ - 0.5. Substituting all the constants to (2.181) we get r

r/(t)), (~(t), ~(t)) ) ~ e 2b(t-t~

(z0, r/0), (z0, ~0) ).

(2.185)

2.4.3 NUMERICAL INTEGRATION

Eqn. (2.175) can be numerically integrated by a method of the first-order as the right-hand side of the differential equation is Lipschitz-continuous. The standard example of the explicit first-order method is the Euler method.

CHAPTER 2 Differential Equations with Hysteresis

252

In general, numerical integration by first-order methods is neither efficient nor precise and this section is devoted to the scheme that allows the application of high-order methods for the integration of Eqn. (2.175). For the purposes of the numerical integration we are going to use algorithmic description of the Preisach nonlinearity given in Subsection 2.4.1 in terms of w. To simplify the notation we assume that r/is equivalent to w and we loosely use the representation r] = W -- {W ( 1 ) , . . . , W (K)}

if there is no ambiguity. We assume that the initial states of the Preisach nonlinearity r/0 for the initial-value problems like (2.175), (2.176) always admit algorithmic representation.

Euler Method The Euler numerical integration method is given by the formula Zn+l tn+l

---

Znf(tn, Zn, Un), tn Jr-'Cn, (F[r/n ] (C, Z n + l ) ) ( t n + l ) ,

r/n+1 Un+l

------ (I)(r/n+ 1).

Here Zn, Un, tin are the values of the numerical trajectory at tn and Zn is the size of the nth time step. Here the state r/n+1 = w and the output Un+l of the Preisach nonlinearity are computed according to the transformation rule (TR*). We denote the result of the z-size numerical integration step of (2.175) with the Euler method as Veul(tO, zo, r/o; to + z) -- (Zeul;1,//eul;1)" The local error of the Euler method with the time step z~h and z0 ~ E can be estimated as h2 ~f ~_~_f/( sup + t,, z, u,) [z(t0 if- Z) -- Zeul;ll ~ -~- z**E,t,,u,

sup zo~E,to,u, z~h

--~

q-h2

'

sup 2fuA~[c[.Lf(t,,z,,u,) [ z, ~E, t,, u,

< eeul;z(h, E). Since the Zeul;1 is linear as a function of z, it is possible to write sup

~( Veul(t0, z0, r/0; to + z)), (z(to + z), r/(t0 + z)) ) < 2[C[eeul;z(h, E).

z ~h, to, rlo,Zo~E

(2.186)

253

2.4 EXAMPLESOF TECHNICAL TOOLBOXES

Thus the number eeul(h, E) = 2]Cleeul:z(h, E) estimates the local error of the method. The algorithm for the numerical integration of (2.175) offered below is going to rely on the Euler method steps in some 'complicated' situations.

Local Error of the Higher-Order Methods Locally Equivalent Systems

Let K(t) denote the length of the sequence {w (i) (t) } that describes ~/(t) and let K denote K(to). Let the initial conditions (2.176) be coherent, i.e. (F[r/0] (c, z))(t0) = r/0. If x'(to) > 0 then by Lemma

2.4.2 the system (2.175) is locally equivalent to dz/dt u(t) w(Kt)(t)

= --

fit, z, u(t)), #K(tO) + #-~o(x), x -- (c, z) .

(2.187)

If x'(t0) < 0, the system (2.175) is locally equivalent to dz/dt u(t) W (K~)(t)

= --

fit, z, u(t)), #K(tO) -- t~qo(x), x -- (c, z).

(2.188)

Both (2.187) and (2.188) are smooth and can be numerically integrated by standard methods. Denote the shift operator along the trajectories of the Eqns. (2.187), (2.176) for the time z by q~t(to, zo, r/0; to + z) = (zt(to + z), ~l~(to + z))

where t/1,(t) = {w(1) . . . . . W (Kt-1), (c, zt(t))}. Denote a step of size z of the numerical integration of the Eqns. (2.187), (2.176) by vt(to, zo, q0; to + z) - (Zvt(to + z), ~lv~(to + z))

where rlv~ (t) = { w (1) . . . . , w ( K i ' - 1 ) , (c, Zv~ (t))}. Notations v4 (to, z0, r/0; to + z) and q~ (to, z0, r/0; to + z) are introduced analogously for Eqns. (2.188), (2.176).

Numerical Step Error for the Equivalent Systems

Let etz(h, E) estimate

the local error of the numerical integration of (2.187) sup zonE, rlo, to~tl ~to+h

[Zvt(tl) - z t ( t l ) l ~ e t z ( h ,

E)

CHAPTER 2 Differential Equations with Hysteresis

254

and let e+z(h, ~) estimate the local error of the numerical integration of (2.188). Let ez(h, E) = max{etz(h, ~), e4z(h, E)}. (2.189) Note that by the definition of r/vt (t) and r/t (t) Q(vt (to, zo, r/o; tl), (Pt (to, zo, r/o; tl)) = max{21c[. Izvt(tl) - zt(ta)[, p,(r/v,(tl), r/t(tl))} ~< max{21cl Izvt(tl) - zt(tl)[, ]2(C, Zvt(tl) - zt(tl))l}" <~ 2[c[]zvt(tl) - z t ( h ) I. Taking s u p r e m u m over ~ 6 E, ~, t0~>0, to<~tl<~to + h on both sides of the inequality above, we may write sup r

t (to, z0, r/0; tl), ~Pt (to, z0, r/0; tl)) < 2]c]etz(h, E).

(2.190)

Similarly for (2.188), sup ~o(v+(to, zo, r/o; h), q~+(to, zo, r/o; h)) < 2[cle+z(h, E).

(2.191)

Thus the number e(h, E) = 2]c[ max{etz(h, E), e4z(h, E)} = 2Iclez(h, ~.) estimates the step error of the numerical integration of both (2.187) and (2.188) in terms of metric Q. With some technical restrictions it is possible to substitute the numerical integration of the system (2.175), (2.176) with that of the appropriate local equivalent. Introduce the notations x(t) = 0 on [to, tl], tl - to~h. If Kt(to) < 3 or xvt(tl) + [clez(h, E) < w(Kt(to)-2)(to) then z t (t) = z(t),

r/t (t) = r/(t)

and

Q((z(h), r/(tl)), vt(to, zo, %; tl)) < e(h, E). Let x'(t) < 0 on [to, tl], tl - to~h. If K4(to) < 3 or Xvj,(tl) - [cIez(h, E) > w(K4(t0)-2) (to) then z~ (t) = z(t), rl~ (t) = ~l(t) and

•((z(tl), r/(tl)), v4 (to, z0, r/0; tl)) < e(h, E).

255

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

Integration Algorithm General Description

The idea of the numerical algorithm is simple. During each cycle of the algorithm we are trying to make a time step h ~ [hmin, hmax] along either (2.187) or (2.188) in such way that the conditions of Lemma 2.4.4 are satisfied. If, as a result of the step, we are too close to the zone w h e r e f of (2.175) is not smooth enough as a function of t, we try to 'jump' over the area of non-smoothness, see Figs. 2.63, 2.64. In this way the local error associated with 'jump' steps is slightly bigger then usual. If some technical restrictions for the 'jump' are not satisfied, the algorithm makes hmin time-step using the Euler method. The advantages of the offered algorithm over the straightforward Euler integration are clear: 'most of the time' we use a high-order integration method thus significantly decreasing global error estimates and the total CPU time for the integration. Since the offered algorithm only 'describes' the conditions in which it is possible to use high-order integration methods, it can be easily coupled with an integration method best suited for the given ODE. The main disadvantage of the algorithm is that the effective global error estimate needs to be computed incrementally along the trajectory, due to the mixture of different integration steps. One particular application of the algorithm, a computer-aided proof of chaotic behavior, is discussed in Section 2.4.4.

Required Constants

To formulate the algorithm we require the follow-

ing constants: 9 the size of the 'jump' zone: Y = Nmax{1, A + Ic[2fu2~} 9e(h, E),

N>I;

9 minimal hmin and maximal hmax possible time steps; 9 set (Wv) of horizontal(vertical) lines on which the Preisach measure under consideration is discontinuous; 9 the maximum length Kmax of the vector w in the algorithmic representation of the Preisach nonlinearity.

Wh

Dit,

dix/dt i

Let i = 1, 2, 3, estimate the derivatives for all solutions of interest z t (t) of the Eqn. (2.187); derivatives are taken along the trajectories of the system. Dlt > to. t. z~E. t/o D2t >

I df(t,z,

sup I C, ~-~ to, t, z~, r/0

+(x))/I ~K(t0 ) + #,o

256

CHAPTER 2

sup

Differential Equations with Hysteresis

I c,=+

+

to, t, z~E, r/o

D31' >

sup

c3u

d2d

9

c3x

cgt

I

'

+ / I.

I c, -d-~(t, z, laK(tO) + #rio (X))

to, t, z~.~., r/o

I n t r o d u c e Di - m a x { D / t , Di4 } w h e r e n u m b e r s DiS are i n t r o d u c e d a n a l o g o u s l y for Eqn. (2.188). Let vl(to, zo, ~/o; to + r) b e d e f i n e d as v 1,(to, zo, r/o; to + ~), Vl(.) v4 (to, zo, ~/o; to + ~),

if if if if

x 1(to) x 1(to) x 1(to) x 1(to)

> < =

0 0 0 0

or a n d x~ (to), x~ (to) > 0 or a n d x "(t0), t x4" (to) < 0.

Finally, as t h e r e are explicit f o r m u l a s for xi, x i,' x i" in t e r m s of locally e q u i v a l e n t s y s t e m s (2.187), (2.188) w e a s s u m e t h a t for e a c h step t h o s e valu e s are c o m p u t e d for t h e e q u i v a l e n t s y s t e m u s i n g c u r r e n t v a l u e s of ti, rli, zi. N o t e , t h a t if a locally e q u i v a l e n t s y s t e m is n o t d e f i n e d , v a l u e x n c a n n o t b e c o r r e c t l y c o m p u t e d , b u t it is n o t u s e d either.

Integration algorithm

INTEGRATE10 Initial values to, r/0, z0; time step range [hmin, hmax]" 'jump' tolerance Y; arrays }'Vh, }iVy. O u t p u t : Result of the integration is an array of {(ti, Zi, rl i, gi) }in_~0 (1) n = 0, t o, z o, r/0 (2) GETNEWZPLANEO (3) while n < = n o (4) hi = ([Xn - Xhyp[- T ) / D 1, h 2 - ([Xn[- Y)/D 2 (5) if hl < h2 then ~n -- min{hmax, h 2 } (6) else ~n = min{hmax, h 1, [x"nl/D3} (7) if TnKhmi n or vI (tn, Zn, tin; .) is not defined (8) EULERSTEP0 (9) else (10) Zn+l = vl (tn, Zn, tin; Zn) (11) ~n=0 (12) if sign(xn - Xhyp) 9 (Xn+ 1 - Xhyp) < T (13) JUMPTOZ0 (14) if sign(x~) 9 Xn+ f 1 ~
Input:

2.4 E X A M P L E S OF T E C H N I C A L T O O L B O X E S

257

GETNEWZPLANE0 I n p u t : Set of planes ]/Vh, ~/Vv on whichf from (2.175) is not differentiable O u t p u t : Xhyp defining the position of the hyperplane (1) Xdisc h -- min{ff; ~ Wh U {~} : ~ > x} (2) Xdiscv "- m a x { ~ E W v U { - cx~} : Co < x} (3) if (c, f(tn, Zn, (I)l 0/; Xn))) < 0 (4) if KS < 3 t h e n Xhyp = Xdisch

(5) (6) (7)

else Xhyp = min{Xdisc h, W(nK*-2) else if K 1, < 3 t h e n Xhyp -- Xdisc v

(8)

else Xhyp -- max{Xdisc v, W(nKt-2)

EULERSTEP0

I n p u t : Current step n O u t p u t : Updated values of Zn, Zn+l, r/n+ 1, qn (1) ~n = 2 Zn -- hmin (2) Zn+l -- Veul(tn, Zn, tin; Zn) JuMPTOZ0 I n p u t : Current step r/ Output: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Updated values of "On, Zn+l, tin+ 1 , ~n if n = = 0 or qn-1 - - = 1 EULERSTEP0 return w h i l e (Xn - Xhyp) 9 (Xn+l - Xhyp)~<0 or ]Xn+ 1 -- Xhyp] > ~I" or ]Xn+ 1 -- Xhyp] < ]c[ez(h, E) Modify Zn Zn+l = vl (tn, Zn; Zn) Project Zn+ 1 orthogonally on x -- Xhyp ;n = 1

JUMPTOZP0 I n p u t : Current step n Output: (1)

Updated values of 77n, Zn+l, Yln+l, gn if n ==

(2) (3) (4) (5) (6)

0 or ~n-1

==

1

EULERSTEP0 return

!

!

0} (7)

I

Choose the largest Zn such that x n 9Xn+ 1 > 0 and IXn+ 1 [~>(A + 21cl2f 2,)lclez(h, E) Compute r/n+1 Letz. betheclosesttOZn+ 1 pointof{(z, r/) : (c, f(tn + Zn, z, rl.)) = if

]Zn+l

-- Z , [ / > F t h e n

EULERSTEP0

258

CHAPTER 2

Differential Equations with Hysteresis

(8) else Zn+ 1 = z,, qn -- 1 UPDATEPREISACH(r/old,X) Input: Old state r/old, new input X Output: Updated value r/ -- {W (i) } and appropriate K (1) r / = r/old, K -- Kol d (2) if Ix[ > = ? (3) K= 1 w (1) = s i g n ( x ) . ? (4) return ~/ (5) if x > w (K~ (6) if K mod 2 = = 0 then K + + (7) while K>~3 and x > w (K-2) (8) K= K- 2 (9) else if x < w (K~ (10) if K mod 2 = = I then K + + (11) while K~>3 and x < w (K-2) (12) K -- K - 2 (13) if K~>Kmax then error (14) w (K) = x (15) return r/ Let us c o m m e n t on the m e t h o d JumpToZPO. In principle, if (c,f(t, z, u)) d e p e n d s o n u it m a y be v e r y h a r d to localize p o i n t s (z,, r/,) s u c h t h a t at a g i v e n t = t,, (c,f(t,, z,, u,)) = 0. In this case JumpToZPO c a n be r e p l a c e d b y the call of the E u l e r S t e p m e t h o d , t h a t w o u l d p e r f o r m the i n t e g r a t i o n at the step n b y the E ul e r m e t h o d .

Properties of Numerical Solution PROPOSITION 2.4.2. Consider a successful step n of the algorithm Integratel O. The

error of the integration step satisfies Q(q)(tn, Zn, r/n; tn+l), (Zn+l,r/n+l)) < e(hmax, E),

/f~n = 0 ;

Q(q~(tn, Zn, r/n; tn+l), (Zn+l, r/n+l)) < e(hmax, E) + 2 1 c l T ,

(2.192)

if qn = 1; (2.193)

Q(q2(tn, Zn, r/n; tn+l), (Zn+l, r/n+1)) < eeul(hrnin, '~"),

if; n = 2.

(2.194)

Error Propagation Let the shift o p e r a t o r ~o(t0, z0, r/0; t) satisfy Q (~a(to, zo, r/o; t), q~(to, zo, rio: t)) ~ e L(t-t~ •((zo, r/o), (z0, ~o)) as l o n g as for all toKtKt,, ~Oz(to,zo, r/o; t), ~Oz(t0, z0, ~0; t) ~ E.

(2.195)

2.4 EXAMPLES OF T E C H N I C A L TOOLBOXES

259

/\x - --~ numerical traj. real traj. Ioc. err.

I

Jumping

Ioc. err.

~

Ioc. err. time

Z> F I G U R E 2.63

Behavior of the a l g o r i t h m close to the h y p e r p l a n e x - Xhyp.

X

--'> numerical traj. -real traj. j

x'=O

-

~

_

~

/// ....-~"

~

~ _

"

"

--

...~._..,.,.;.::.~ ..... L . . . . . . . . . . . . . . ~-.--: " Ioc. err.

"~'~'~"~' \ Ioc. err. ".(~ ::> ti m e

~x'

o,.

11! Jumping

-y

F I G U R E 2.64

"'-

Ioc. err.

oc;er:

f

time

Behavior of the a l g o r i t h m close to h y p e r s u r f a c e x t = 0.

260

CHAPTER 2 Differential Equations with Hysteresis Let 8appr estimate

sup [~z(t,, z,, r/,; t , + ~ z ) - ( 1 - a ) z , - a v z ( t , , O<<.z<<.h,t,,z,~G,rl,

z,, r/,; t,+z)l < 8appr. (2.196)

PROPOSITION2.4.39 Let the algorithm Integratel O terminate successfully and let 110 the output of the algorithm be a set {(tn, Zn, qn, ~n)}n=0" Let ~ be a convex open set. Let e > 0 be a given number. Introduce 811 by the following recursive formula:

80 = en+l 8n+1 8n+l

8, -- 8ne L~n + e(hmax, ~) = 8ne Lzn 4- e(hmax, E) 4- 21cIY = 8ne Lzn 4- eeul (hmin, E)

if q n - - 0 if gn -- 1; if qn - 2.

Let the points Zn satisfy Zn ~ E(Sn 4- 8appr), where E(a) = {z ~ E ' l z -

z,I > a for all z, ~ ~9E}.

Then for each (zo, rlo) such that Q( (zo, flo), (zo, rlO) ) ~ e ~Oz(to, zo, q0; t) ~ E,

for all to<<.t<~tn

(2.197)

and Q( ~p(t0, z0, q0; tn), (Zn, rln) ) < en,

for all O<<.n<<.no.

(2.198)

PROOF. Introduce the notation r ;50, q0; t) = (~(t), ~(t)). Let us s h o w the statement of the l e m m a b y induction on n. The statement of the l e m m a holds for n = 0. A s s u m e that the statement of the l e m m a holds for some n. Let us s h o w that it holds for n 4- 1. A s s u m e that (2.197) is not true. Then there is a m o m e n t t, such that t,-

min {t'f:(t) ~ 0E}. tE[tn,tn+l]

Let v(t) = (1 - a)Zn + ~Zn+l where ~ = (t - tn)/Zn. Then by the triangle inequality [~(t,) -- v(t,)l ~ ICpz(tn, Z(tn), ~(tn); t,) -- r

4.1r

Zn, tln; t,)[

Zn, tin; t,) -- v(t,)l.

By (2.195) and (2.196)

ICPz(tn, Z, tl; t,) -- v(t,)l<~ene L(t*-tn) 4. eappr.

2.4

EXAMPLES OF TECHNICAL

261

TOOLBOXES

Let d(., 0E) denote the distance to the b o u n d a r y of E. By the convexity of we can write

d(v(t.), cgE)~>(1-c0d(zn, OE)q-c~d(Zn+l, OE) > e.apprq-en((1-cOq-o~e L(tn+l-tn)) where c~ - (t-tn) /Zn. Since (1--CO-}-o~eL(tn+l-tn) >/e L(t*-tn), point qOz(tn , ~:, fl; t. ) belongs to E. We have arrived at a contradiction and (2.197) holds. The inequality (2.198) for n + I holds by (2.197) and Proposition 2.4.2. []

Plane Intersection Algorithm To construct a numerical analogue of the first return map on the hyperplane, one needs a procedure to estimate the area in which the intersection of the solution with the hyperplane happens. Setting aside the question of whether it is the first intersection or not, we offer a simple algorithm that allows us to localize the point of intersection on a given hyperplane using the information on the numerical solution and integration errors. Let (p, z) - a be a hyperplane in R a. Let {(tn, Zn, r/n, ~n)}n~ 0 be a numerical solution produced by IntegratelO. Let (z(t), r/(t)) denote the solution of (2.175) with the initial conditions at to being (z0, r/0). Let there exist 0~
9 Iz(tn) -

Znl~81

9 a - ( p , znl)

;> 81

for all nl<~n<<.n2, and (p, Zn2)- a > el.

Let np be defined as

np -

min {n " (p, Zn) ~ a<<.(p, Zn+l)}. nl <<.n<<.n2

Let us find the intersection point of the line segment O~Znpif- ( 1 - - 0 0 Z n p + l , 0 ~ C t ~ 1 with the hyperplane (p, z) - a: Zbase - -

(XZnp -}- (1

-

~)Znp+l

where,- /P Znp+ll (p Znp+l Znpl For all Zn, n ~ [nl, n2] we compute the projection ~:n on the hyperplane (p, z / = a along the vector Zproj - - Znp+l -- Znp"

Zn = Zn -Jr-flZproj,

fl -- (a -- (p, Zn})/ (p, Znp+l -- Znp).

Let 82 m maXnl<<.n<<.n2 IZn -- Zbasel.

CHAPTER 2 Differential Equations with Hysteresis

262

LEMMA2.4.5. In the conditions of this subsection, z(t) intersects the hyperplane (p, z) = a at t, ~ (tnl, tn2) in the direction 'from (p, z) < a to (p, z) > a'. The

intersection point z(t,) belongs to the open ball Bo of radius

81

(C,C)(Zproj,Zproj)

(C,Zproj)

-}-

82 q- 8appr centered at Zbase, where 81, 82 are given above.

If, additionally, 0<

inf

rl,zEBo,tn 1 ~t~tn 2

(p,flt, z, (I)(F[r/0] (c, z) (t)))),

then t, ~ (tnl, tn2) s u c h that (p, z(t,)) - a is unique. PROOF. The proof follows from the construction method.

[]

Computation of the Topological Degree of a Plane Vector Field Consider a continuous map F 9 ~2 ~ ~2 defined on Jordain set f2 c R 2 with the piecewise smooth boundary. Since we are mainly interested in fixed points of F we restrict ourselves to the problem of computing deg(f2, F - id, 0) = 7(f2, F - id), where id stands for identity function. There are various algorithms for computing two-dimensional rotation of the vector field. Here, just for consistency, we offer one of them. Let B(v; r) ~ ~2 be an open ball satisfying 3r2 > 0 such that F(B(v; r)) c B(F(v); r2); F(B(v; r)) n B(F(v); r2) -- f~. (2.199) Then for every v, ~ B(v; r) the angle between the vectors F(v,) - v, and F(v) - v is strictly less than n/2. Let F(s), OKsKs,, F(0) = F(s,) be a 'natural' parameterization of 8fl; (i.e. s is length of the curve 0f2 between the points F(0) and F(s) measured counterclockwise).

(1) (2) (3) (4) (5) (6) (7)

Choose so = O, v0 = F(s0) and rmin; n = O. while (true) if B(vn; rmin) not satisfies (2.199) then error Choose rn~rmin such that Bn = B(vn; rn) satisfies (2.199) if Of2 ~ U~=0Bk then terminate successfully Choose Sn+l > = Sn + rn such that F([Sn; Sn+l]) C Bn Xn+l --

F(Sn+l); n + +

LEMMA2.4.6. If the algorithm above terminates successfully, then

/(F(vn)_Vn, 7(f2, F) = [ 1~nn ~i=1

f(Vn+l)_Vn+l)q_~l,

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

263

where /(v, v.) stands for the angle between the vectors v, v. measured counterclockwise. PROOF. The proof easily follows from the algorithm.

Proofs System Equivalence Lemmas LEMMA2.4.7. Let (z(t), rl(t)) be the solution of (2.175), (2.176) and let x(t) = (c,z(t)). Let x'(t) > Ofor all t ~ (to, tl]. If Kt(to) < 3 or if Kt(to)~3 and x(t) < w(Kt (t~ then the solution (z(t), ~/(t)) coincides with the solution of (2.187), (2.176) on the time interval [to, tl]. Let x'(t) < Ofor all t ~ (to, t]]. IrKs(to) < 3 or if K4(to)~3 and x(t) > w(K4 (t~ then the solution (z(t), ri(t)) coincides with the solution of (2.188), (2.176) on the time interval [to, t]]. LEMMA2.4.8. Let (z(t), ~l(t)) be the solution of (2.187), (2.176) and let x(t) =

(c, z(t)). Let x'(t) > 0 for all t ~ (to, tl]. If K t (to) < 3 or if K t (to) >>,3and x(t) < W (Kt(t~ then the solution (z(t), rl(t)) coincides with the solution of (2.175), (2.176) on the time interval [t0, tl]. Let (z(t), rl(t)) be the solution of(2.188), (2.176) and let x(t) = (c, z(t)). Let x'(t) < Ofor all t ~ (to, tl]. IrKs(to) < 3 or if K(to)~3 and x(t) < w (K~(t~ then the solution (z(t), ~l(t)) coincides with the solution of(2.175), (2.176) on the time interval [to, t]]. Proofs are omitted. PROOF OF LEMMA 2.4.4. As cases with x'(t) > 0 and xt(t) < 0 are similar we consider only the first one. If Kt(t0) < 3, the statement of the lemma immediately follows from the first part of Lemma 2.4.7. By (2.189) Ixvt (tl) - xt(tl)l -- I Ic, zvt(tl) - z$(tl))I~iclizvt(tl) - z$(tl)l~lclez(h, E). The last inequality means that

x t ( t l ) ~ X v t ( t l ) + Iclez(h, E)~w(t) (Kt(t~ and solutions (zt(t), r/t(t)) and (z(t), rl(t)) coincide on the time interval [to, tl] by the first part of Lemma 2.4.7. The error estimate ~((z(tl), r/(tl)), vt(t0, z0, ~/0; tl)) < e(h, E) is n o w trivial. The lemma is proved.

[]

264

CHAPTER 2 Differential Equations with Hysteresis

PROOFOF PROPOSITION2.4.2. The case w h e n qn the description of the Euler method. Let us consider the situation w h e n qn E nient to formulate the following lemma.

--

{0,

2 is trivial and follows from 1}. Technically it is conve-

LEMMA2.4.9. Let the algorithm Integratel O terminate successfully. Then each step n of the algorithm such that ~n E {0, 1} satisfies Lemma 2.4.4. PROOF. Consider a step n. This m a y be either a step along (2.187) or (2.188). We are going to consider only the case of a step along (2.187) as the situation with (2.188) is exactly the same. Firstly, we show that the sign of x~ (t) is constant on the interval (tn, tn+l]. If x~ (tn) > D2"cn then the condition holds by the fact that D2 estimates sup Ix~ (t)I. If x~ (tn) <<,D2h there are two situations possible: there is a call to JumpToZPO on lines 14, 15 and qn = 1, or there is no such call. In the first c a s e tn+l is chosen so that

X~v(tn+l ) > (A q- 21clAfuA~)lClez(h, E), see lines 4, 5 of the

(2.200)

JumpToZPO; while in the second case

Xl, v' (tn+l) > Y > (A if-21Cl2du2r

E)

(2.201)

see line 14 of IntegratelO. Note that by the choice of ~ n , lines 4-6 of IntegratelO, the call of JumpToZO is impossible. Let us estimate [X~v(tn+l) - x 't (tn+l)[. Since "Cn~
[X~v(tn+l ) - x~ (tn+l)[

<~(c,flt, Ztv(tn+l), ~/1.v(tn+l)) -fit, Ztv(tn+l), r/tv(tn+l))) E).

~
~< x~(tn+l) q-[X~v(tn+l) - x~(tn+l)[ <~ x~(tn+l) q-IcI(A q- 21cl2fu2r E).

(2.202)

Inequalities (2.200), (2.201) and (2.202) imply that x~ (tn+l) > 0. From line 6 of IntegratelO IXnl > D3"cn, which means that x ft (t) is monotone on the > 0 since x 't (tn+l) > 0 and x~>0. interval [tn , tn+l] and x '(t) t

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

265

Secondly, if K t ~>3 we need to show Xv~(tn+l) -Jr-JcJez(h, E) < Xhyp

(2.203)

where Xhyp stores the value of W (Kt-2) . If Xhyp -- X~ (tn) > D l h m a x q- Y then the condition holds by the fact that D1 estimates sup Ix~ (t)l. If x '1,(tn)<<,Dlhmax there are two situations possible: there is a call to JumpToZO on lines 12, 13 and qn -- 1 or there is no such call. In the first case "Cn+l is chosen so that Xhyp-

xtv(tn+l) > Iciez(h, •),

see lines 4-6 of the JumpToP(); while in the second case Xhyp- xtv(tn+l) > Y > Iclez(h, E),

(2.204)

see line 12 of IntegratelO. Note that by the choice of rn, lines 4-6 of IntegratelO, the call of JumpToZP() here is impossible. Inequalities (2.200), (2.201) imply that (2.203) holds.

[]

We can n o w complete the proof of the proposition. Again since cases w h e n x' (t) > 0 and x'(t) < 0 are similar, we consider only the former case. Consider the step n. If ~n -- 0 then v1,(tn, Zn, YIn; tn+l) -- (Zt (tn+l), r/1,(tn+l)) -(Zn+l, r/n+ 1) and the proposition is proved. If qn -- 1, which only h a p p e n s due to a call from either JumpToZO or JumpToZP(), that w o u l d result in

IZn+l - zvt i~Y. In the case of JumpToZPO the condition above is explicitly checked, see lines 6-8, while in the case of JumpToZ() it follows from the orthogonal projection to the hyperplane (line 9) and choice of/:n (lines 5, 7). By the properties of the metric

O(vt(tn, Zn, tin;

t n + l ) , (Zn+l, r/n+l))~<21ci Y

and the proposition is proved.

[]

Example: Integration of Eqn. (2.9) Pendulum Example

To avoid confusion we rewrite Eqn.

notation of this section

dz(1)/dt dz(2)/dt u(t) rl(t)

--

Z(2)

= = =

--Z (1) + sin(v/2t) + b(2u(t) - 1), ~0/(t)), (F[~/.]z(1)) (t).

(2.9) in the

CHAPTER 2 Differential Equations with Hysteresis

266

The measure of the Preisach nonlinearity #(a, fl) is defined as #(a, fl) -- (a, -- fl,)2;

a, -- max{a, --0.5}, fl, -- min{fl, 0.5}.

Thus we have a one-parameter (b) family of equations. Similarly to (2.187), (2.188) we need to introduce locally equivalent systems. If x'(t0) > 0 then by L e m m a 2.4.2 the system (2.9) is locally equivalent to

I

dz/dt u(t) w(Kt)(t)

(Z (2), --Z (1) q- sin(~/2t) + b(2u(t) - 1)), x

--

--

,i

+ (x), ~K (t0) + #,0 X -- (C, Z) -- Z(1).

(2.205)

If x I (to) < 0, the system (2.9) is locally equivalent to

I

dz/dt u(t) w(K,)(t)

(z (2) , --Z (1) -}- sin(~/2t) + b(2u(t) - 1)),

--

#K(tO) -- I~o (X),

"-

X-- (C,Z) = Z (1).

(2.206)

From the formula for the Preisach measure # it follows that u(t) has the form

u(t) = kl q- k2(x - k3) 2

in (2.205),

(2.207)

u(t) - k l - k2(x - k3) 2

in (2.206);

(2.208)

where kl E [0, 1], k2 -- {0, 1}, k3 E [-0.5, 0.5]. Note that ki are determined based on #K(t0), #~0 (x), and u(t) is always smooth. Note that (2.205) and (2.206) are equivalent for Iz(1)l~>0.5. R u n g e - K u t t a M e t h o d of Fourth Order Let q~(t0, z0; t) denote a shift operator for a time t - to along the trajectories of

z' = g(t, z),

(2.209)

with the initial conditions z(to) = zo. We assume that all the trajectories of interest z(t) belong to an open convex set E. Let hi~hmax be the time-step size for numerical integration at the step i and let ti = to + Y~=0 hk. We approximate (p(t0, z0; ti) by vi that are calculated recursively by the Runge-Kutta m e t h o d of fourth order: v0 = z0,

h 4 Vi+l =- Vi -t- -~ E bkkk(vi' ti; hi) -Jr coi. k=l

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

267

Here bl - b4 -- 1, b2 = b3 = 2, coi is the total error of the c o m p u t a tions a c c u m u l a t e d on the step i due to the usage of IEEE double-precision floating-point n u m b e r s . M a p s ki are c o m p u t e d according to the following formulas k] (z, k2(z, k3(z, k4(z,

t; t; t; t;

2:) 2:) 2:) 2:)

= = = =

g(t,z), g(t + 0.52:, z + 0.52:k1(z, t; 2:)), g(t + 0.52:, z + 0.52:k2(z, t; 2:)), g(t + Z, z + )r t; Z)).

We will refer to the numerical integration m e t h o d described above as RK4. General estimates of the local error for R u n g e - K u t t a m e t h o d s can be found, for example, in T h e o r e m 3.1 from [183], p.157, and w e reformulate t h e m for our particular case. LEMMA 2.4.10. The local error e(h, E) of the RK4 with the step size h and initial condition z(t) ~ E at time t can be estimated as h5 ~o(z5) h 5 ~ Ibil (4) e(h, E)<~-~. sup l (t, z; t + z)l + ~ i=1 --6-- sup Iki (z, t; 2:)1 + co. (2.210) Here suprema are taken over all t, z ~ E, 2: E [0, h]; bi are from the definition of RK4: bl = b4 = 1, b2 -- b3 = 2; and co estimates m a x iogii for the given range of parameters. Note that all derivatives in the l e m m a above are taken w i t h respect to the system (2.209). For example, d egg (t, z) + -~z c~g (t, z)g(t, z). k~ 1) (z, t; X) = -~g(t, z) - --~

Poincare Section M e a s u r e # is equal to 0 outside the triangle {(~, fl) : -0.5~fl~0.5}. Thus (I)0/(t)) = 0 (or 1) for every z(1)(t)~ - 0.5 (z(])(t)~ 0.5), m o r e than that, the history of z (1) (t) w h e n Iz(1) (t)l ~0.5 does not affect the behavior of the system w h e n Iz(1) (t) i < 0.5. We a s s u m e that (F[~/]x) (t) = r/0.5 = {0.5} for all t : x(t) > 0.5 and that (F[~]x)(t) = ~/-0.5 = {-0.5} for all t: x(t) > 0.5. We can introduce a Poincare section for Eqn. (2.9) on any plane z (]) = a, lal~0.5. The full state of the s y s t e m on such a plane is described by z (2), t m o d ~/2~ since the r i g h t - h a n d side of (2.9) is ~/2~-periodic in time. We restrict ourselves to the situation w h e n z (2) ~0.

CHAPTER 2 Differential Equations with Hysteresis

268

We introduce a discrete mapping Fb from the semi-infinite cylinder (z (2) ~>0, t mod ~/2=) to itself as a first return map of the trajectories of (2.9); of course, such a mapping may be partially defined. Global Error Propagation for Eqn. (2.9) Note that the surface dz (1)/dt=0 is given by the plane z (2) = 0. We assume that 8n Jr eappr < 0.01 (2.211) Later we will be interested not in the behavior of a particular trajectory but rather the behavior of the set of trajectories in the e-neighborhood of a given one. Numerical construction of the Fb consists of the three phases: 9 numerical integration of (2.9) until Zn < 0.5 - 0.01 and Zn+m > 0.5 + 0.01; 9 error estimates for the set of solutions from e-neighborhood of the initial conditions of (2.9) 9 computation of Fb based on the data from the previous two steps Introduce s

E_, E0.5 as s

= E n {z 9z(1)~>0.5 + 0.01},

9"~,_ -- E n {z" z (1) N< -- 0.5 -- 0.01}, E0.5 -- E n {z. [z(1)[~0.5 -+- 0.01}.

LEMMA2.4.11. Let the array of (tn, Zn, ~]n, gn), n = 0 . . . . . no be the result of successful termination of lntegratelO and let the assumption (2.211) hold. If zi ~ s for all il <.i<.i2, then

z(t)

e E n {z" Z(1)/>0.5}(, "~. n {Z" Z(1)
for all til 0 be a given number. Introduce en by the following recursive formula: 80 - - g ;

Ln =

0, 2b,

iff Zn, Zn+l E ~+ U ~ _ , iff Zn or Zn+l ~ ,'~,+ O E_;

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

8n+1 "-

{ Ene Lnrn q- e(hmax, E), ~n eLnzn q- e(hmax, E) q- 21clY, ~neLnzn if- eeul (hmin, E)

269

if-f ~n = O, iff ;n -- 1, ill;n--2.

Let the points Zn satisfy Zn E E(~n + Eappr), where E(~) - {z ~ E" l z - z l > a for all ~ ~ OF,}. Then for each (z0, ~0) such that r

~0), (z0, r/0))~e

q0z(t0, z0, ~0; t) ~ E for all to~t~tn

(2.212)

and p(q~(to, zo, ~0; tn), (Zn, tin)) < en for all OKnKno.

(2.213)

PROOF. The proof follows from the previous lemma, Lipschitz constant estimates (2.184), (2.185), and Proposition 2.4.3. [] Estimates for the Integration of (2.9) w i t h b - 1 In this subsection we fix the parameter value b = 1. In particular, for this value of the parameter we observe chaotic-like behavior of the map F1 = F, see Subsection 2.2.4. The estimates of this subsection will be used in the computer-assisted proof of Theorem 2.3.5. We define E to be an open set (-3, 3) x (-2.5, 2.5).

LEMMA2.4.13. Let h~0.01 and b - 1. Consider the RK4 method applied to (2.205) and (2.206). A local error of the numerical integration of both (2.205) and (2.206) is estimated by ez(h, E) - 59h 5 + 10 -13 < 6-10 -9.

(2.214)

PROOF. To show the statement of the lemma we need to estimate the terms of the inequality (2.210) of Lemma 2.4.10. To do so we follow the computeraided technique described in [2.153]. The basic idea is simple: using a powerful symbolic computations package (like 'Mathematica') one can obtain the formulas for the time derivatives of ki, that contains hundreds of thousands of terms, see [161] for details. Next, using the computer implementation of the interval arithmetics one could 'fold up' those gigantic formulas and obtain surprisingly reasonable estimates. The Mathematica program to perform such computations can be downloaded from http://phys.ucc.ie / - oll/hysteresis/. To reduce the estimates for k~4) , ~p(5)we consider three separate situations, and in the case of each estimate we present only the maximum over

CHAPTER 2 Differential Equations with Hysteresis

270

three values. The situations are 9 numerical step along ODE (2.205) for the initial condition z ~ • 9 Iz(~) 1~<0.5; 9 numerical step along ODE (2.206) for the initial condition z ~ E 9 IZ(1) I~0.5;

9 numerical step along ODE (2.205)/(2.206) for the initial condition Z E E " ]Z(1)i~0.5.

The estimate (2.214) follows immediately from (2.210) and the inequalities (2.215)-(2.220) that can be verified by a program in 'Mathematica': Ik~4)(z, t; Z)I < 1179,

z ~ E, 0 ~ ; ~ h ,

(2.215)

Jk(24)(z, t; ;~)l < 1 070,

z ~ E, 0KzKh,

(2.216)

Ik(34)(z, t; X)I < 1070,

z ~ E, 0 ~ z ~ h ,

(2.217)

Ik4(4)(z, t; Z)I < 1101,

z ~ E, 0~x~h,

(2.218)

Icp}s)(t, z; Z)I < 1039,

z ~ E, 0 ~ ; ~ h ,

(2.219)

o ~ 10 -13.

(2.220)

In particular, the inequality (2.220) follows from the description of IEEE arithmetic, and the manner in which trigonometric and exponential functions are realized in the computer (See [184] and bibliography therein). The lemma is proved. [] In the lemma we give the estimates for the other constants required for the numerical integration of (2.9) for b = 1. We choose hmin - 5 . 1 0 -5, hmax - 0.01, Kmax = 10 and Y = 10e(hmax, E). LEMMA2.4.14. Thefollowing estimates hold: D1 < 3, D2 < 5, D3 < 14, eeul(5" 10 -5, E) < 8 . 1 0 -8.

2.4.4 PROOFSOF SUBSECTION 'CHAOTIC BEHAVIOR' Topological Hyperbolicity In this section we remind the reader of the definition of some topological tools that play a major role in the proof of chaotic behavior.

2.4 EXAMPLESOF TECHNICAL TOOLBOXES

271

Let 19 denote the closure of the set V. Fix two positive integers du, ds with du + ds = d. Let V and W be bounded, open and convex product-sets V = V (u) x V (s) C R clu x ~Cts,

W = W (u) x W (s) C ~'~" x R ';is,

satisfying the inclusions 0 ~ V, W and letg 9 17 ~ ~au x Ras be a continuous mapping. It is convenient to treat g as the pair (g(U), g(S)) where g(U). V R~u and g(S). V ~ R as . The mapping g is topologically hyperbolic on the pair of sets (V, W), if the equations g(U) (SV(U) x V(S)) N lTV(U) - O, (2.221) hold, and deg (g(U)(., 0), V (u), O) 4= O.

(2.222)

Here S denotes the closure of a set S. The first relationship in (2.221) means geometrically that the image of the 'u-boundary' c~V(u) x 12(s) of V does not intersect the infinite cylinder C - 17V(u) x R~as;similarly, the second part of (2.221) means that the image g(l/) of the whole set V can intersect the cylinder C only by its central fragment I,V(u) x W (s). Thus the first equation (2.221) means that the mapping expands in a rather weak sense along the first coordinate in the Cartesian product Ran x •as, whereas the second one confers a type of contraction along the second coordinate (the indices' (s)' and'(u)' refer to the adjectives 'stable' and'unstable', respectively). Figure 2.65 illustrates the geometrical meaning of the relationships (2.221) in the two-dimensional case. The application of the concept of topological hyperbolicity is simplified significantly if du = 1. In this case the mapping g(U)(x(U), 0) is onedimensional, V (u) is an interval (a, fl) with 0r fl < 0, and verification of the inequality (2.222) is trivial. LEMMA2.4.15. The inequality (2.222) holds if and only if g (u) (~, 0)g (u) (fl, 0) < 0.

The concept of topological hyperbolicity allows one to show the existence of chaotic behavior. Here we rephrase Corollary 3.1 from [155] for our particular notation. Recall that the matrix is called k-transitive, if all entries of its kth power are positive.

272

CHAPTER 2

Differential Equations with Hysteresis

FIGURE 2.65 The horizontal axis represents R du and the vertical one represents R ds for du = ds = 1. The rectangle represents the set W = W (u) x W (s). The shaded areas are parts of the infinite cylinder C = 17 x Rds . This cylinder cannot be intersected by the image g(17); the images of OV(u) x 17(s) also cannot intersect W. The deformed quadrilateral represents an admissible location of g(V). The solid parts of its border are images of the sets OV(u) x V (s), whereas images of V (u) x OV(s) are dashed.

COROLLARY 2.4.1. Let a matrix A be k-transitive and let F : ~a __, Ra be a continuous mapping. Let there exist homeomorphisms "}'[i and the product set V such that c/-[71FC]-[i is (V, V)-hyperbolic if ai,j -- 1, and let the family ld of connected components of the union set ~ c/-li(V) have more than one element. Then any mapping F~, which is sufficiently close to F in the uniform metric, is (b/, k)-chaotic.

Proof of Theorem 2.3.5 (V, W)-Hyperbolicity and Poincare Map F1 To p r o v e t h e t h e o r e m w e n e e d to satisfy the c o n d i t i o n s of C o r o l l a r y 3.1 f r o m [155]. D e n o t e V = {(~(u) ~(s)). i~(U)l~0,003, i~(S)l~0.01} w h e r e i - 0 . . . . . 8. I n t r o d u c e ~ i " (~(u)~(s)) ~ va(i) + ~(U)va(i) + ~(s)vb(i) w h e r e i - 0 . . . . . 8 a n d vc(i), vb(i), Va(i) are g i v e n in Table 2.2. N o t e t h a t "t-li(V) - R(i) for all

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

273

i = 0 . . . . . 8. It is easy to see that each "]-/i is a h o m e o m o r p h i s m so we can introduce m a p s gi,j - 9r{71F19r{i. Finally let us define (9 x 9)-matrix A = (ai,j) i, j = 0 . . . . . 8: I1

a

~.

1

...

0

0

0

0

...

0

0

9

9

9

o

0

0

...

0

1

1

0

...

0

0

.9

.

(2.223)

LEMMA2.4.16. The matrix A is 16-transitive and not k-transitive for k < 16. PROOF9 The proof consists of straightforward calculations. Note that according to L e m m a 2.3.6 there are more than two connected components of b/, matrix A is 16-transitive according to L e m m a 2.4.16 and to finalize the proof we need the following statement. LEMMA2.4.17. The mapping F1 is defined and continuous on Ri and the mappings gi,j - "]-//1Fl'}-[i are (V, V)-hyperbolic if ai,j = 1. The proof is computer assisted and given in the next subsection.

Computer-Assisted Proof of Lemma 2.4.17 Let us fix i and j. To prove that F1 is defined on R(i) and that gi,j is topologically hyperbolic for (V, V) we cover R(i) with a n u m b e r of balls of the given radius and show the required properties of F1 and gi,j for each ball separately. Let Vc, Va, Vb be some vectors in S and the parallelogram R = R(vc, Va, Vb) C S be given by

R -- {Vc + O~Va if- fly b

:

I~1~1, Ifll~l},

see Fig. 2.66. Denote an open (closed) ball with the radius r centered at by B(~; r) (B[~; r]). We define the parallelogram grid with the step hg, G(R) = {~(k, l) = Vc + khgva + lhgVb E S}, k E [-N1, N1], l E [-N2, N2]

CHAPTER 2 Differential Equations with Hysteresis

274

z//'"

. . . . . ~ ... -..

-../

"-- . ~ / .

-.. -~ / /

/

;" Y

-.-

Va

/

FIGURE 2.66 Parallelogram R = R(c, y, z) and the grid G(R).

where N1 = [[Va[/hg] + 1, N2 = [IVb]/hg] + 1 and the operation [.J stands for the maximum integer that is not greater than the number in the 'floor brackets'. Introduce A I ( j ) = {V ~--- Vc(j) if- O~Va(j) nt- flVb(j) : [fl[~l} and A2(j) = {v = vc(j) + ~va(j) + flvb(j) : [~[~>1}. Let dist(v, A) = supo,A Iv - ~[ be the Euclidean distance between the point v and the set A. LEMMA2.4.18. Letr > Oand 13(v; r) ~ H. Ifthealgorithm Getlmage(v, r, v,, r,) returns 'True" then the map F1 is defined on 13(v; r) and FI (B(v; r)) c/3(v,; r,) where the values v,, r, are returned by the algorithm. PROOF. Here we just sketch the proof as there is no formal definition of some steps of the algorithm GetlmageO. By line 1 the point v is transformed into the initial conditions for the problem (2.9). There exists a numerical trajectory described by the set no { (tn, Zn, ~ln, qn)}n=0 which is an output of the algorithm IntegratelO. By lines 5, 6 and Proposition 2.4.3, the error in numerical trajectory at the moment tn satisfies en~eno ~0.01 - eappr. This, together with lines 3, 4 guarantees that the numerical trajectory in the extended phase, space including cyclic time, satisfies conditions of Lemma 2.4.5 and FI(B(v; r)) is well defined. More than that, there exist v,, r, such that FI(B(v; r)) c 13(v,; r,). The lemma is proved. []

275

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

Let R be a parallelogram satisfying ~ (R) c V. If the algorithm VWHypPartl (R, hg, j) returns 'True', then F1 is defined and continuous on R and

LEMMA 2.4.19.

7-/71F I(R) r3

(V -(u) x (~ \ V(S))) = 0.

(2.224)

GETIMAGE(V,r,v,,r,) Input: Ball on the plane FI centered at v with radius r Output: True if F 1 is well defined on/3(v; r)" in this case F 1 (B(v; r)) C/3(v,; r,) (1) Lett 0 = v(1),z0 = (0.5, v(2)), r/0 = 0 . 5 . no (2) Use IntegratelO to compute (tn, Zn, rln, qn)n= 0 (3) (4) (5) (6) (7) (8) (9) (10)

Value n o should satisfy Zno _(1) > 0.5 + 0.01 , and 3n, < no " z n, (1) < 0.5 -- eappr. if there is no such n o then return false Compute sequence en according to Lemma 2.4.12 if value eno + eappr > 0.01 then return false Use plane intersection algorithm in the extended phase space (including cyclic time) for the plane (z, (1, 0)) = 0.5 to compute B(v,; r,) if Plane intersection algorithm fails then return false Make sure that this is the first intersection and intersection before tnl is not possible return true

VWHYPPARTI(R, hg, j) Input: Parallelograms R, R(j) mesh hg Output: True if part 2 of (2.221) 'partially' holds Let G(R) be the grid (2N 1 + 1) x (2N 2 + 1) on R with the mesh size hg (1) (2) for k = - N 1 to N 1 for l = - N 2 to N 2 (3) if GETIMAGE(~(k,l),hg,v,,rad)==false then return false (4) if dist(v,, E2 \ AI(j)) < rad t h e n return false (5) return true (6)

PROOF. First w e m e n t i o n t h a t R c a n b e c o v e r e d b y balls of r a d i u s at t h e n o d e s of t h e grid: R C

U /3(~;hg). ~e~(R)

hg c e n t e r e d (2.225)

I n d e e d , m e s h e s of t h e g r i d G (R) are r h o m b u s e s w i t h s i d e s of l e n g t h hg a n d a r h o m b u s e s w i t h s i d e s of l e n g t h a c a n b e c o v e r e d b y t h e f o u r c l o s e d balls w i t h c e n t e r s in its c o r n e r s a n d r a d i i of 4 a . Let t h e a l g o r i t h m VWHypPartl(R, hg, j) r e t u r n 'True'. By line 4 of t h e a l g o r i t h m a n d L e m m a 2.4.18 t h e m a p F1 is d e f i n e d o n all ~ ~ ~(R) (lines 1, 2, 3). T h u s F1 is d e f i n e d a n d c o n t i n u o u s o n R b y (2.225) a n d

gi,j is d e f i n e d a n d c o n t i n u o u s o n 7/I -1) (R).

276

CHAPTER 2 Differential Equations with Hysteresis

By line 5 of the algorithm B(~; hg) C .AI(j) for all ~ e G(R). Thus F1 (a) C ,,41 (j). Since, by the definition 71~ 1 (A1 (j)) C [~ x W(s) and ~ 1F1 (a)

C R x V (s), formula (2.224) follows. The lemma is proved. LEMMA2.4.20. Let R

[]

- "}-[i(V). If the algorithm VWHypPart2(hg, i, j) returns

"True', then gi,j ((9

x

F]

: 0.

(2.226)

VWHYPPART2(hg,i, j) Input: ParallelogramsR(i), R(j) mesh hg Output: True if part 1 of (2.221) holds for gi,j (1) Let G(R(i)) be the grid (2N1 + 1) x (2N2 + 1) on R(i) with the mesh size hg (2) for l = - N 2 to N 2 (3) if GETIMAGE(~(--N1,l),hg,v,,rad)==false t h e n return f a l s e (4) if dist(v,, R2 \ A2(j)) < rad t h e n return f a l s e (5) (6) if GETIMAGE(~(N1,l),hg,v,,rad)==false t h e n return f a l s e (7) if dist(v,, •2 \ A2 (j)) < rad t h e n return false (8) return true

PROOF. Let ~, = {~(k, l) 9 k e {-N1, N1}, l e [-N2, N2]}. We notice that by the definition of the grid ~(R),

c}-~i(~V(U) x V (s)) C U B(~,hg).

(2.227)

~e~, Let the algorithm VWHypPart2(R, hg, j) return 'True'. By lines 3, 6 of the algorithm, Lemma 2.4.18 and (2.227) the map F1 is defined on r (cOR(i)(u) x R(i)(s)). By lines 4, 7 of the algorithm B(~; hg) c A2(j) for all ~ e G,. Thus

"]'{i (0V(u) x

W(S)) C ,A2(j).

Since, by the definition of A2 (j)

"]-{)-I)(A2(j)) C ( ~

\ V (u)) x [~

formula (2.226) follows. The lemma is proved.

2.4 EXAMPLES OF TECHNICAL TOOLBOXES TABLE 2.3

Computational details:

277

hg(i), #({Rk(i)}), average time

to process

Rk(i)

(i, j)

(0,0)

(0,1)

(1,2)

(2,3)

(3,4)

(4,5)

(5,6)

(6,7)

(7,8)

(8,0)

hg (i)

10 -4 1 59

10 -4 1 59

10 -4 1 62

10 -4 1 67

10 -4 1 74

5.10-7 4 694 996

10 -6 1381 951

10 -4 1 56

10 -4 1 54

10 -4 1 57

Pieces Average time (s)

LEMMA2.4.21. Let {Rk} be a finite set of parallelograms such that R(i) C UkRk.

Let hg be fixed. The successful terminations of VWHypPartl (Rk, hg, j)for all k and VWHypPart2(hg(i), i, j) together with g!U) z,] ( - 0 . 0 0 3 , "" u)gi,(u) j (0.003, 0) < 0

imply that gi,j is defined on V and is (V, V)-hyperbolic. PROOF. By Lemma 2.4.19, the map gi,j is defined on all arLiRk, which means that gi,j is defined on the whole of V. By (2.224)

gi,j(~7) n (fz ~u) x (~, \ V(S))) - 0, that is, exactly the second part of (2.221). The first part of (2.221) holds due to successful termination of VWHypPart2(hg, i, j) and Lemma 2.4.20. Finally (2.222) follows from the supposition of the lemma and Lemma 2.4.15. [] Our implementations of VWHypPartl(Rk(i),hg(i),j) and VWHypPart2 (h~(i), i, j) terminated successfully for all i,j such that ai,j -- 1 and appropriate partitions Rk(i) of R(i). The additional verification shows that (2.221) holds for all i, j such that ai,j -- 1. This, together with Lemma 2.4.21, proves Lemma 2.4.17. Theorem 2.3.5 is proved. The value hg(i) and total number of Rk(i) for all i can be found in Table 2.3. On the intuitive level the proof may be seen in Fig. 2.67. However, to justify that picture proof we used parallel computations on a cluster containing 100 CPU P-III 1.2GHz. The total processor time used for each R(i) is given in Table 2.3 as well. The total computational time is just below 6 000 000 seconds, that is, 70 days of computing on one PC. []

278

CHAPTER 2 Differential Equations with Hysteresis

0.01

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FIGURE 2.67 The first picture: g0,0(V) against V; other eight pictures are gi, i+l (V) against V for i = 0 . . . . . 7 and the final one is g8,0(V) against V.

ACKNOWLEDGEMENT This work was partially supported by the Enterprise Ireland, Grant S C / 2000/138, grant S C / 2 0 0 3 / 3 7 6 and Post-doctoral fellowship P D / 2 0 0 4 / 2 6 . D. Rachinskii was partially supported by the Russian Science Support Foundation, grant MD-87.2003.01 of the President of Russia and grants 03-01-00258, 04-01-00330 of the Russian Foundation for Basic Researches.

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134. M. Brokate and J. Sprekels, 'Existence and optimal control of mechanical processes with hysteresis in viscous solids', IMA J. Appl. Math. 43(3), 219-229 (1989). 135. C. Cavaterra and F. Colombo, 'Automatic control problems for reaction-diffusion systems', J. Evol. Equ. 2(2), 241-273 (2002). 136. S. Gatti, 'Automatic control of the temperature in phase change problems with memory', Z. Anal. Anwendungen 20(4), 883-914 (2001). 137. M. Grasselli, 'Automatic control of the temperature in thermoviscoelasticity', Math. Methods App1. Sci. 22(16), 1447-1464 (1999). 138. K.-H. Hoffmann, J. Sprekels, and A. Visintin, 'Identification of hysteresis loops', J. Comput. Phys. 78(1), 215-230 (1988). 139. M.A. Krasnosel'skii and A. V. Pokrovskii, 'Simulation of transformers with hysteresis by means of continual relay systems', Dokl. Akad. Nauk SSSR 227(3), 547-550 (1976). 140. R. C. Smith, 'Hysteresis modeling in magnetostrictive materials via Preisach operators', J. Math. Systems Estim. Control 8(2), 23 pp. (1998). 141. P. Colli, M. Grasselli, and J. Sprekels, 'Automatic control via thermostats of a hyperbolic Stefan problem with memory', Appl. Math. Optim. 39(2), 229-255 (1999). 142. W. Desch and J. Turi, 'The stop operator related to a convex polyhedron', J. Differential Equations 157(2), 329-347 (1999). 143. L. R. Dupr6, R. Van Keer, and J. A. A. Melkebeek, 'Complementary 2-D finite element procedures for the magnetic field analysis using a vector hysteresis model', Int. J. Numer. Methods Engrg. 42(6), 10051023 (1998). 144. M.A. Krasnosel'skii and A. V. Pokrovskii, 'The absolute stability of systems with several nonlinear components', Dokl. Akad. Nauk SSSR 271(6), 1314-1316 (1983). 145. R.E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill Book Co., New York (1969). 146. M.A. Krasnosel'skii, V. S. Burd, and Y. S. Kolesov, Nonlinear Almost Periodic Oscillations, Halsted Press (A division of John Wiley & Sons), New York, Toronto (1973). (Translated from the Russian by A. Libin, translation edited by D. Louvish). 147. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985). 148. M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin (1984) (Translated from the Russian by Christian C. Fenske).

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149. N. A. Bobylev, Y. M. Burman, and S. K. Korovin, Approximation Procedures in Nonlinear Oscillation Theory, de Gruyter Series in Nonlinear Analysis and Applications, vol. 2, Walter de Gruyter & Co., Berlin (1994). 150. A. V. Pokrovskii, 'The limit norm of a linear operator and its applications', Dokl. Akad. Nauk SSSR 249(3), 548-551 (1979). 151. A. V. Pokrovskii, 'Limit norm of a linear element', Avtomat. i Telemekh. (11), 62-69 (1985). 152. A. V. Pokrovskii and M. E. Semenov, 'Stable periodic regimes in control systems with monotone nonlinearities', Avtomat. i Telemekh. (2), 31-37 (1990). 153. M.A. Krasnosel'ski/, The Operator of Translation Along the Trajectoriesof Differential Equations, American Mathematical Society, Providence, RI (1968) (Translations of Mathematical Monographs, Vol. 19. Translated from the Russian by Scripta Technica). 154. A. Pokrovskii, 'Shuttle algorithm in investigation of systems with hysteresis nonlinearities', in Models of Hysteresis, Trento, 1991, Longman Sci. Tech., Harlow (1993), pp. 124-142. 155. A. Pokrovskii, S. J. Szybka, and J. G. McInerney, 'Topological degree in locating homoclinic structures for discrete dynamical systems', Preprints of INS, UCC, Ireland (January 2001). 156. M. Krasnosel'ski/, J. Mawhin, and A. Pokrovskii, 'New theorems on forced periodic oscillations and bounded solutions', Soviet Physics Doklady 36(11), 743-745 (1991). 157. P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho, Tokyo (1996). 158. J. Hale, Theory of Functional Differential Equations, second edition, Applied Mathematical Sciences, vol. 3, Springer-Verlag, New York (1977). 159. L. Padulo and M. A. Arbib, System Theory. A Unified State-space Approach to Continuous and Discrete Systems, W. B. Saunders Co., Philadelphia, PA (1974). 160. A. PokrovskiL 'Topological shadowing and split-hyperbolicity', Funct. Differ. Equ. 4(3-4), 335-360 (1998), (1997). 161. O. Rasskazov, 'Methods of geometrical analysis of complicated dynamics, with applications to models of semi-conductor lasers', PhD thesis, University College Cork, Cork, Ireland (2003). (Webpage: http: //phys.ucc.ie/~oll/phdthesis.pdf).

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162. D. RachinskiL 'On natural continua of periodic solutions of systems with hysteresis', Automat. Remote. Control. 64(3, part 1), 420-438 (2003). 163. D. Rachinskii and A. M. Krasnosel'skii, 'On a nonlocal condition for existence of cycles of systems with hysteresis', Automat. Remote. Control. 64(2, part 1), 231-251 (2003). 164. V. S. Kozjakin and M. A. Krasnosel'skii, 'The method of parameter functionalization in the Hopf bifurcation problem', Nonlinear Anal. 11(2), 149-161 (1987). 165. J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Applied Mathematical Sciences, vol. 19, Springer-Verlag, New York (1976). (With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale). 166. A.M. Krasnosel' skii and M. A. Krasnosel'ski/, 'Large amplitude cycles in autonomous systems with hysteresis', Dokl. Akad. Nauk SSSR 283(1), 23-26 (1985). 167. M. A. Krasnosel'skiL D. I. Rachinskii and M. G. Yumagulov, 'Hopf bifurcation for autonomous systems with hysteresis', Dokl. Akad. Nauk 355(4), 446-449 (1997). 168. P. Diamond, N. Kuznetsov, and D. RachinskiL 'On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity', J. Differential Equations 175(1), 1-26 (2001). 169. T. Y. Li and J. A. Yorke, 'Period three implies chaos', Amer. Math. Monthly 82(10), 985-992 (1975). 170. P. Diamond, P. E. Kloeden, M. A. Krasnosel'skiL and A. PokrovskiL 'Chaotic dynamics in nonsmooth perturbations of bishadowing systems', Arab J. Math. Sci. 6(1), 41-74 (2000). 171. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge (1995) (With a supplementary chapter by A. Katok and L. Mendoza). 172. E. Cox, M. Mortell, A. PokrovskiL and O. Rasskazov, 'On chaotic and recurrent travelling wave patterns in a periodically forced and extended kdev', Preprints of School of Math., Appl. Math. and Stat., UCC, Ireland (February 2002). 173. O. Rasskazov, G. Huyet, J. G. McInerney, andA. PokrovskiL 'Rigorous analysis of complicated behaviour in a truncated Lang-Kobayashi

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CHAPTER 3

Scalar Preisach Models of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 3.1 3.2

The Classical Preisach Model of Hysteresis Generalized Scalar Preisach Models of Hysteresis References

293 360 443

3.1 T H E C L A S S I C A L P R E I S A C H M O D E L OF HYSTERESIS 3.1.1 DEFINITION OF THE CLASSICAL PREISACH MODEL The phenomenon of hysteresis has been with us for ages and has been attracting the attention of many investigators for a long time. The reason is that hysteresis is ubiquitous. It is encountered in many different areas of science. Examples include magnetic hysteresis, ferroelectric hysteresis, mechanical hysteresis, superconducting hysteresis, adsorption hysteresis, optical hysteresis, electron beam hysteresis, economic hysteresis, etc. However, the very meaning of hysteresis varies from one area to another, from paper to paper and from author to author. As a result, a stringent mathematical definition of hysteresis is needed in order to avoid confusion and ambiguity. Such a definition will serve a twofold purpose: first, it will be a substitute for vague notions, and, second, it will pave the road for more or less rigorous treatment of hysteresis. We begin with the definition of scalar hysteresis and, for the sake of generality, we adopt the language of control theory. Consider a transducer that can be characterized by an input u(t) and an output flt). This transducer is called a hysteresis transducer (HT) if its input-output relationship is a multibranch nonlinearity for which branch-to-branch transitions occur after input extrema. For the most part, the case of rate-independent hysteresis nonlinearity will be discussed. The term 'rate-independent' The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

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means that branches of such hysteresis nonlinearities are determined only by the past extremum values of input, while the speed (or particular manner) of input variations between extremum points has no influence on branching. It is important to stress that the notion of rate-independent hysteresis implies three distinct time scales. The first is the time scale of fast internal dynamics of the transducer. The second is the time scale on which observations (measurements) are performed. This time scale is much larger than the time scale of internal transducer dynamics so that every observation can be identified with a specific output value of the transducer. The third is the time scale of input variations. This time scale is much larger than the observation time scale so that every measurement can be associated with a specific value of input. In the existing literature, the hysteresis phenomenon is, by and large, linked with the formation of hysteresis loops (looping). This may be misleading and create the impression that looping is the essence of hysteresis. In this respect, the given definition of hysteresis emphasizes the fact that history-dependent branching constitutes the essence of hysteresis, while looping is a particular case of branching. Indeed, looping occurs when the input varies back and forth between two consecutive extremum values, while branching takes place for arbitrary input variations. From the given definition, it can also be concluded that scalar hysteresis can be interpreted as a nonlinearity with a memory which reveals itself through branching. In the given definition of hysteresis, the physical meanings of the input u(t) and the outputf(t) were left unspecified. It was done deliberately, for the sake of mathematical generality. However, it is not difficult to specify the meanings of u(t) and fit) in particular applications. For instance, in magnetism u(t) is the magnetic field and fit) is the magnetization, in mechanics u(t) is the force andf(t) is the displacement (length), in adsorption u(t) is the gas pressure and fit) is the amount of material adsorbed. The notion of hysteresis transducer may have different interpretations as well. For instance, in magnetism the HT can be construed as an infinitesimally small volume of magnetic material, and the corresponding input-output hysteresis nonlinearity can be interpreted as a constitutive equation for this material. All rate-independent hysteresis nonlinearities fall into two general classifications: (a) hysteresis nonlinearities with local memories, and (b) hysteresis nonlinearities with nonlocal memories. The hysteresis nonlinearities with local memories are characterized by the following property. The value of output f(t0) at some instant of time to and the values of input u(t) at all subsequent instants of time t>~to uniquely predetermine the

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295

value of outputf(t) for all t > to. In other words, for hysteresis transducers with local memories the past exerts its influence upon the future through the current value of output. This is not the case for hysteresis transducers with nonlocal memories. For such transducers, future values of output fit) (t~to) depend not only on the current value of outputflt0) but on past extremum values of input as well. Our discussion is solely concerned with mathematical models of hysteresis with nonlocal memory. The question arises, why are these models needed? The answer is that the hysteresis transducer is usually a part of a system. As a result, its input is not known beforehand, but is determined by the interaction of the transducer with the rest of the system. Since the input of HT is not predictable a priori, it is impossible to specify ahead of time the branches of hysteresis nonlinearity that will be followed in a particular regime of the system. This is the main impediment as far as selfconsistent mathematical descriptions of systems with hysteresis are concerned. To overcome the difficulty mentioned above, mathematical models of hysteresis are needed. These models represent new mathematical tools that themselves (due to their structure) will detect and accumulate input extrema and will choose appropriate branches of the hysteresis nonlinearity according to the accumulated histories. Coupled together with the mathematical description of the rest of the system, these models will constitute complete and self-consistent mathematical descriptions of systems with hysteresis. Without such models, the self-consistent mathematical descriptions of systems with hysteresis are virtually impossible. Next, we proceed to the discussion of the classical Preisach model of hysteresis. This model has a long and instructive history that can be best characterized by the following eloquent statement of J. Larmor made in his preface to the book [1]: ...scientific progress, considered historically, is not a strictly logical process, and does not proceed by syllogisms. New ideas emerge dimly into intuition, come into consciousness from nobody knows where, and become the material on which the mind operates, forging them gradually into consistent doctrine, which can be welded on to existing domains of knowledge. This is exactly what has happened with the Preisach model. The origin of this model can be traced back to the landmark paper of F. Preisach [2] published in 1935. Preisach's approach was purely intuitive. It was based on some plausible hypotheses concerning the physical mechanisms of magnetization. For this reason, the Preisach model was first regarded as a physical model of hysteresis. It has remained primarily known in the area of magnetics where this model has been the focus of considerable research for many years. This has resulted in the further development of

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the Preisach model, and many improvements and valuable facts have been accumulated in this area. These developments and results are recorded, for instance, in the publications [3-12,51]; it is important to note that the given list of references is not complete by any standards. Somewhat in parallel with the mentioned developments in magnetics, the Preisach model was independently invented and then extensively studied and tested for adsorption hysteresis by D. H. Everett and his collaborators [13-16]. This clearly indicated that the applications of Preisach's model were not limited only to the area of magnetics. The next decisive step in the direction of better understanding of the model was made in the 1970s and 1980s when the Russian mathematician M. Krasnoselskii and his colleagues undertook a comprehensive mathematical study of systems with hysteresis. It was then gradually realized that the Preisach model contained a new general mathematical idea. As a result, this model was separated from its physical connotation and was represented in a purely mathematical form that is similar to a spectral decomposition of operators [17]. In this way a new mathematical tool has evolved that can now be used for the mathematical description of hysteresis of various physical nature. At the same time, this approach has strongly revealed the phenomenological nature of the Preisach model. This has put this model into a new perspective and has led to the clarification of many controversies that have long surrounded the Preisach model. We next proceed to the purely mathematical description of the Preisach model. Consider an infinite set of simplest hysteresis operators ~aB. Each of these operators can be represented by a rectangular loop on the input-output diagram (see Fig. 3.1). Numbers c~and fl correspond to 'up' and 'down' switching values of input, respectively. It will be assumed in the sequel that a~>fl. Outputs of the above elementary hysteresis operators may assume only two values, +1 and -1. In other words, these operators can be interpreted as two-position relays with 'up' and 'down' positions corresponding to ~ u ( t ) = +1 and }~u(t) - -1, respectively. As the input, u(t), is monotonically increased, the ascending branch abcde is followed. When the input is monotonically decreased, the descending branch edfba is traced. It is clear that the operators ~1~ represent hysteresis nonlinearities with local memories. Along with the set of operators ~a]/consider an arbitrary weight function #(a, fl) that is often referred to as the Preisach function. Then, the Preisach model can be written as follows: A

fit) - Fu(t) -

/~(c~, fl)~a~u(t) de dfl.

(3.1)

A

Here F is used for the concise notation of the Preisach hysteresis operator that is defined by the integral in (3.1).

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

297

A

f +1

d

e

-

~U

-1

FIGURE 3.1

It is apparent that the model (3.1) can be interpreted as a continuous analog of a system of parallely connected two-position relays. This interpretation is illustrated by the block diagram shown in Fig. 3.2. According to this diagram, the same input u(t) is applied to each of two-position relays. Their individual outputs are multiplied by #(~, fl) and then integrated over all appropriate values of a and ft. As a result, the output, fit), is obtained. Discrete approximation to the above block diagram can be used as device realizations of the Preisach model (3.1). It is clear from the above discussion that the Preisach model is constructed as a superposition of simplest hysteresis operators ~ . These operators can be construed as the main building blocks for the model (3.1). The idea that a complicated operator can be represented as a superposition of simplest operators is not entirely new and was exploited before in mathematics, particularly in the functional analysis [17]. For instance, according to the spectral decomposition theory for self-adjoint operators, any self-adjoint operator can be represented as a superposition of projection operators that are, in a way, the simplest self-adjoint operators. The above analogy shows that the Preisach model (3.1) can be interpreted from the mathematical point of view as a spectral decomposition of a complicated hysteresis operator F into the simplest hysteresis operators ~ . There is also an interesting parallel between the Preisach model and wavelet transforms that are currently very popular in the area of signal processing.

CHAPTER 3 ScalarPreisach Models of Hysteresis

298

A

A

u(t)

/ FIGURE 3.2

Indeed, all rectangular loop operators ~//can be obtained by translating and dilating the rectangular loop operator ~1,-1, that can be regarded as the 'mother loop operator'. Thus, the Preisach model can be viewed as a 'wavelet operator transform'. The Preisach hysteresis nonlinearity (3.1) is constructed as a superposition of elementary hysteresis nonlinearities ~a/~with local memories, nevertheless, it usually has a nonlocal memory. (This fact will be proved in the next section.) It is remarkable that a new qualitative property of nonlocal memory emerges as a collective property of a system having a huge (infinite) number of simple and qualitatively similar components. Having defined the Preisach model, it is appropriate to describe the directions along which the further discussion will proceed. The subsequent discussion will be centered around the following five topics. 1. How does the Preisach model work? In other words, how does this model detect local input extrema, accumulate them and choose the appropriate branches of hysteresis nonlinearity according to the accumulated histories? The answer to this question will reveal the mechanism of memory formation in the Preisach model. It will require the development of a special diagram technique that will constitute the mathematical foundation for the analysis of the Preisach model. 2. What experimental data are needed for the determination of the Preisach function, #(a, fl), for a given hysteresis transducer?

3.1 THE CLASSICALPREISACH MODEL OF HYSTERESIS

299

This is the so-called identification problem. The solution to this problem is very important as far as the practical applications of the Preisach model are concerned. 3. What are the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model? The significance of this problem is that its solution will clearly establish the limits of applicability of the Preisach model. 4. How can the Preisach model be implemented numerically? This is an important question because it seems at first that the numerical evaluation of two-dimensional integrals is required for the numerical implementation of the Preisach model. However, it will be shown in the sequel that the evaluation of the above integrals can be completely avoided. 5. How can the Preisach model be useful for the computation of hysteretic energy dissipation? It is well known that the hysteretic energy dissipation can be easily evaluated for the case of periodic (cyclic) input variations. However, the problem of computing hysteretic energy losses for arbitrary (not necessary periodic) input variations has remained unsolved. It will be shown (see Section 3.1.5) that the Preisach model can bring about the solution to this problem. This solution can be useful for the computation of entropy production and, in this way, may facilitate the development of irreversible thermodynamics of hysteretic media. There is another problem which has not been mentioned above but has been extensively studied by Krasnoselskii and his colleagues. This problem is related to the fact that hysteresis nonlinearities are naturally defined on the set of continuous and piecewise monotonic inputs. However, the above set of functions does not form a complete function space. This constitutes the main difficulty as far as the rigorous mathematical treatment of differential (or integral) equations with hysteresis nonlinearities is concerned. Thus, the problem of continuous extension of hysteresis operators from the above set of piecewise monotonic inputs to some complete function spaces presents itself. The essence of the problem is in finding (or description) of such complete function spaces. The solution to this problem is important because these function spaces form the natural 'environment' for the rigorous mathematical study of equations with hysteresis nonlinearities. Nevertheless, this problem is more or less of a purely mathematical nature. It is not directly related to practical applications, and for this reason will not be discussed here. The reader interested in the

CHAPTER 3 ScalarPreisach Models of Hysteresis

300

discussion of this problem is referred to Krasnoselskii and Pokrovskii [18] as well as more recent publications [19-24]. It is apparent from the above discussion that the Preisach model has been defined without any reference to a particular physical origin of hysteresis. This clearly reveals the phenomenological nature of the model and its mathematical generality. To further emphasize this new approach to the Preisach model, it is instructive to compare the definition of the model given above with the definition regularly used in magnetics. In magnetics, separate magnetic 'particles' (or 'domains') are introduced. These particles have rectangular hysteresis loops and they play the same role as elementary hysteresis operators ~/~. A loop typical for such particles is shown in Fig. 3.3. Here Hu and Ha are 'up' and 'down' switching magnetic fields, respectively, and ms is the magnetic moment (magnetization) that is identical for all particles. The notation ~(Hu, Ha) is used for the particle having the hysteresis loop shown in Fig. 3.3. The magnetic material is considered to be composed of many such particles. It is also assumed that the different particles have some distribution of reversal field Hu and Ha that can be characterized by the distribution function dp(Hu, Hcl). This function plays the same role as ~(~, t) in (3.1). It is typical to speak about statistical nature of the distribution function dp(Hu,Hi), although the author is not aware of any rigorous justification or experimental evidence for this statistical interpretation. By using the magnetic particles and their distribution function, the Preisach model is usually defined in M

+m

s

-

Hd

-m

iHu

s

FIGURE 3.3

=H

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

301

magnetics as follows

M(t) - ffUu>Ha

dp(Hu, Hd)~(Hu, Hd)H(t) dHu dHd,

(3.2)

where M is the magnetization, while

~(Hu, Ha)H(t) = +ms

(3.3)

if the particle is switched up, and

~(Hu, Ha)H(t) = -ms

(3.4)

if the particle is switched down. The given definition is framed in terms of magnetics and can naturally be called the 'magnetic' definition. It is obvious that the 'magnetic' definition of the Preisach model is mathematically equivalent to the previous definition (3.1). The'magnetic' definition suggests some relation of the Preisach model to certain physical realities such as magnetic particles, their distribution, etc. For this reason, the Preisach model has long been regarded as the physical model. However, there are some intrinsic difficulties with respect to this interpretation. Indeed, if the magnetic particles are to have some features of reality, then their geometric shapes and mutual spatial locations should be important and affect the value of magnetization. But, in the 'magnetic' definition of the Preisach model there is no reference whatsoever to the spatial locations of the particles or to their shapes. In the mathematical definition (3.1), this difficulty does not appear because there is no need to relate the operators, ~ , to some real physical objects. Another difficulty comes from the fact that many (almost all) particles in the Preisach model have asymmetrical rectangular loops (Hu ~: -Ha). Such loops are not physical for single particles. The use of asymmetrical loops is commonly justified on the grounds that asymmetry is caused by the interaction between the particles, and the amount of asymmetry is a measure of the interaction. More precisely, this means that each particle 'feels' not only the applied (external) magnetic field H(t), but also the interaction magnetic field that is due to the magnetization of adjacent particles. The interaction magnetic field may, of course, vary from one particle to another. These interaction magnetic fields result in shifted (asymmetrical) hysteresis loops when these loops represent the dependence of magnetization on the applied field alone. Although the given explanation sounds plausible, it is not completely satisfactory. This explanation does not account for the fact that the

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CHAPTER 3 Scalar Preisach Models of Hysteresis

~U

FIGURE 3.4

interaction fields depend on particular states of magnetization, shapes, and lbcation of the particles. This information is not represented in the Preisach model at all. In the case of purely mathematical definition (3.1) of the Preisach model, the above difficulty does not appear. This is because there is a straightforward phenomenological explanation for the need of operators ~a/~ with asymmetrical loops. These operators are needed in order to describe asymmetrical minor loops exhibited by actual hysteresis nonlinearities (see Fig. 3.4). Indeed, if only operators ~ _ ~ with symmetrical loops were used, then the model (3.1) being constructed as a superposition of such ~-operators would also describe only symmetrical minor loops. In other words, ~-operators with symmetrical loops do not form a complete set of operators in the sense that hysteresis nonlinearities with asymmetrical minor loops cannot be represented as superpositions of ~-operators from this set. From the discussion presented above the following conclusion can be reached. The 'magnetic' definition of the Preisach model is possible and historically was developed first. However, this definition obscures the model, conceals its mathematical and phenomenological nature, and narrows the area of applicability of this model to the field of magnetics. The definition (3.1), on the other hand, interprets the Preisach model as a new mathematical tool whose importance may well extend beyond the area of magnetics. For this reason, the purely mathematical definition (3.1) seems to be more attractive and will be used throughout our discussion.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS (O~o,~o)

(X

303

A

/

FIGURE 3.5 3.1.2 GEOMETRIC INTERPRETATION A N D MAIN PROPERTIES OF THE PREISACH MODEL

The mathematical investigation of the Preisach model is considerably facilitated by its geometric interpretation. This interpretation is based on the following simple fact. There is a one-to-one correspondence between operators ~/~ and points (~, fl) of the half-plane ~>~fl (see Fig. 3.5). In other words, each point of the half-plane a~ fl can be identified with only one particular ~-operator w h o s e ' u p ' and 'down' switching values are respectively equal to a and fl coordinates of the point. It is clear that this identification is possible because both ~-operators and the points of the half-plane ~ f i are uniquely defined by pairs of numbers, a and ft. Consider a right triangle T (see Fig. 3.5). Its hypotenuse is a part of the line ~ = fl, while the vertex of its right angle has the coordinates ~0 and fl0 with fl0 = -~0. In the sequel, this triangle willbe called the limiting triangle and the case when #(a, fl) is a finite function with a support within T will be discussed. In other words, it will be assumed that the function #(a, fl) is equal to zero outside the triangle T. This case covers the important class of hysteresis nonlinearities with closed major loops. At the same time, the above case will not essentially limit the generality of our future discussions. To start the discussion, we first assume that the input u(t) at some instant of time to has the value that is less than rio- Then, the outputs of all ~-operators which correspond to the points of the triangle T are equal to -1. In other words, all ~-operators are in the 'down' position. This corresponds to the state of 'negative saturation' of the hysteresis nonlinearity represented by the model (3.1). Now, we assume that the input is monotonically increased until it reaches at time tl some maximum value Ul. As the input is being increased, all ~-operators with 'up' switching values a less than the current

CHAPTER 3 Scalar Preisach Models of Hysteresis

304

input value u(t) are being turned into the 'up' position. This means that their outputs become equal to +1. Geometrically, this leads to the subdivision of the triangle T into two sets: S + (t) consisting of points (a, fl) for which the corresponding ~-operators are in the 'up' position, and S-(t) consisting of points (a, ]/) such that the corresponding ~-operators are still in the 'down' position. This subdivision is made by the line a = u(t) (see Fig. 3.6) that moves upwards as the input is being increased. This upward motion is terminated when the input reaches the maximum value Ul. The subdivision of the triangle T into S + (t) and S- (t) for this particular instant of time is shown in Fig. 3.7. Next, we assume that the input is monotonically decreased until it reaches at time t2 some minimum value u2. As the input is being decreased, all )-operators with 'down' switching values ]/above the current input

m

FIGURE 3.6

/

S-(t)

/(X-- U1 / ~ ~

s§

/

FIGURE 3.7

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

305

(x

s§

[~ = u(t)

/

FIGURE 3.8 (x

FIGURE

3.9

value, u(t), are being turned back into the 'down' position. This changes the previous subdivision of T into positive and negative sets. Indeed, the interface L(t) between S + (t) and S-(t) now has two links, horizontal and vertical. The vertical link moves from right to left and its motion is specified by the e q u a t i o n / / = u(t). This is illustrated by Fig. 3.8. The above motion of the vertical link is terminated when the input reaches its m i n i m u m value u2. The subdivision of the triangle T for this particular instant of time is shown in Fig. 3.9. The vertex of the interface L(t) at the above instant of time has the coordinates ~ = Ul and fl = u2. Now, we assume that the input is increased again until it reaches at time t3 some m a x i m u m value u3 that is less than Ul. Geometrically, this increase results in the formation of a new horizontal link of L(t) which

CHAPTER 3 Scalar Preisach Models of Hysteresis

306

/

s*

FIGURE

3.10

/ ( u~, u 2) s§

i (u~u,)

FIGURE

3.11

moves upwards. This upward motion is terminated when the maximum u3 is reached. This is shown in Fig. 3.10. Next, we assume that the input is decreased again until it reaches at time t4 some minimum value u4 that is above u2. Geometrically, this input variation results in the formation of a new vertical link that moves from right to left. This motion is terminated as the input reaches its minimum value u4. As a result, a new vertex of L(t) is formed that has the coordinates = u3 and fl = u4. This is illustrated by Fig. 3.11. By generalizing the previous analysis, the following conclusion can be reached. At any instant of time, the triangle T is subdivided into two sets: S+ (t) consisting of points (a, fl) for which the corresponding ~-operators are in the 'up' position, and S-(t) consisting of points (a,//) for which the corresponding ~-operators are in the 'down' position. The interface

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

I

307

/

/ F I G U R E 3.12

s-(t)

F I G U R E 3.13

L(t) between

S + (t) and S-(t) is a staircase line whose vertices have a and fl coordinates coinciding respectively with local maxima and minima of input at previous instants of time. The final link of L(t) is attached to the line a = fl and it moves w h e n the input is changed. This link is a horizontal one and it moves u p w a r d s as the input is increased (see Fig. 3.12). The final link is a vertical one and it moves from right to left as the input is decreased (see Fig. 3.13). Thus, at any instant of time the integral in (3.1) can be subdivided into two integrals, over S + (t) and S-(t), respectively: A

fit)

= F#(t)=

+(t)

#(o~, fl)~u(t) d~ dfl

q - / ~ s - (t)

#(a,

fl)~u(t) da dfl.

(3.5)

CHAPTER 3 Scalar Preisach Models of Hysteresis

308

Since

~&~u(t) = +1,

if (~, fl) ~ S+(t)

(3.6)

~u(t)

if (~, fl) ~ S-(t),

(3.7)

and - -1,

from (3.5) we find

fit) = fS +(t) #(a, fl) da dfl - f L

-(t)

#(a, fl) d0r dfl.

(3.8)

From the above expression, it follows that an instantaneous value of output depends on a particular subdivision of the limiting triangle, T, into positive and negative sets S+ (t) and S-(t). This subdivision is determined by a particular shape of the interface L(t). This shape, in turn, depends on the past extremum values of input because these extremum values are the coordinates of the vertices of L(t). Consequently, the past extremum values of input shape the staircase interface, L(t), and in this way they leave their mark upon the future. To make the above point perfectly clear, consider two inputs ul(t) and u2(t) with two different past histories for t < t'. This means that they had different local extrema for t < t t. It is next assumed that these inputs coincide for t~t'. Then according to (3.8), the outputs fl (t) andf2(t) corresponding to the above inputs are given by the formulas fl(t) -/fs~-(t) #(0r fl) d~ dfl - / L 1 (t) #(a, fl) da dfl,

(3.9)

f/" f2(t) = U /z(a, fl)da d f l - ] ] /~(0r fl)d0r dfl, dds -~(t) d l S2 (t)

(3.10)

where S + (t) and S l(t), S+ (t) and S2 (t) are positive and negative sets of two subdivisions of T associated with ul(t) and u2(t), respectively. The above two subdivisions are different because they correspond to two different input histories. Thus, from (3.9) and (3.10) we conclude

fl(t) ~-f2(t)

fort > t'.

(3.11)

It is clear that inequality (3.11) holds even if the outputsfl(t') andf2(t') are somehow the same at time t I. This means that the Preisach model (3.1) describes, in general, hysteresis nonlinearities with nonlocal memories. The above discussion reveals the mechanism of memory formation in the Preisach model. The memory is formed as a result of two different rules for the modification of the interface L(t). Indeed, for a monotonically increasing input, we have a horizontal final link of L(t) moving upwards,

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

309

while, for a monotonically decreasing input we have a vertical final link of L(t) moving from right to left. These two different rules result in the formation of the staircase interface, L(t), whose vertices have coordinates equal to past input extrema. It is apparent from the previous analysis that the Preisach model can be defined in purely geometric terms, without any reference to the analytical definition (3.1). Indeed, the formula (3.8), along with the above two rules for the modification of L(t), can be interpreted as an independent definition of the Preisach model. This definition is fully equivalent to the previous one. However, the geometric definition may be convenient for further generalization of the Preisach model. For instance, new and more general rules for the subdivision of T into positive and negative sets, S+ (t) and S-(t), may be introduced. In these rules, the links of L(t) may not necessarily be the segments of straight lines parallel to coordinate axes. Furthermore, different functions/~+ (~, fl) and #-(~, fl) may be defined on the positive and negative sets, respectively. All these modifications may result in some meaningful generalizations of the Preisach model. However, the above possibilities have not been examined enough in the existing literature. Having described the geometric interpretation of the Preisach model, we are now well equipped for the discussion of the main properties of this model. We begin with the simplest property, which expresses the fact that the output value, f+, in the state of positive saturation is equal to the minus output value, - f - , in the state of negative saturation. In the state of positive saturation, the input u(t) is more than a0 and all ~-operators are in the 'up' position. Hence, according to (3.8), we find: f + - JJT/~(a' fl) da dfl.

(3.12)

Similarly, in the state of negative saturation the input u(t) is less than fl0 and all ~-operators are in the 'down' position. As a result, we obtain

f - - - ffT l.t(~x, fl) da dfl.

(3.13)

From (3.12) and (3.13), we have f+= -f-.

(3.14)

It is important to keep in mind that the saturation valuesf + and f - remain constant for any value of input u(t) above ~0 and below fl0, respectively. In other words, after ascending and descending branches merge together, they become flat. Partly for this reason, it is often said that the Preisach

CHAPTER 3 Scalar Preisach Models of Hysteresis

310

(~ / .

~ .

t

.

.

u 1 __

u3

I

U8

/

FIGURE 3.14

model does not describe reversible components of hysteresis nonlinearities. These components are regarded as being responsible for finite slopes of ascending and descending branches after they merge together. The inability of the Preisach model to describe the reversible components of hysteresis nonlinearities has long been viewed as a deficiency of the model. It will be shown later in the book that this deficiency along with some others can be removed by the appropriate generalization of the Preisach model. We next proceed to the more interesting property which further elucidates the mechanism of memory formation in the Preisach model. It turns out that this model does not accumulate all past extremum values of input. Some of them can be wiped out (erased) by subsequent input variations. To make this property clear, consider a particular past history that is characterized by a finite decreasing sequence {Ul, U3, U5, U7} of local input maxima and an increasing sequence {u2, u4, u6, us } of local input minima. A typical a - ] / d i a g r a m for this kind of history is shown in Fig. 3.14. Now, we assume that the input u(t) is monotonically increased until it reaches some maximum value u9 that is above u3. This monotonic increase of input u(t) results in the formation of a horizontal final link of L(t) that moves upwards until the maximum value u9 is reached. This results in a modified a-//diagram shown in Fig. 3.15. It is evident that all vertices whose a-coordinates were below u9 have been wiped out. It is also clear that the wiping out of vertices is equivalent to the erasing of the history associated with these vertices. Namely, the past input maxima and minima that were respectively equal to ~- and fl-coordinates of the erased vertices have been wiped out. We have illustrated how the wiping out of vertices occurs for monotonically increasing inputs. However, it is obvious that the wiping out of vertices occurs in a similar manner for monotonically

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

311

U 1

/

FIGURE 3.15

decreasing inputs as well. Thus, we can formulate the following property of the Preisach model. WIPING-OUTPROPERTY. Each local input maximum wipes out the vertices of L(t) whose a-coordinates are below this maximum, and each local minimum wipes out the vertices whose fl-coordinates are above this minimum. The wiping-out property is asserted above in purely geometric terms. This makes this property quite transparent. However, the same property can also be described in analytical terms. The analytical formulation complements the geometric one because it is directly phrased in terms of time input variations. Consider a particular input variation shown in Fig. 3.16 for the time interval toKt~t I. We assume that at the initial instant of time to the input value u(to) was below fl0. This means that the initial state is the state of negative saturation. Consequently, the whole history has been written by the input variation after time to. We would like to specify explicitly local input extrema that will be stored by the Preisach model at time Y. Consider the global maximum of the input at the time interval [to, t']. We will use the notation M1 for this maximum and t~- for the instant of time the maximum was reached: M1 - max u(t), u(t-~) - M1. (3.15) [t0,Y] It is clear that all previous input extrema were wiped out by this maximum. Now, consider the global minimum of the input at the interval [t +, Y]. We will use the notation ml for this minimum and t I for the time itwas reached: ml - min u(t), [t~-,t']

u ( t l ) - ml.

(3.16)

312

CHAPTER 3

Scalar Preisach Models of Hysteresis

u(t)

mk m2 ml

FIGURE 3.16

It is apparent that all intermediate input extrema that occurred between t + and t I were erased by the minimum ml. Next, consider the global m a x i m u m of the input at the interval [t 1, tl]. The notations M2 and t~- are appropriate for this maximum and the time it occurred: M2 - max u(t), [t~-,t']

u(t~)

-- M2.

(3.17)

It is obvious that this maximum wiped out all intermediate input extrema that occurred between t I and t~. As before, consider the global minimum of input at the time interval [t +, t'] and the notation m2 and t 2 will be used for this minimum and the time it was achieved: m2 = min u(t),

u(t2) = m2.

(3.18)

[t~-,t']

It is clear that this minimum wiped out all intermediate input extrema. Continuing the above line of reasoning, we can inductively introduce the global maximum, Mk, and global minimum, mk"

Mk -- max u(t), [tk_l ,t']

u(t-~) = Mk

(3.19)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

313

and

mk = min u(t),

u(t-~) -- mk.

(3.20)

[ t k+ ,t']

Only these input extrema were accumulated by the Preisach model, while all intermediate input extrema were erased. It is natural to say that Mk and mk (k = 1, 2 . . . . ) form an alternating series of dominant maxima and minima. It is evident from the above analysis that a- and//-coordinates of vertices of the interface L(t') are equal to Mk and mk, respectively. It is also clear that the alternating series of dominant extrema is modified with time. This means that new dominant extrema can be introduced by the time-varying input, while the previous ones can be erased. In other words, Mk and mk are functions of t f as it is clearly suggested by their definitions (3.19) and (3.20). Now, the wiping-out property can be stated in the following form. WIPING-OUTPROPERTY. Only the alternating series of dominant input extrema are stored by the Preisach model. All other input extrema are wiped out. It is worth noting that the wiping-out property is somewhat natural and consistent with experimental facts. Indeed, experiments in the area of magnetics show the existence of major hysteresis loops whose shapes do not depend on how these loops are approached. In other words, the major hysteresis loops are well defined. It means that any past history is wiped out by input oscillations of sufficiently large magnitude. This is in complete agreement with the wiping-out property. Consider another characteristic property of the Preisach model that is valid for periodic input variations. Let ul(t) and u2(t) be two inputs that may have different past histories (different alternating series of dominant extrema). However, starting from some instant of time to, these inputs vary back and forth between the same two consecutive extremum values, u+ and u_. It can be shown that these periodic input variations result in minor hysteresis loops. Let Figs 3.17 and 3.18 represent ~-fl diagrams for the inputs ul(t) and u2(t), respectively. As the inputs vary back and forth between u+ and u_, the final links of staircase interfaces Ll(t) and L2(t) move within the identical triangles T1 and T2. This results in periodic shape variations for L1 (t) and L2(t), which in turn produce periodic variations of the outputs, fl (t) and f2(t). This means that some minor hysteresis loops are traced in t h e f - u diagram for both inputs (see Fig. 3.19). The positions of these two loops with respect to the f-axis are different. This is because the above two inputs have different past histories that lead to different shapes for staircase interfaces, Ll(t) and L2(t). As a result, the values of

314

CHAPTER 3

Scalar Preisach Models of Hysteresis

/

FIGURE 3.17

___

U+_

FIGURE 3.18

2

~U U

U+

FIGURE 3.19

315

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

outputs for the same values of inputs are different. This is easily seen from the formula (3.8). However, it can be proved that the above two hysteresis loops are congruent. This means that the coincidence of these loops can be achieved by the appropriate translation of these loops along the f-axis. The proof of the congruency of the above loops is equivalent to showing that any equal increments of inputs ul(t) and u2(t) result in equal increments of outputs fl(t) and f2(t). To this end, let us assume that both inputs after achieving the same minimum value u_ are increased by the same amount: Aul = Au2 = Au. As a result of these increases, the identical triangles AT1 and AT2 are added to the positive sets S~-(t) and S~-(t) and subtracted from the negative sets S l(t) and S2 (t) (see Figs 3.17 and 3.18). Now, using the formula (3.8), we find that the corresponding output increments are given by

afl a f2

-

II #(~, fl) d~ dfl, J,I AT1

(3.21)

2 ]/ g(c~,/~)dor dfl. JJA T2

(3.22)

2

Since AT1 -- AT2, we conclude All = Af2.

(3.23)

The equality (3.23) has been proved for the case when inputs Ul(t) and u2(t) are monotonically increased by the same amount after achieving the same m i n i m u m value u_. Thus, this equality means the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (3.23) holds when the inputs Ul (t) and u2(t) are monotonically decreased by the same amount Au after achieving the maximum value u+. This means that the descending branches of the above minor loops are congruent as well. Thus, we have established the following property of the Preisach model. CONGRUENCYPROPERTY. All minor hysteresis loops corresponding to back-andforth variations of inputs between the same two consecutive extremum values are congruent.

Possible Applications of the Preisach Model as a Neural Network We conclude this section with a remark on possible applications of the Preisach model beyond the area of hysteresis modeling. It is clear from the previous discussion that the Preisach model has the ability to detect

316

CHAPTER 3 Scalar Preisach Models of Hysteresis

and store the alternating series of dominant extrema of input. In other words, the Preisach model is endowed with memory. For this reason, the Preisach model might have appeal as the mathematical model of memory with some interesting properties, and its device realization (see Fig. 3.2) might be utilized as an unusual storage device. We will discuss below only a few peculiarities of this memory. First, the mechanism of memory formation in the Preisach model is quite simple and results from the parallel connection of qualitatively similar elements (two-position cells) ~a/~. Since this model employs very simple elements and has little structure, its memory formation can be interpreted as a 'spontaneously emerged collective effect'. (This kind of terminology is in vogue in the area of neural networks; see, for instance, [25].) Second, storage of information is not localized. Indeed, the model (3.1) stores the information (extremum values of input) not in particular separate cells (as in the case of computer storage devices), but some ensembles of the cells ~/~ participate in storage of each bit of information. As a result, if some of the cells are destroyed, the stored information might still be preserved. This opens the way for cell replenishment without affecting the currently stored information. Third, some subsequent events may erase the information previously stored in the memory (wiping-out property). This erasing occurs only if new events make stronger impacts (characterized by larger input extrema) than previous ones. Fourth, it can be shown that the Preisach model exhibits 'fading' memory. This means that the stored information is slowly and gradually erased in the presence of small random perturbations (intrinsic noise). The properties discussed above are somewhat similar to those being observed (or suspected) for memories in biological systems. However, it would be imprudent to speculate how far this similarity goes. Nevertheless, it may be expected that the mathematical tool (3.1) might find some interesting applications beyond the conventional area of hysteresis modeling. 3.1.3 IDENTIFICATION PROBLEM FOR THE PREISACH MODEL REPRESENTATION THEOREM

Next, we proceed to the discussion of the identification problem for the Preisach model. The essence of this problem is in the determination of weight function #(~, fi). The set of first-order transition curves will be used for this purpose. These curves can be defined as follows. First, the input u(t) should be decreased to the value that is less than rio- This brings a hysteresis nonlinearly to the state of negative saturation. Next, the input is

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

317

/

FIGURE 3.20

I I

"I

I

,

,.u

FIGURE 3.21

monotonically increased until it reaches some value c~'. The corresponding a-fl diagram is shown in Fig. 3.20. As the input is increased, an ascending branch of a major loop is followed (see Fig. 3.21). This branch will also be called the limiting ascending branch because usually there is no branch below it. The notationfa, will be used for the output value on this branch that corresponds to the input value u = a t. Afirst-order transition (reversal) curve is formed as the above monotonic increase of the input is followed by a subsequent monotonic decrease. The term 'first-order' is used to emphasize the fact that each of these curves is formed after the first reversal of input. The notationf~,/~, will be used for the output value on the transition curve attached to the limiting ascending branch at the pointfa,. This output value corresponds to the input value u = fl' (see Fig. 3.21). The above

318

CHAPTER 3 Scalar Preisach Models of Hysteresis (x / o~'

---

T (0~,, 13, )

i

FIGURE 3.22

monotonic decrease of input modifies the previous a-fl diagram shown in Fig. 3.20. A new a-fl diagram for the instant of time when the input reaches the value fl' is illustrated by Fig. 3.22. Now, we define the function 1

(3.24)

=

This function is equal to one-half of the output increments along the firstorder transition curves. The next step is to express this function in terms of the Preisach function #(a, fl). To this end, we compare the a-fl diagrams shown in Figs 3.20 and 3.22. It is clear from these diagrams that the triangle T(a', fl') is added to the negative set S- and subtracted from the positive set S + as a result of the monotonic input decrease from the value u = a' to the value u = fl'. Using the above fact and the formula (3.8), we find that the Preisach model will match the output increments along the first-order transition curves if the function #(~, fl) satisfies the equation

fa, fl, - fa, = -2 f/T(a',fl') #(a, fl) da dfl.

(3.25)

By comparing the formulas (3.24) and (3.25), we obtain

F(a', fl') = [I /~(a, fl)da dfl. .liT (~,,~')

(3.26)

The integral over the triangle T(a ~, fl~) can be written as the following double integral: F(a', fl') =

,

#(a, fl)da

dfl.

(3.27)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

319

By differentiating the last expression twice (first with respect to fiit and then with respect to air), we find

#(ait fiit) = _

a2F(ait, fiit) 0ait0fl it .

(3.28)

Invoking (3.24), the expression (3.28) can be written in another equivalent form: 1r ' (3.29) #(a', fl') - 2 0a'0fl'" The formula (3.29) allows for a simple geometric interpretation of the function #(a', fl'). Indeed, for the first derivative of fa,~, with respect to fl' we have ~f~,/~' (3.30) @' = tan 0(a', fl'), where 0(a it, fiit) is the angle b e t w e e n the axis u and the tangent to the firstorder transition curvefa, fl, at the point u = fiit. From (3.29) and (3.30), we find 1 0 tan 0(a it, fiit)

(3.31)

-

From (3.31), we conclude that the Preisach function #(a', fl') is positive if tan 0(a', fl') is a monotonically increasing function of a' for any fixed fiit. The last condition is satisfied if all first-order transition curves are monotonically increasing functions of fiit, and they do not intersect inside the major loop but merge together at the point where the descending and ascending branches of the major loop meet one another (see Fig. 3.23). 1 To secure the above merge of first-order transition curves, tan 0(a it, fiit) should increase as a function of a it for any fixed fit in order to compensate for larger values off~,/~,. It is worthwhile noting that in the 'magnetic' definition of the Preisach model (see Section 3.1.1) the function #(a, fl) is treated as a distribution function with some statistical connotation. For this reason, this function is tacitly a s s u m e d to be positive. In our treatment, the last property is directly related to experimental facts. For positive #(a, fl), F(a it, fiit) is a monotonically increasing function of a it for any fixed fit and a monotonically decreasing function of fl' for any fixed a it. This follows directly from (3.26).

lit is useful to keep in mind that the descending branch of the major loop can be interpreted as a particular case of the first-order transition curve for which a I = a 0.

CHAPTER 3 Scalar Preisach Models of Hysteresis

320

f

U

i F I G U R E 3.23

If the Preisach function is determined from (3.29) (or (3.28)), then the expression (3.25) is satisfied. This means that the Preisach model matches the output increments along the first-order transition curves. We next show that this also implies that the first-order transition curves themselves are matched. To this end, consider the particular case when fl' - fl0. For this case, we have f=,& = f - . (3.32) From (3.25) and (3.32), we obtain: f/" f - -f~, = - 2 I I

,liT (~',3o)

#(e, fl) da dfl.

(3.33)

Thus, the Preisach model matches the output increment, f - - f ~ , , along the limiting ascending branch. For a t - ~0, we have f~0 - f + . (3.34) From (3.34) and (3.33), we find

f - - f + = - 2 f f T #(o~, fl) dcz dfl,

(3.35)

where T = T(a0, rio) is the limiting triangle. Since major hysteresis loops are usually symmetric, the following equality holds: f+= -f-. (3.36)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

321

From (3.35) and (3.36), we derive:

f - - - ffT #(~' fl) da dfl.

(3.37)

This means that the Preisach model matches the output value in the state of negative saturation. From the last fact and (3.33), we conclude that the model matches the limiting ascending branch. Since the limiting ascending branch and the output increments along first-order transition curves are matched, we deduce that the first-order transition curves themselves are matched. The Preisach function, #(~, fl), has been found by using the first-order transition curves. These curves are attached to the limiting ascending branch, and each of them is formed when a monotonic increase along this branch is followed by a subsequent input decrease. For this reason, these curves can be named first-order decreasing transition curves. However, by almost literally repeating the previous reasoning, a similar expression for #(~, fi) can be found by using the first-order increasing transition curves. These curves are attached to the descending branch of the major loop. Each of these first-order increasing transition curves is formed as a monotonic decrease along the limiting descending branch is followed by a subsequent input increase. The notationf]~,, will be used for the output value on the limiting descending branch. This value is achieved when the input is monotonically decreased from some value above a0 to the value u = fit,. The corresponding a-/J diagram is shown in Fig. 3.24. The notation f~,,e, will be used for the output value on the first-order increasing transition curve that is attached to the limiting descending branch at the point f/~,,. This output value corresponds to u - :r (see Fig. 3.21). The corresponding a-fl diagram is shown in Fig. 3.25. By using the function #(~, fl) d0{ dfl,

=

(3.38)

from Figs 3.24 and 3.25 and the formula (3.8), we obtain F(a", fl") - 1 (/,,,=,, -f,,,)

(3.39)

From (3.38), as before we find cq2F(o~", ]~') _

_

&,,@,,

(3.40)

CHAPTER 3

322

Scalar Preisach Models of Hysteresis

FIGURE 3.24 / s+

T (o~",ly' )

/

FIGURE 3.25

It is clear on the physical grounds (symmetry considerations) that firstorder decreasing and increasing transition curves are congruent. In mathematical terms, this means that if fl"=-~'

and

~"=-fl',

(3.41)

then f//,,- -f~,

and f/~,,~,,- -f~,/~,.

(3.42)

From (3.24), (3.39), and (3.42) we find that if (3.41) holds, then F(a", fl") = F(a', fl').

(3.43)

By substituting (3.41) into (3.43), we obtain

F(-fl', -o() - F(o(, #').

(3.44)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

323

/

,, ,,~,,(.u;~,) X (- u',- ~') " , , , ~

FIGURE 3.26

Next, substituting (3.43) and (3.41) into (3.40), we derive #(-fl" -~') - -

c92F(~', fl') a]/'a~, "

(3.45)

By comparing (3.45) and (3.28), we find # ( - f l ' , - a ' ) = #(a', fl').

(3.46)

The formulas (3.44) and (3.46) express the mirror symmetry of functions F(~, fl) and #(a, fl) with respect to the line ~ = -fl (see Fig. 3.26). This symmetry is a consequence of the congruency of the first-order decreasing and increasing transition curves. If the line a = -fl is the interface between the positive and negative sets, S + (t) and S-(t), then, according to the above symmetry and (3.8), we find that the output is equal to zero:

fit) = 0.

(3.47)

For this reason, the state corresponding to the above interface is called in magnetics 'the demagnetized state'. However, this state cannot be exactly achieved. This is because an interface L(t) is always a staircase line whose links are parallel to the a and//axes. As a result, an actual staircase interface can only approximate the line a = -ft. Such an approximation can be achieved, for instance, if the input u(t) is an oscillating function whose amplitude is slowly decreased to zero starting from some value above a0 (see Fig. 3.27). The corresponding staircase interface is shown in Fig. 3.28. We next proceed to the formulation and the proof of the fundamental theorem which gives the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model.

CHAPTER 3 Scalar Preisach Models of Hysteresis

324

U

I

FIGURE 3.27 fx /

/

FIGURE 3.28

The wiping-out property and the congruency property constitute the necessary and sufficient conditions for a hysteresis nonlinearity to be represented by the Preisach model on the set of piecewise monotonic inputs.

REPRESENTATION THEOREM.

PROOF. Necessity: Let a hysteresis transducer be representable by the Preisach model. Then, this transducer should have the same properties as the model. In particular, it should have the wiping-out and congruency properties. Sufficiency: Consider a hysteresis transducer that has both the wipingout property and the congruency property. It can be recalled that the wiping-out property means that the future values of output do not depend on all past extremum values of input but only on those that form an alternating series of dominant input extrema. The congruency property, on the other hand, suggests that all minor hysteresis loops formed as a result of back-and-forth input variations between the same two consecutive input extrema are congruent. We intend to prove that the above properties

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

325

f

I

FIGURE

7

~u

3.29

imply that the hysteresis transducer can be represented by the Preisach model. The proof of the last statement is constructive. First, it is assumed that the weight function, #(~, fl), is found for the given transducer by matching its first-order transition curves. This can be accomplished by using the formula (3.28). This formula is equivalent to (3.25), which means that the integrals of/~(~, fl) over triangles T(~', fl') are equal to one-half of the output increments, (1/2)Af = (1/2)(f~, -f~,F), along the first-order transition curves. Next, it will be proved that if the above weight function is substituted in (3.1), then the Preisach model and the given transducer will have the same input-output relationships. This statement is true for the first-order transition curves due to the very way the weight function, /~(~, fl), is determined. The induction argument will be used to prove that the same statement holds for higher-order transition curves as well. Let us assume that the above statement is true for transition curves with numbers 1, 2 . . . . . k. Then, for the induction inference, we need to prove that this statement holds for a transition curve number k 4- 1. Let a be a point at which the transition curve number k 4- 1 starts (see Fig. 3.29). The point, a, corresponds to some input value u = a'. According to the induction assumption, the output values of the transducer and the Preisach model coincide at this point. Thus, it remains to be proved that the output increments along the transition curve number k 4-1 are the same for the actual transducer and for the Preisach model. Consider an arbitrary input value u = fl' < ~t. The output increment for the transducer will be equal to the increment off along some curve ab (see Fig. 3.29). Let Figs 3.30 and 3.31 represent a-fl diagrams for the Preisach model at the time instants

CHAPTER 3 Scalar Preisach Models of Hysteresis

326

/ s-

FIGURE 3.30

/ s-

/ FIGURE 3.31

when u = ~ and u - ]/~, respectively. From these diagrams we find that the input decrease from ~' to ]/' results in adding the triangle T(~', ]/') to the negative set, S-, and subtracting the same triangle from the positive set, S +. Using the above fact and the formula (3.8), we find that for the Preisach model the output increment along the transition curve number k + 1 is given by

af

2 [[ #(~, fl) d~ dfl. d.l Z(~',3')

(3.48)

However, according to the way the function, #(a, ]/), is defined, the righthand side of (3.48) is equal to the increment of the transducer output along the first-order transition curve cd (see Fig. 3.29). Thus, it remains to be shown that the output increments along the curves ab and cd are the same. It is here that the wiping-out and congruency properties will be used. The

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

327

proof proceeds as follows. If starting from point b we monotonically increased the input value from fl' back to a', then, according to the wiping-out property, we would arrive at the same point a by moving along some curve ba (see Fig. 3.29). Indeed, the wiping-out property implies that as soon as the input exceeded the value a', the history associated with back-and-forth input variations between ~' and fl' should be erased and the subsequent output variation should follow the transition curve number k. But, this would be possible only if for u = ~' we arrived back at the point a. Similarly, if starting from the point d we monotonically increased the input value from fl' to a', then, according to the same wiping-out property, we would arrive at the point c moving along some curve dc (see Fig. 3.29). Now, by invoking the congruency property, we conclude that the hysteresis loops bab and dcd are congruent. This is true because both loops result from back-and-forth input variations between the same two consecutive input extrema, a' and fl'. From the congruency of the above loops, we find that the output increments along the curves ab and cd are the same. Consequently, the output values of the transducer and the Preisach model coincide along the transition curve number k + 1. The last fact has been proved for any fl'~?' (see Fig. 3.29). However, according to the wipingout property, the transition curve number k 4- 1 should coincide with the transition curve number k - 1 for fl' < 7'. Thus, the case//' < 7' falls in the domain of the induction assumption. In the above discussion, we have considered the transition curve number k + 1 corresponding to the monotonically decreasing input. However, the case when the transition curve number k 4- 1 is formed as a result of monotonically increasing input can be treated similarly. The only difference in the proof will be that the first-order increasing transition curves must be employed instead of the first-order decreasing transition curves used in the above reasoning. This completes the proof of the theorem. [] Discussion

It is easy to see that the essence of the given proof is in the reduction of higher-order transition curves to the first-order transition curves. This reduction rests on both the wiping-out property and congruency property. The proven theorem is important because it clearly establishes the limits of applicability of the Preisach model. These limits are formulated in purely phenomenological terms, without any reference to the actual physical nature of hysteresis. This reveals the physical universality of the Preisach model. The theorem also explicitly indicates the factors which affect the accuracy of the Preisach model. Indeed, deviations from the

328

CHAPTER 3 ScalarPreisach Models of Hysteresis

conditions of the representation theorem may serve as the measure of model accuracy. The above theorem allows for a simple explanation of so-called statistical instability of the Preisach model. This instability has been a very popular topic in the 'magnetic' literature. It usually means that weight functions, #(a,//), determined from different experimental data are not identical. Since these functions have been construed in magnetics as 'particle distributions' (with some statistical connotation), the dependence of /x(a,//) on a particular experimental way of its determination has been termed as 'statistical instability'. The origin of this 'statistical instability' can be easily understood from the following discussion. The formula (3.28) for #(a,//) has been derived from the expression (3.26). This means that the weight function, #(a, fl), can always be determined if the integrals of #(a, fl) over the triangles T(a', fl') are somehow experimentally found. It has been proposed before (see (3.25)) to find these integrals by matching the first-order transition curves. However, this is not the only way it can be done. Indeed, according to the formula (3.48), the same integrals can be found by matching the output increments along any particular transition curves. But, we will end up with the same values for these integrals if and only if higher-order transition curves and the first-order transition curves are congruent. According to the proof of the representation theorem, this congruency takes place if and only if the wiping-out and congruency properties are valid for hysteresis transducers. If these properties are not valid, then, by matching different transition curves, we will end up with different values for the integrals over the triangles T(~',//') and, consequently, with different values of #(~, fl). This is exactly what happened when 'statistical instability' was discovered (see [3.8]). Thus, from the phenomenological point of view, the origin of 'statistical instability' comes from the fact that the wiping-out and congruency properties may not be exactly satisfied for actual hysteresis nonlinearities. Under these circumstances, the Preisach model cannot serve as an absolutely accurate representation for actual hysteresis nonlinearities, but it can still be used as an approximation. The quality of this approximation will depend on the extent to which the wiping-out and congruency properties are satisfied. It has been mentioned before that the weight function, #(~,//), can be found by matching output increments along any transition curves. Thus, the question arises: 'Which transition curves should be used for the above purpose?' If the wiping-out and congruency properties are valid, then it really does not matter which transition curves are used for the determination of/~(~, fl). All of them will theoretically lead to the same result. However, from the practical point of view, the first-order transition curves have some

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

329

clear advantages. First, it is easier to find these curves experimentally than higher-order transition curves. Second, measurements of these curves start from a well-defined state, namely, the state of negative (or positive) saturation. This is not the case for some experimental techniques described in the literature. For instance, it has been suggested in the literature to use a demagnetized state as a starting state in the experimental determination of #(e, fl). But, as was pointed out before, the demagnetized state is not well defined for the Preisach model. This state depends on the particular way it has been prepared. As a result, some errors and discrepancies may be introduced if the demagnetized state is used as the initial state for the experimental determination of/2(e, fl). It is often asserted in the literature that the Preisach model describes reversal curves with zero initial slopes. This statement is based on the following reasoning. Let f~,/~, be a reversal curve which is traced for a monotonically decreasing input. This curve starts at the point f~,, which corresponds to the local input maximum u - e'. The difference f~,fl, -f~, is then given by (3.25) and the current slope of this reversal curve at the point fl = fl' can be found from (3.30). By using (3.25), (3.26), (3.27), and (3.30), it is easy to derive the following expression for the initial slope lim tan O(e', fl') = 2 lim

#(e, if)de.

In the literature, the function #(e, fl) is usually construed as a distribution function for magnetic particles and, for this reason, it is tacitly assumed to be bounded. This assumption and the last formula result in zero initial slopes for the reversal curves f~,fl,. This property is often regarded as an intrinsic property of the Preisach model. However, this is not true. Indeed, if the phenomenological approach is adopted, then there is no need to interpret/2 as a distribution for magnetic particles and the assumption that /2 is bounded can be removed. The freedom of choice of function #(e, fl) should be used in order to match as many experimental facts as possible. It is easy to show that experimentally observed nonzero initial slopes of reversal curves f~,~, can be matched by the Preisach model if we allow for delta-type (Dirac function) singularities of #(e, fl) along the line e = ft. It is easy to see that these singularities can be interpreted as an additional (and fully reversible) term in the Preisach model consisting of degenerate (zero width) rectangular loop (or step) operators ~,~:

fit) =

fl(e, fl)~flu(t) de dfl +

k(e)~au(t) de,

where function fl(e, fl) is now free of singularities.

CHAPTER 3 Scalar Preisach Models of Hysteresis

330

f

ms I

I fo~13

/

'/Li I I I

I I I

w

v

1~

t~ 2

~u

-m/ r

w F I G U R E 3.32

This form of the Preisach model leads to the following modification of formulas (3.25)-(3.27)-

f~'B' -f~'

= -2

,

#(a, fl) da

dfl - 2

, k(a) da.

From the last formula we derive: ~t

tan O(e', fl') =

c~fl'

= 2

ffl

, #(a, fl')da + 2k(fl'),

and lim tan 0(a', fl') = 2k(a'). Thus, the actual nonzero initial slopes of the first-order reversal curves can be matched by the appropriate choice of function k(a). This section will be concluded by the discussion of a simple example that illustrates the application of the formula (3.29). Consider a hysteresis nonlinearity with a local memory shown in Fig. 3.32. This hysteresis nonlinearity has a set of inner curves that are parallel to the u-axis. The above hysteresis nonlinearity is completely characterized by the following parameters: ms, ~Xl, 0~2, W. O u r goal is to find the Preisach representation

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

331

(3.1) for this nonlinearity. To this end, we have to find the weight function #(~, fl). This requires the specification of first-order transition curves f~/~ and the subsequent application of the formula (3.29). It is clear from Fig. 3.32 that for any a such that 0r < 0r < OO the corresponding first-order transition curve f~/~ consists of two parts: (a) a particular inner curve, and (b) a part of the limiting descending branch if al - w ~ ] ~ 0 ~ 2 - w , tiff- w ~ 0 r K oo. It is apparent thatf~/~ varies only along the second part. From the above remarks, we find 2ms

f~ - -ms + ~ ( f l -

o~1 +

W),

(3.49)

0r -- 0~1

0r1 -- W ~ f l ~ O ~ 2 -- W,

~

+ w~cz~oo,

(3.50)

and f~/~ -- const

(3.51)

for all other possible values of ft. It is worthwhile to keep in mind that, depending on a particular value of fl, the constant in (3.51) may assume one of the following two values: - m s , - m s + ((2ms)/(~z2 - 0r162 - 0r From (3.49), (3.50), and (3.51), we derive: c~f~

2ms

=

~

,

(3.52)

0~2 -- 0~1

if fl + w ~ a ~ o o ,

0~1 -- W ~ f l ~ O ~ 2 -- W,

(3.53)

and 8f~/~ = 0

(3.54)

for all other values of a and ft. The last three formulas can be written in the following equivalent form: c~f~

2ms

=

~fl

0r

,

if (a, fl) ~ f~,

(3.55)

- - 0r

and ~f~/~ = 0,

M

if (~, fl) ~ f~,

(3.56)

where f] is the region defined by the inequalities (3.53) (see also Fig. 3.33).

CHAPTER 3 Scalar Preisach Models of Hysteresis

332

0~ /

/

//

/

/ /

I !

I i

.p

3.33

FIGURE

From (3.55) and (3.56), we obtain:

c92f ~ 630~6~~

2ms

~&(a-

O~2 -- O~1

f l - w),

(3.57)

(3.58)

~1 ~ 0 r

and (32f~/~)/ (cgacgfl)is equal to zero otherwise. From (3.29), (3.57), and (3.58), we find #(~, fl) = m------L--s&(~- f l - w) (3.59) 0r -- 0r

for (3.60)

~1 ~ ~ 2

and #(a, fl) is equal to zero otherwise. Thus, the weight function #(a, fl) has the support on the line a - fl- w = 0. This support is the bold segment (see Fig. 3.33) of the above line. By substituting (3.59) and (3.60) into (3.1), after simple transformations we obtain fit)

-

ms 0~2 -- ~1

~~2 1

~,~_wU(t) d~.

(3.61)

This is the Preisach representation of the hysteresis nonlinearity shown in Fig. 3.32. It is left to the reader as a useful exercise to verify that the formula (3.61) indeed describes all branches of the above hysteresis nonlinearity.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

333

,' , [ -

U m

"',2 x4 I I I I I I

Urn

U

I I I I I t

FIGURE 3.34

It is clear from Fig. 3.33 that by varying 0r 0r and w we can freely move around the support of the function # in the half-plane a~fl. This fact and the expression (3.61) suggest that the equivalent representation for the Preisach model can be obtained as a superposition (parallel connection) of an infinite number of hysteresis nonlinearities shown in Fig. 3.32. However, the representation (3.1) is preferable due to the most elementary nature of hysteresis operators ~a/~. It is not accidental that in the above example the function #(a, fl) has a 'line support'. This is related to the fact that the above hysteresis nonlinearity has a local memory. It can even be proved in general that if we have a hysteresis nonlinearity with a local memory that can be represented by the Preisach model, then the weight function #(~, fl) has a support along some particular curve. This curve may intersect all possible staircase interfaces L(t) only once. It was emphasized before that the Preisach model can describe (and usually does describe) hysteresis with nonlocal memories. These nonlinearities are much more complex than those with local memories. However, it would be a mistake to think that any hysteresis nonlinearity with local memory can be represented by the Preisach model. In fact, there are very simple hysteresis nonlinearities with local memories that cannot be described by the model. For instance, this is true for the hysteresis nonlinearity shown in Fig. 3.34. Indeed, let the input vary back and forth between two consecutive extremum values a' and fl',--Um < fi' < ~' < Um. Then, depending on the past history, the bold reversible parts of the descending and ascending branches will be traced back and forth. These parts can

334

CHAPTER 3 ScalarPreisach Models of Hysteresis

be construed as degenerate minor loops. Since these parts are not congruent; the congruency property of the representation theorem is not satisfied. Consequently, the hysteresis nonlinearity shown in Fig. 3.34 cannot be represented by the Preisach model. However, this hysteresis nonlinearity is typical for single Stoner-Wohlfarth magnetic particles. It is known from the theory of these particles that the reversible parts of their hysteresis loops result from uniform rotations of magnetization within these particles, that is, from reversible processes. Incongruence of the above parts is the reason why the Preisach model cannot describe the hysteresis loops of single Stoner-Wohlfarth particles. For this reason, it is often asserted that the Preisach model does not fully account for reversible processes. It will be shown in Section 3.2 that this deficiency of the Preisach model can be removed by its appropriate generalization.

3.1.4 NUMERICAL IMPLEMENTATION OF THE PREISACH MODEL

The Preisach model can be numerically implemented by using the formula (3.8) for the computation of the output, fit), and the formula (3.29) for the determination of the weight function, #(~, fl). Although the above approach is straightforward, it encounters two main difficulties. First, it requires the numerical evaluation of double integrals in (3.8). This is a time-consuming procedure that may impede the use of the Preisach model in practical applications. Second, the determination of the weight function #(a, fl) by employing the formula (3.29) requires differentiations of experimentally obtained data. These differentiations may strongly amplify errors (noise) inherently present in any experimental data. It turns out that another approach can be developed for the numerical implementation of the Preisach model. This approach completely circumvents the above two difficulties. It is based on the explicit formula for the integrals in (3.8). This formula directly involves (without any differentiation) the experimental data used for the identification of #(~, fl). Moreover, the above formula will be a valuable tool for the theoretical investigation of the Preisach model, and it is also useful in the discussion of vector models of hysteresis. The starting point for the derivation of the explicit formula for fit) is the expression (3.8). It is worthwhile to remind here that the positive (S + (t)) and negative (S-(t)) sets in (3.8) are separated by the staircase interface L(t). This interface has vertices whose a and fl coordinates are equal to Mk and mk, respectively (see Fig. 3.35). As discussed in Section 3.1.2, numbers Mk and mk form the alternating series of dominant input extrema.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

335

/

~-~1 (Mk, mk_~)

(t)

FIGURE 3.35

By adding and subtracting the integral of/~(a, fl) over S + (t), the expression (3.8) can be represented in the form:

f(t) - - ffz#(O~, fl) do~dfl + 2 fL+(t) #(~, fl) da dfl,

(3.62)

where, as before, T is the limiting triangle. According to (3.26), we find:

ffTlz(~z, fl) dcz dfl - F(ao, rio).

(3.63)

The positive set, S + (t), can be subdivided into n trapezoids Qk (see Fig. 3.35). As a result, we have n(t)

fL

+(t)

p(a, fl)dcz dfl - ~

k=l

fro k(t)

fl)da dfl.

(3.64)

It is clear that the number, n, of these trapezoids and their shapes may change with time. For this reason, n and Qk are shown in (3.64) as functions of time. Each trapezoid Qk can be represented as a difference of two triangles, T(Mk, mk-1) and T(Mk, mk). Thus, we obtain

ffQ k(t)

fi) dfl = ff r(Mk,mk-1) p(a, fl) d~ dfl ff - ]/

JJT (Mk,mk)

/z(~, fi)da dfl,

where, for the case of k - 1, m0 in (3.65) is naturally equal to rio.

(3.65)

CHAPTER 3

336

Scalar Preisach Models of Hysteresis

According to (3.26), we find:

ffT

(Mk,mk-1)

fl) da dfl = F(Mk, mk-1),

(3.66)

and

fIT

[1(~, fl) dc~ dfl - F(Mk, ink). (Mk,mk) From (3.65), (3.66), and (3.67), we derive:

/Q

fl(a, [1) da dfl = F(Mk, mk-1) -- F(Mk, mk).

(3.67)

(3.68)

k(t)

From (3.62), (3.63), (3.64), and (3.68), we obtain n(t)

f(t) = -F(~o, rio) 4- 2 ~ [ F ( M k , mk-1) -- F(Mk, mk)].

(3.69)

k=l

It is clear from Fig. 3.35 that mn is equal to the current value of input

mn = u(t).

(3.70)

Consequently, the expression (3.69) can be written as n(t)-I

fit) = -F(~o, rio) 4- 2 ~

[F(Mk, ink-l) -- F(Mk, mk)]

k=l

4- 2[F(Mn, ran-l) - F(Mn, u(t))]. (3.71) The last expression has been derived for monotonically decreasing input, that is, when the final link of interface L(t) is a vertical one. If the input u(t) is being monotonically increased, then the final link of L(t) is a horizontal one and the a-fl diagram shown in Fig. 3.35 should be slightly modified. The appropriate diagram is shown in Fig. 3.36. This diagram can be considered as a particular case of the previous one. This case is realized when mn(t) = Mn(t) -- u(t). (3.72)

According to the definition (3.26) of F(a, fl), we find

F(Mn, mn) - F(u(t), u(t)) - O.

(3.73)

From (3.69), (3.72), and (3.73), we derive the following expression for fit) in the case of monotonically increasing input: n(t)-I

fit) - -F(~o, rio) 4- 2 ~_, [F(Mk, mk-1) -- F(Mk, mk)] 4- 2F(u(t), mn-1). k=l

(3.74)

3.1 THE CLASSICAL P R E I S A C H M O D E L OF HYSTERESIS

337

/

]il F I G U R E 3.36

The function F(a, fl) is related to experimentally measured first-order transition curves by the formula (3.24). Using this formula, expressions (3.71) and (3.74) can be written in terms of the above experimental data as follows: n-1

fit) - -f +

+ ~(fMkm k --&kmk_l ) "J-&nu(t) --&nmn-1,

(3.75)

k=l n-1

fit)-

-f +

+ ~--~(fMkmk --&kmk_X ) -4-f--mn_l -- f--mn_x,--u(t)"

(3.76)

k=l

Here, f+ is, as before, the positive saturation value of output, and the last term in (3.74) has been transformed using the formulas (3.24) and (3.44). Thus, we have derived the explicit expressions (3.75) and (3.76) for f i t ) in terms of experimentally measured data. These expressions constitute the basis for the numerical implementation of the Preisach model. Using the above expressions, the digital code that implements the Preisach model has been developed. This code determines the output of a hysteresis transducer from an input, a set of first-order transition curves and an input history that are all specified by the user. The code also allows the user the flexibility of specifying an analytical expression for the weight function,/~(~, fl), instead of first-order transition curves. In this case, the function F(~, fl), defined by (3.26), is precomputed and then used in the formulas (3.71) and (3.74) for the evaluation of output values. As usual, the code performs three main functions: preprocessing, processing, and postprocessing. During the preprocessing stage, a square mesh covering the limiting triangle T (see Fig. 3.37) is created and a discretized set of first-order transition curves is entered. This set consists of mesh values of the functionf~fl. In the case when an analytical expression

338

CHAPTER 3 ScalarPreisach Models of Hysteresis

-§247247 _#-,-§

FIGURE 3.37

for #(a, fl) is specified, the mesh values of F(a, fl) are computed during the preprocessing. At the processing stage, an input history and current values of input are entered. Using these data and the formulas (3.19) and (3.20), the alternating series of dominant input extrema {Mk, ink} is first determined and then continuously updated for each new instant of time. Using Mk, mk, and mesh values off~]~ (or F(~fl)), all terms in the formulas (3.75) and (3.76) (or (3.71) and (3.74)) are computed. This is done by first determining particular square (or triangular) cells to which points (Mk, mk), (Mk, mk-1), and (Mn, u(t)) belong, and then computing the values off~/~ (or F(a, fl)) at these points by means of interpolation of the mesh values off~]~ (or F(a, fl)) at the vertices of the above cells. In the code, the interpolation is based on the following local approximation to fa~ in each square mesh cell: f~fl "~" CO if- C10~ q-- C2fl -}- C30~fl. (3.77) The C-coefficients in (3.77) are found by matching the values of f~/~ in the cell vertices. For triangular cells that are adjacent to the line a = fl, the linear local approximations to f~/~ are used: fail "" CO q- Clo~ q- C2~.

(3.78)

After computing all necessary values of f~/~, the current values of output are evaluated by employing the formulas (3.75) and (3.76). At the postprocessing stage, the computed output values are stored in a special file along with the corresponding input values and the values of Mk and ink. The user has the option to plot the output and input as functions of time, the path traced on t h e f - u plane and the set of first-order transition curves used in computations.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS .....i ' ' ' ' I

....

1 ' ' ' ' i ' ' ' ' 1

....

I ' ' ' ' 1

....

339

I ' ' ' ' I ' ' ' ' [ ' ' ' ' l

-

.04

.02

- -

-

J 4.J

:

0

o=

-.02

-

-.04

-

-2500

-2000

- 1500

- 1000

-500

0

500

1000

1500

2000

2500

Input

FIGURE 3.38

Below, we present some examples computed by using the above code. Figures 3.38, 3.39, and 3.40 show the set of first-order transition curves, the initial staircase interface L(t), and the input u(t), respectively. These data have been used to compute the outputflt). The computed outputflt) and t h e f - u path of the transducer are given in Figs 3.41 and 3.42, respectively. As was pointed out before, the derived formulas (3.75) and (3.76) (or (3.71) and (3.74)) are useful not only for the numerical implementation of the Preisach model but for its theoretical investigation as well. For instance, by using (3.75) and (3.76), the following proposition can easily be proven. PROPOSITION. A n y f -u path of hysteresis transducer representable by the Preisach model is piecewise congruent to first-order transition curves. In other words, any f - u path of such a transducer consists of pieces, which are congruent to some particular first-order transition curves. PROOF. Suppose that the input u(t) is being monotonically decreased from its previous maximum value u = Mn until it reaches the value u = mn-1. Then, the formula (3.75) is valid. In this formula, all terms exceptfMmu(t) do not vary. Consequently, the f - u path traced during the above input variation is congruent to the first-order transition curvefMnu that is attached to the limiting ascending branch at the point fMn" As the input reaches

CHAPTER 3

340

,

w

I

~

~'

~ ...... '

"i' ''~

~

'

'

Scalar Preisach Models of Hysteresis

I

~

'~

~

'~

J

i.

J

l

,

"r

,

~ ............ I '

i

I

2000

1000

I

/

- 1000 -2000 l.

I

i ..... i

,

,

-2000

1

,

~

J

,

I

- 1000

I

0

1

~

I

I...,

1000

I

,

,

2000

F I G U R E 3.39

i-,

2000

.... ,

.., .... i ..... i . ,

~

,

,

i

,

,

.v

,

I.

,.

,

,

-T

i..~.,.~

...... ,

~

9i

!

1000

-1000

-2000

0

100

200

300 Time

FIGURE 3.40

400

500

,

341

3.1 T H E C L A S S I C A L P R E I S A C H M O D E L OF HYSTERESIS

.04

.02

o

-.02

-.04

0

lO0

200

300

400

500

Time

F I G U R E 3.41

'

'

I

. . . .

I

'

'

'

'

I

'

'

'

i

I

'

'

'

'

1

.04

.02

.a

0

o

-.02

-.04

-2000

- 1000

0 Input

F I G U R E 3.42

1000

2000

'

'

CHAPTER 3 ScalarPreisach Models of Hysteresis

342

the value mn-1, the cancellation of the last two terms in (3.75) occurs. If the input is being further decreased, remaining between u = ran-1 and u -~ ran-2, then the formula (3.75) is modified as follows n-2

fit) - - f + + ~_, (fMkmk -- fMkmk_l) -}- fMn-lU(t) -- fMn-lmn-2 9 k-1

(3.79)

From this formula, as before, we conclude that the f - u path traced during the above input variation is congruent to the first-order transition curve fM,_lU that is attached to the limiting ascending branch at the point f = fMn-l" By continuing the same line of reasoning, we find that if the input is being monotonically decreased between u = mk and u = mk-1, then the corresponding f - u path is congruent to the first-order transition curve fMkU. For the case when the input is monotonically increased, the formula (3.76) is valid. From this formula, we find that if the input is monotonically increased between the values u = Mk and u -- Mk-1, then the time-varying part of the output is described by the t e r m f_mk_l_u(t). Thus, the corresponding f - u path is congruent to the first-order transition curvef-mk_l.-u that is attached to the limiting ascending branch at the point f = f-mk-l" This completes the proof. [] It is evident from the proven theorem that initial slopes of reversal curves described by the Preisach model coincide with initial slopes of experimentally measured first-order transition curves and, consequently, in general they are not equal to zero. This is consistent with our discussion of initial slopes of reversal curves presented in the previous section. Another important application of the formulas (3.75) and (3.76) is the derivation of the expressions for linear extrapolations of output corresponding to small input increments. Such extrapolative expressions can be useful for numerical analysis of systems with hysteresis by time-marching techniques. The derivation of these expressions proceeds as follows. Consider the case when the input has been monotonically increased for t t~t and this monotonic increase continues just after the time t. Then, the formula (3.76) is valid for the output before and after the time t. From this formula, by using the local Taylor expansion at time t for t t > t, we find f ( t ' ) - f i t ) "" -

ff-mn-l ~

[u(t') - u(t)].

(3.80)

fl=-u(t)

It is convenient to represent the last formula in terms of input and output increments: A f -- --

(~f--mn-lfl ] AU. Off fl=-u(t)

(3.81)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

343

It is important to keep in m i n d that mn-1 in (3.81) m a y change with time. This occurs because of the wiping out property. It is easy to see that, at any instant of time, mn-1 in (3.81) is the largest mk in the series of Mk and mk. If a monotonic increase of input for t ~Kt is followed by a monotonic input decrease for t t > t, then the input value u(t) can be treated as the last m a x i m u m , Mn, and from (3.75), we find: n-1

fit) -- --f+ -t- ~-~(fMkmk -- &kmk+l) q- fu(t)u(t) -- fu(t)mn-1

(3.82)

k=l

and

n-1 f (t') = - f + + ~_, (fMkmk -- &kink-1 ) q- fu(t)u(t') -- fu(t)mn-l"

(3.83)

k=l

From (3.82) and (3.83), we obtain f(t')

-

fit) --fu(t)u(t,)

--

fu(t)u(t).

(3.84)

Using the last expression, we derive

Af "~" Cgfu(t)fl Au. ~fl fl=u(t)

(3.85)

Now, we consider the case w h e n the input has been undergoing a monotonic decrease for t'~ t. If this decrease continues for t' > t, then the formula (3.75) is valid. From this formula as before we find

'

Af ~"

(~fMnfl I Au. ~fl fl=u(t)

(3.86)

In the last formula, Mn m a y change with time. It is clear that at any instant of time Mn in (3.81) is the smallest Mk in the alternating series of Mk and

mk. If the monotonic decrease of input for t IK t is followed by a subsequent monotonic input increase for t I > t, then the input value u(t) can be considered as mn-1 and the formula (3.76) can be used for t' > t. From this formula, as before we derive

Af "~ -

~f-u(t)fl I AU. ~fl fl=u(t)

(3.87)

Thus, for the previously increasing input, the output can be linearly extrapolated by using the formulas (3.81) and (3.85) for Au > 0 and Au < 0,

CHAPTER 3 Scalar Preisach Models of Hysteresis

344

respectively. For the previously decreasing input, the output can be linearly extrapolated by using the formulas (3.86) and (3.87) for Au < 0 and Au > 0, respectively. It is important to note that the above formulas are given in terms of slopes of experimentally measured first-order transition curves. The formulas (3.81), (3.85), (3.86), and (3.87) give local linearizations of output increments at time t. In other words, these linearizations are fairly accurate only for small time difference t - t'. However, by employing the formulas (3.75) and (3.76), it is possible to find global linearizations of output increments which are valid for finite (not necessary small) time intervals. The simplest way to do this is to use a special set of continuous inputs u(t). Consider an arbitrary time interval [to, tl]. We define the set K[to,t'] of inputs that are continuous on this interval and have the properties: (a) any two inputs from the set assume their maximum and minimum values at the same time instances, (b) maxima and minima that constitute alternating series of dominant extrema are also assumed at the same times. For two inputs u(t) and ~(t), the last condition can be expressed as

u(t-~) = Mk,

~(t~-) =/VIk,

(3.88)

u(t~) -- mk,

~(t~) = Tnk,

(3.89)

where the same notation as in Section 3.1.2 is used above. It is obvious that K[to,t'] is a convex set of functions. Indeed, if the above two properties are satisfied for inputs u(t) and ~(t), they are also satisfied for the input 2u(t) + (1 - 2)~(t) where 0~2K1. It is also clear that if u(t) belongs to K[to,t,], then 2u(t) also belongs to K[to,t'] for any 2~0 and it does not belong to K[to,t,] for 2 < 0. This means that the set K[to,t,] is a cone in the Banach space of continuous functions. It is clear that the set of inputs K[to,t'] depends on the alternating sequence of t k+ and tk- By specifying different times t k+ and t~-, we end up with different sets of inputs. For this reason, it would be more precise to adopt the n o t a t i o n K[to,t-~,t-~ ..... t+tn,t,]. But for the sake of conciseness, we preserve the previous notation. From the above definition of K[to,t'], it follows that any two inputs from K[to,t'] are monotonically decreased (or increased) on the same time intervals. This means that at any instant of time the same formula (3.75) (or (3.76)) can be applied to the computations of outputs. Thus, assuming that at time t both inputs, u(t) and ~(t), are monotonically decreased, we can use (3.75) for u(t) and the similar expression for ~(t): n-1

(3.90) k=l

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

345

By subtracting (3.75) from (3.90) and retaining only linear terms of Taylor expansions, we derive

n-1 ( ~fafl

bMk q- (~fafl

- - k=l ~(~]M

k'mk-1(~Mkq-~lMk,m k 1~mk-1)

8f(t)=~\ k-1 8~ Mk,mk

8f~

(~mkt

- ~ Mk,mk

(~Mn + 8f~

-'~-~ Mnu(t)

bu(t),

(3.91)

- ~ Mnu(t)

where the following notations are used:

6f(t) =d?(t) - fit),

6u(t) - ~(t) - u(t),

6 M k - Mk - M k ,

6mk -- ~nk -- mk.

(3.92) (3.93)

If both inputs are monotonically increased at time t, then the formula (3.76) is applied and, as before, we derive n-1

( Sf~fl

~Mk + 8f~

af(t)= k=l ~_.,--~ Mkmk

~mk

--~ Mkmk

8f~fl ~fafl 80~ IMkmk_1bMk ----~-IMkmk_l (~mk-l)

(Sfa

~ --mn-1

Ofa~

Of~fl

~(~mn-1

~0~ --mn-1,--u(t)Y 8u(t).

if- - ~ -mn-1,-u(t)

(3.94)

The expressions (3.91) and (3.94) represent a global linearization of output increment on the interval [to, t']. These expressions may be useful in the analysis of periodic regimes in systems with hysteresis by perturbation techniques. We conclude this section by discussing another example that demonstrates the usefulness of the above explicit formulas for the output f(t). Consider an input u(t) that has an infinite number of oscillations as time progresses to infinity. We assume that the amplitude of these oscillations (around some value u(cx~)) monotonically diminishes to zero. In other terms, this means that l i m M k = lim m k = u ( r k--,~ k~c~

(3.95)

346

CHAPTER 3 Scalar Preisach Models of Hysteresis

The described input variations are typical for anhysteretic magnetization processes. We are going to prove that there exists the limit for the output flt) as time goes to infinity. By using (3.75), we find oo

lim fit) = -F(ao, rio) + 2 ~ [ F ( M k , mk-1) -- F(Mk, mk)].

(3.96)

t--~ o o

k=l

Thus, it remains to be proven that the infinite series in (3.96) converges. To this end, we introduce the following notations: F(Mk, mk-1) = a 2 k ,

F(Mk, mk) -- a2k+l

(3.97)

and rewrite (3.96) as follows: oo

lim fit) = -F(ao, rio) + 2 y~(--1)kak .

(3.98)

t---~ o o

k=2

Since usually p(a, fl)>~0, from (3.26) and (3.97) we find that ak > 0. Moreover, since mk-1 < mk and Mk > Mk+l, (3.99) we conclude that a2k -~ F(Mk, ink-l) > F(Mk, ink) -- a2k+l,

(3.100)

a2k+l -- F(Mk, mk) > F(Mk+I, ink) = a2k+2.

(3.101)

In addition, from (3.95) and (3.26) we have lim a2k -- lim a2k+l -- F(~, ~) = 0.

k--+oo

k--,oo

Thus, the expression (3.98) is the alternating series with monotonically decreasing to zero coefficients. According to the well-known theorem, this series converges. This proves that there exists the limit off(t) for t ~ ~ . 3.1.5 THE PREISACH MODEL AND HYSTERETIC ENERGY LOSSES A hysteresis phenomenon is associated with some energy dissipation, which is often referred to as hysteretic energy losses. The problem of determining hysteretic energy losses is a classical one. It has been attracting considerable attention because the hysteresis energy loss is an important component of 'core losses' occurring in almost all electromagnetic power devices as well as in many high frequency microwave devices. For this reason, the

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

347

~U

FIGURE 3.43

means for accurate predictions of hysteresis losses and their reduction are important for optimal design of various equipment. The solution to the above problem has long been known for the particular case of periodic (cyclic) input variations. In magnetics, this solution is most often associated with the name of C.P. Steinmetz. This solution shows that a hysteretic energy loss per cycle is equal to an area enclosed by a loop resulting from periodic input variations (see Fig. 3.43). However, energy dissipation occurs for arbitrary (not necessary periodic) variations of input. The problem of computing hysteretic energy losses for arbitrary input variations has remained unsolved. A solution to this problem would be of both theoretical and practical importance. From the theoretical point of view, the solution to the above problem will allow for the calculation of internal entropy production that is a key point in the development of irreversible thermodynamics of hysteretic media. In magnetics, it may also allow one to separate a dissipated energy from an energy stored in a magnetic field. This eventually may lead to expressions for electromagnetic forces in hysteretic media. From the practical viewpoint, the solution to the above problem may bring new experimental techniques for the measurement of hysteretic energy losses occurring for arbitrary input variations. It should not be surprising that the expression for hysteretic energy losses has been found only for the case of periodic input variations. The reason behind this fact is that the hysteretic energy losses occurring for periodic input variations can be easily evaluated by using only the energy conservation principle; no knowledge of actual mechanisms of hysteresis

348

CHAPTER 3 ScalarPreisach Models of Hysteresis

or its model is required. The situation is much more complicated when arbitrary input variations are considered. Here, the energy conservation principle alone is not sufficient, and an adequate model of hysteresis should be employed in order to arrive at the solution to the problem. It turns out that the Preisach hysteresis model is very well suited for this purpose. In this section, the Preisach model will be used for the derivation of general expressions for hysteretic energy losses. These expressions will be given in terms of the weight function, #(c~, fl), as well as in terms of experimentally measured first-order transition curves. Furthermore, a formula that relates the hysteretic energy losses occurring for arbitrary input variations to the losses occurring for certain periodic input variations will be found. This formula may result in simple techniques for the measurement of hysteretic losses occurring for arbitrary input variations. The application of the mentioned results to the irreversible thermodynamics of hysteretic media will be discussed as well. We begin by defining the input, u(t), and the output, fit), as work variables. This means that the infinitesimal energy supplied to the transducer (media) in the form of work is given by 6W = u dr.

(3.102)

In magnetics, u is the magnetic field H, f is the magnetization M, and the formula (3.102) becomes the classical expression for the work done in magnetizing a unit volume of magnetic media: 6W = H dM.

(3.103)

Similarly, in mechanics, u is the force F, f is the specific length L, and from (3.102) we find the standard formula 6W = F dL.

(3.104)

Now, we proceed to the derivation of expressions for hysteretic energy losses. We first consider the case when a hysteresis nonlinearity is represented by a rectangular loop shown in Fig. 3.44. If a periodic variation of input is such that the whole loop is traced, then the hysteretic energy loss for one cycle, Qcycle, is equal to the area enclosed by the loop Qcycle = 2 ( a - fl).

(3.105)

It is clear that the horizontal links of the loop are fully reversible and, for this reason, no energy losses occur as these links are traced. Thus, it can be concluded that only' switching-up' a n d ' switching-down' result in energy losses. It can be assumed (on physical grounds) that there is symmetry

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

-}-11

349

--

~U

-I

T. 1

FIGURE 3.44

between the above two switchings. In other words, these switchings are identical as far as energy losses are concerned. Consequently, the same energy loss occurs for each of these switchings. As a result, we conclude that the energy loss per switching, q, is given by q = ~ - ft.

(3.106)

The product # ( a , / / ) ~ can be construed as a rectangular hysteresis loop with output values equal to 4-#(a, fl). For this reason, switchings of such loops will result in energy losses equal to #(a, fl)(a - fl). In the Preisach model, any input variation is associated with switchings of some rectangular loops #(~, fl)~fl. These switchings represent irreversible processes occurring during input variations. Consequently, it is natural to equate the hysteretic energy loss occurring for some input variation to the sum of energy losses resulting from the switching of rectangular loops during this input variation. Since in the Preisach model we are dealing with continuous ensembles of rectangular loops, the above summation should be replaced by integration. Thus, if f2 denotes the region of points on ~fl diagram for which rectangular loops were switched during some input variation, then the hysteretic energy loss, Q, for this input variation is given by Q - ~.to #(~' fl)(~ - fl) d~ dfl.

(3.107)

This is the fundamental formula for hysteretic energy losses, and all subsequent results will follow from this expression. It is clear from the above

CHAPTER 3

350

Scalar Preisach Models of Hysteresis

/

tS FIGURE 3.45

reasoning that the derivation of the formula (3.107) rests on the following two facts: (a) for rectangular hysteresis loops hysteretic losses can be evaluated for arbitrary input variations, (b) the Preisach model represents complicated hysteresis nonlinearities as superpositions of rectangular loops. The above two facts make the Preisach model a very convenient tool for the solution of the problem at hand. However, it has to be kept in mind that the formula (3.107) cannot be applied to any hysteresis nonlinearity. It has certain limits of applicability that are the same as for the Preisach model itself. A typical shape of the region f~ is shown in Fig. 3.45. It is clear from this figure that f2 can be always subdivided into a triangle and some trapezoids. The trapezoids, in turn, can be represented as differences of triangles. Thus, if the integral in (3.107) can be evaluated for any triangular region, then it will be easy to determine this integral for any possible shape of f2. For this reason, it makes sense to compute the values of the above integrals over various triangles. By using these values, hysteresis losses can be easily found for any input variations. In the case when f2 is a triangle, the integral (3.107) can be evaluated in terms of first-order transition curves. The derivation proceeds as follows. Consider the function (a - fl)F(a, fl). By differentiating this function, we find (92 02F(a, 3a3]/[(a - ]/)F(a, fl)] = 0F(a, fl) _ 3F(a, ]/) + ( a - ]/)]/-------~)

(3.108)

By using (3.28) and (3.108), we derive:

~F(a, ~) #(a, ]/)(~- ]1) --

~fl

~F(a,fl) -

~

~2 8e~fl[(a- ]/)F(a,fl)].

(3.109)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

351

/

i

/~/ T(u§ ] FIGURE 3.46

Let T(u+, u_) be a triangle (see Fig. 3.46) swept during the monotonic increase of input from u_ to u+. According to (3.107), the above input variation results in the hysteretic loss Q ( u _ , u+) that is given by ?f Q (u _ , u + ) - ] ] la ( oe, fl ) ( oe - fl ) d oed fl. .]J T (u+,u_)

(3.110)

Substituting the expression (3.109) into (3.110), we find: c')F ( oe, fl ) doe d fl Q ( u _ , u+)

-

ff:r(u+,u_) -

lfT

(u+,u_ )

0F(oe, fl) doe dfl cgoe

02 [(oe - fl)F(oe, fl)] doe dfl. - ffT (u+,u_) cgoeOfl

(3.111)

The first integral in (3.111) can be evaluated as follows:

aF(~, fl) d~ dfl

dfl d~

-

M

_

-

f(e, ~) d~ -

1 u+

- since F(oe, a) = O.

_

F(oe, u_) de

/uu+

fuu+F(oe, u_) doe,

(3.112)

CHAPTER 3 Scalar Preisach Models of Hysteresis

352

Similarly, for the second integral in (3.111) we obtain U+

OF(a, fl) da dfl 0~

/T(u+,u_ )

F(u+, fl) dfl.

(3.113)

Finally, for the third integral in (3.111) we derive

//T(u+,u_) 3~fl ~2 [(a - fl)F(a, fl)] d~dfl u+

(~2

-- ~u_ ( f u+ O~-Ofl[(~x- fl)F(~' fl)] d0r -

luU+

[(u+ - fl)F(u+, fl)] dfl - - ( u + - u_)F(u+, u_).

(3.114)

Substituting (3.112), (3.113), and (3.114) into (3.111), we find

Q(u_, u+) = (u+ - u_)F(u+, u_) -

luu+

f(u+,

fl) dfl -

/uu+f(~,

u_) d~.

(3.115)

The last formula has two main advantages over (3.110). First, its application requires the evaluation of one-dimensional integrals. Second, this formula expresses the losses directly in terms of experimentally measured firstorder transition curves. By using the expression (3.115), the hysteretic energy losses can be evaluated for arbitrary input variations. Consider some input variation for which the region f2 has the shape shown in Fig. 3.45. Then, the corresponding energy loss is given by (3.107) that can be written as follows: k

Q - Q(Mn, mn) q- ~,[Q(Mn-i, mn-i) 1 Q(Mn-i, mn-i-1)],

(3.116)

i=1

where it is assumed that

mn_k_ 1 -- u(t)

(3.117)

and each term in (3.116) can be evaluated by employing the formula (3.115). Next, we discuss some interesting qualitative properties of energy losses occurring in hysteresis transducers described by the Preisach model. Consider a cyclic variation of input between two consecutive extremum values u_ and u+. During the monotonic increase of input from u_ to u+, the final horizontal link of L(t) sweeps the triangle T(u+, u_) (see Fig. 3.46). Consequently, the losses occurring during this monotonic increase

3.1 THE CLASSICALPREISACH MODEL OF HYSTERESIS

353

I

,

lu2; /

FIGURE 3.47

I

u2

FIGURE 3.48

are given by (3.110). On the other hand, during the monotonic decrease of input from u+ to u_, the final vertical link of L(t) sweeps the same triangle, and the corresponding losses Q(u+, u_) will be given by the same integral in (3.110). Thus, Q(u_, u+) -- Q(u+, u_) (3.118) and we obtain the following result.

For any loop, the hysteretic losses occurring along ascending and descending branches are the same.

CHAPTER 3 Scalar Preisach Models of Hysteresis

354

The above result can be used to find the formula which relates a hysteretic loss occurring for arbitrary input variations to certain cyclic hysteretic losses. Suppose that the input u(t) is monotonically increased from some minimum value u_ and it reaches successively the values Ul and u2 with u2 > Ul (see Fig. 3.47). We are concerned with the hysteretic loss Q(ul, u2) during the monotonic input increase between Ul and u2. For this input increase we have (see Fig. 3.48)

f2 = T(u2, u_) - T(Ul, u_).

(3.119)

Q(Ul, u2) -- Q(u_, u2) - Q(u_, Ul).

(3.120)

Consequently, Using the above result (see formula (3.118)), losses Q(u_, U2) and Q(u_, Ul) can be expressed in terms of cyclic losses

Q(u_, Ul)

1 m

-

~Q(u_, Ul),

Q(u_,

z

1--

U2) "-

~Q(u_, A

U2),

(3.121)

where Q(u_, ul) is the hysteretic loss per cycle when the input is periodically varied between u_ and Ul; the notation Q(u_, u2) has a similar meaning. By substituting (3.121) into (3.120), we obtain

Q(Ul, u2)

-- x [ Q ( u _ , z ~

u2) -

Q(u_, Ul)].

(3.122)

The last formula expresses the loss occurring during the monotonic increase of input. By literally repeating the same line of reasoning a similar formula can be derived for the case of monotonic input decrease between Ul and u2 (Ul > u2)" m

Q(Ul, U2)

= X[ A"

Q(u+,

m

U2) --

Q(u+,

Ul)],

(3.123)

where u+ is the last input maximum (see Fig. 3.48). The formulas (3.122) and (3.123) may be useful from the practical point of view, because it is much easier to measure cyclic losses than those occurring for nonperiodic input variations. It is instructive to show that the derived expressions for hysteretic energy losses are consistent with the classical result:

A hysteretic energy loss occurring for a cyclic input variation is equal to the area enclosed by the loop resulting from this cyclic input variation.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

355

Consider a cyclic input variation between u_ and u+. According to (3.118) and (3.115), we find that the hysteretic loss per cycle for the above input variation is given by

Q(u_, u+) - 2Q(u_, u+)

[

= 2 (u+ - u_)f(u+, u_) -

L

f(u+, fl) dfl

f(e, u_) de .

(3.124)

On the other hand, the area enclosed by the corresponding loop is given by /.8

W = $

u df.

(3.125)

d u _It+U_

Since

~u_u+u udf + ~u_u+u f dU- ~u_u+u_ d(uf)

- 0,

(3.126)

f du.

(3.127)

we find

w=-

fu

_U+u_

f du--

fu

_U+

f d u - fu

+lt_

To evaluate the last two integrals, we shall use the formulas (3.71) and (3.74). In the case of monotonic input increase, the formula (3.74) is appropriate. This formula can be written in the form

fit) = C + 2F(u(t), u_), where u_ is used instead of

mn-1

(3.128)

and the constant C is given by

n-1 C - -F(~o, rio) + 2 ~ [ F ( M k ,

ink-l) -- F(Mk, mk)].

(3.129)

k=l

From (3.128), we find

/u

f du = C(u+ -

-U+

u_) + 2

L

F(u, u_) du.

(3.130)

_

In the case of monotonic input decrease from u+ to u_, the formula (3.71) is appropriate. This formula can be rearranged as follows

fit) - C + 2[F(u+, u_) - F(u+, u(t))],

(3.131)

CHAPTER 3 Scalar Preisach Models of Hysteresis

356

where the notation u+ is used instead of Mn, and C is the same historydependent constant as in (3.128). From (3.131), we obtain: u

f du - C(u_ - /,/+) + 2(u_ - u+)F(u+, u_)

+U_

- 2

F(u+, u) du.

f +u_

(3.132)

Using (3.127), (3.130), and (3.132), we derive W = 2[(u+ - u_)F(u+ , u _ ) -

fuu, F(u+, u) d u - /uu, F(u, u_) du

. (3.133)

It is apparent that the expressions (3.124) and (3.133) are identical. This proves that the expressions for hysteretic energy losses derived above are consistent with the classical result (3.125). We next discuss the applications of the above results to the irreversible thermodynamics of hysteretic media. It is clear that any hysteresis phenomenon is accompanied with energy dissipation. This means that hysteretic processes are irreversible, and consequently they fall in the domain of irreversible thermodynamics. Irreversible thermodynamics is the farreaching extension of classical thermodynamics that describes reversible processes. For this reason, it is appropriate to begin with the brief review of the formal structure of classical thermodynamics. Classical thermodynamics is based upon three main principles. The first principle of classical thermodynamics is the law of energy conservation. According to this principle, there exists a function of state, called the internal energy, U, of the closed system. This state function is such that its infinitesimally small change d U may occur as a result of energy exchange with the surroundings in the form of heat cl Q, as well as a result of energy cl W added to the system (or spent by the system) in the form of work. Mathematically, the first principle is expressed by dU = fl Q 4- fl W,

(3.134)

where a stroke is put across the symbol d in (3.134) to emphasize that the quantities d Q and cl W are path-dependent infinitesimals, which are sometimes called imperfect differentials. In other words, the above quantities depend on a particular path traced by the system during its transition from one equilibrium state to another. The second principle of classical thermodynamics postulates the existence of another state function, called the entropy S. This principle also

3.1 THE CLASSICALPREISACH MODEL OF HYSTERESIS

357

relates cl Q to the differential of entropy dS for reversible processes by the formula dQ dS- T ' (3.135) where T is the absolute temperature. For irreversible processes, the above equality is replaced by the inequality dQ dS > ~ . (3.136) T The formulas (3.134) and (3.135) are often combined into one formula: 1

dS = ~ ( d U - d W) 1

(3.137)

that constitutes the mathematical foundation of classical thermodynamics. The factor 1/T in (3.137) and (3.135) can be mathematically interpreted as an integrating factor for the imperfect differential cl Q. The first and second principles of classical thermodynamics are complemented by the Nernst-Plank postulate which is sometimes called the third principle of thermodynamics. According to this principle, the entropy of any system vanishes at zero temperature. This principle provides some useful information concerning the asymptotic behavior of entropy. However, the bulk of phenomenological thermodynamics does not require this principle. For this reason, the above postulate cannot be compared in importance with the first and second principles of classical thermodynamics. Classical thermodynamics has by and large been extended in two main directions. The first extension is based on the introduction of new variables describing the composition of the system. This approach has been very successful in applications of thermodynamics to chemical reactions. It has led to the development of chemical thermodynamics. The second extension is based on the generalization of the second principle of classical thermodynamics; it has led to the development of thermodynamic theory of irreversible processes that is often called irreversible thermodynamics. This theory has been mainly developed by I. Prigogine and his collaborators. As mentioned above, the basic difference between classical and irreversible thermodynamics lies in the way in which the second principle is stated. In irreversible thermodynamics, the second principle is formulated as follows. 1. The change in entropy dS can be split into two parts: dS = ~e S if- ~i S,

(3.138)

358

CHAPTER 3

Scalar Preisach Models of Hysteresis

where de S is due to the flow of entropy into the system from its surroundings, while di S is the generation of entropy by irreversible processes within the system. The term di S is often called internal entropy production. 2. The internal entropy production di S is never negative. It is zero if only the system undergoes a reversible process and positive if it undergoes an irreversible process. Thus, ~i S/~0.

(3.139)

3. For closed systems, the term de S is related to the energy de Q received in the form of heat by the formula that is similar to (3.135):

de Q

d e S -- ~ ,

T

(3.140)

where T is the absolute temperature that is assumed to be definable (at least locally) for nonequilibrium situations. It is also assumed in irreversible thermodynamics that the formula (3.137) holds with dS meaning the total change in entropy. From (3.138) and (3.140), we find T dS = d e Q + T ~i S.

(3.141)

The last expression allows one to relate the entropy production to the dissipated energy within the system. Indeed, from (3.137) and (3.141) we find dU = d W 4- ~e Q 4- T di S. (3.142) The first two terms in (3.142) have the same meaning as in (3.134) while the last term can be interpreted as the energy supplied to the system as a result of dissipating (irreversible) processes. This interpretation is especially clear for adiabatically isolated (de Q = o) systems that undergo cyclic changes ( / d W = 0). The increase in internal energy during one cycle ( / d U ) of such systems is only due to dissipating processes within the system ( / T di S). From (3.142) and the foregoing discussion, we find that the entropy production can be related to the dissipated energy di Q by the formula: ~i Q d i S -- ~ . (3.143) T It is apparent that the novel part of the above formulation of the second principle is the introduction of internal entropy production di S. However,

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

359

this new quantity is useful only if it can be evaluated (in a mathematical form) for different irreversible processes. This is the central problem of irreversible thermodynamics and it has been emphasized in many books on the subject. For instance, in the book [26; p. 90] by I. Prigogine we find: 'The main feature of the thermodynamics of irreversible processes consists of the evaluation of the entropy productions...' Similarly, in [27; p. 21] we read: 'In thermodynamics of irreversible processes, however, one of the important objectives is to relate the quantity di S, the entropy production, to the various irreversible phenomena which may occur inside the system'. Finally, in [28; p. 69] it is noted: 'A central problem of irreversible thermodynamics, in fact, is the development of formulas for the entropy production ~i S in specific cases'. The above problem has been resolved for various irreversible processes that are caused by macroscopic non-uniformities of the system. Examples of such processes include heat flows due to temperature gradients, diffusion flows due to density gradients, electric current flows due to potential gradients, and so forth. For the above processes, the so-called entropy balance Eqn. (3.138) is written in local (differential) form in which the entropy production, eli S, is replaced by the entropy source. The entropy source is then found as a sum of several terms, each being a product of a flux characterizing a particular irreversible process and a quantity, called thermodynamic force, which is related to a particular macroscopic nonuniformity of the system (temperature gradient, for instance). In this way, many useful results have been established. The celebrated Onsanger reciprocity principle for phenomenological coefficients is the most known example of these results. However, the above developments cannot cover hysteresis phenomena. This is because hysteresis is not caused by macroscopic nonuniformities and therefore cannot be linked to gradients of some physical quantities. As a result, different approaches should be developed for the calculation of entropy production in the case of irreversible hysteretic processes. It is logical to expect that mathematical models of hysteresis may help to solve the above problem. It is shown below that by using the Preisach model and the expressions for hysteretic energy losses derived on the basis of this model, the entropy production for hysteresis processes can indeed be found. Consider the input u(t), which is monotonically increased from its previous local minimum value u_. If the current input value, u(t), is lower than M n , then by using (3.115), for the dissipated energy we find

Q ( u _ , u) = (u - u _ ) F ( u , u _ ) -

fun F(u, fl) dfl - fuu F(a, u) da.

(3.144)

360

CHAPTER 3 Scalar Preisach Models of Hysteresis

Employing (3.144), it is easy to conclude, that the energy dissipation, ~i Q, that occurs as a result of the input increase from u to u 4- du, is given by ~i Q 1

3Q(u_, u) du. Ou

(3.145)

From (3.145) and (3.143), we derive: 1 OQ(u_, u) di S - ~ cgu du.

(3.146)

By differentiating (3.144) with respect to u and substituting the result into (3.146), after simple transformations we obtain du fli g - ---Z- fu _n ~0 [F(u , u_) - F(u, fl)] dfl.

(3.147)

If the input is monotonically decreased from its previous local maximum value, then the internal entropy production occurring as a result of the input decrease from u to u - du is given by {1i S -

du {u+ ~ ~UU[F(u+, u) - F(a, u)] da.

(3.148)

The derivation of the last formula is similar to that for (3.147). It has been tacitly assumed in the previous derivation that the monotonic input variations are such that no previous history is wiped out. However, by using (3.116), it is easy to extend (3.147) and (3.148) to the most general case. The details of this extension are left to the reader. Very interesting discussion of thermodynamic aspects of hysteresis can be found in [29,30]. The presentation of the material in this section is largely based on references [31-36].

3.2 G E N E R A L I Z E D S C A L A R P R E I S A C H M O D E L S OF HYSTERESIS 3.2.1 'MOVING' PREISACH MODEL OF HYSTERESIS The classical Preisach model of hysteresis has been discussed in detail in the Section 3.1. It has been repeatedly emphasized that this model has some intrinsic limitations. The most important of them are the following: 1. The classical Preisach model describes hysteresis nonlinearities that exhibit congruency of minor loops formed for the same reversal

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

361

values of input. However, many experiments show that actual hysteresis nonlinearities may substantially deviate from this property. 2. The classical Preisach model is rate-independent in nature and does not account for dynamic properties of hysteresis nonlinearities. However, for fast input variations these properties may be essential. 3. The classical Preisach model describes hysteresis nonlinearities with the wiping-out property. This property is tantamount to the immediate formation of a hysteresis loop after only one cycle of back-andforth variation of input between any two reversal values. However, experiments show that hysteresis-loop formation may be preceded by some 'stabilization process' that may require a large number of cycles to achieve a stable minor loop. This process is also called in the literature the'accommodation' or'reptation' process. 4. In the classical Preisach model, a scalar output exhibits hysteretic variations with respect to only one scalar input. However, in the case of magnetostrictive hysteresis, the strain is a hysteretic function of two variables: magnetic field and stress. Thus, the problem of developing the Preisach-type models with two inputs presents itself. These models may find applications beyond the area of magnetostrictive hysteresis, for instance in the modeling of piezoelectric hysteresis. To remove (or relax) the above-mentioned limitations, essential generalizations of the classical Preisach model are needed. These generalizations are discussed in this section. In this section, experimental testing of various scalar Preisach-type models of hysteresis is presented as well. The presentation of the material is largely based on our publications [31,32,37--46]. We begin with an interesting modification of the classical Preisach model. This modification will reveal that the Preisach model does describe, to a certain extent, reversible properties of hysteresis nonlinearities. This fact has been overlooked in the existing literature. Apart from the mentioned fact, this modification will also be instrumental in the further generalizations of the classical Preisach model that are discussed in subsequent sections. The classical Preisach model has been defined as

f(t)

A

= Fu(t) -

#(e,

fl)~u(t) de d[3,

(3.149)

where T is the limiting triangle specified by inequalities flo~fl~e~eo. This triangle is the support of the function #(e,/~) and does not change with input variations.

CHAPTER 3 Scalar Preisach Models of Hysteresis

362

(~o

FIGURE 3.49

We next subdivide the triangle T into three sets Fig. 3.49), that are defined as follows:

Su+(t) (ce,fl) ~ Ru(t) (ce,fl) ~ Su(t) (ce, fl) E

Su+(t),Ru(t) and Su(t) (see

if fl0~flKceKu(t),

(3.150)

u(t)~ce~ceo, if u(t)KflKceKceo.

(3.151)

iffl0~fl~u(t),

(3.152) By using the above subdivision, we can represent (3.149) in the form

fit) =//R

#(ce'fl)~u(t)dcedfl +/fs+ /a(ce,fl)~u(t)dcedfl u(t)

u(t)

+//s

#(ce,fl)~u(t)

dce dfl.

(3.153)

u(t)

Since

+ u(t)~ce for any (ce,fl) ~ Su(t), then ~u(t)

/fs

#(ce'fl)~u(t)

ace 4]3 = / f s

#(ce']3) ace aft.

(3.154)

u(t) +

u+(t)

Similarly,

= +1 and

u(t)Kfl for any (ce,fl) E Su(t) and

/fs

/a(ce,fl)~&~u(t)dceaft -

-/fs

u(t)

#(ce']3)dce dfl.

(3.155)

u(t)

By substituting (3.154) and (3.155) into (3.153), we obtain

f(t)=ff nu(t) U(~,fl)G,u(t)d~dfl+ffs+u(t) #(ce,fl)dcedfl - f/s

#(ce'fl) dcedfl. u(t)

(3.156)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

363

We next find a simple expression for the last two terms in (3.156). Consider a monotonic increase of input from some value below fl0 (state of negative saturation) to some value u(t). Then the output will change along the ascending branchf,+L,, and according to (3.156) we find u~,t) .pp

:-

PP

11 #(~, fl)d~x dfl + 1 1 #(oc,fl)da dfl JJR u(t) dd Su+(t) - ff

diS

/~(~,fl)da dfi.

(3.157)

Similarly, if we consider a monotonic decrease of input from some value above a0 (state of positive saturation) to the same value u(t), then by using (3.156) we obtain

-- H #(a, fl)d0~ dfl + H /~(a, fl)d~ dfl J JR u (t) dds+u(t) - ff #(~, fl)d~ dfl. JJs ~lt) By summing up (3.157) and (3.158), we derive

f~Su+(t)

fl)

aft - ffSu ,) #(~

fl) da aft = ~'1-(fu~t) + f~t)).

(3.158)

(3.159)

By substituting (3.159) into (3.156), we finally obtain

f(t) =

la(o~, fl)~flu(t) dadfl

1

+ -~(fu+((t)+ f~t))"

(3.160)

u(t)

The last expression is formally equivalent to the classical one (3.149). However, in this expression the integration is performed not over the fixed limiting triangle T but over the rectangle Ru(t) which changes along with input variations. For this reason, the expression (3.160) is termed here as a ,moving , Preisach model. It is also clear from (3.160) that 2(fu(t) 1 + -}-f~t)) represents a fully reversible component of hysteresis nonlinearity described by the classical Preisach model. In this respect, the first term in the right-hand side of (3.160) can be construed as an irreversible component of the classical Preisach model. To make the last point transparent, consider some output increment Afu corresponding some input value u (see Fig. 3.50). This increment depends on a particular history of input variations and can be regarded as a measure of irreversibility of hysteresis nonlinearity for this particular history. It is clear from the diagram shown in Fig. 3.51 and the expression (3.160) that the above increment is given by

Afu - 2 f f #(~, fl) d~ dfl. JJ f~

(3.161)

364

CHAPTER 3 ScalarPreisach Models of Hysteresis f

L

i ~U

FIGURE 3.50

FIGURE 3.51

The region of integration f~ belongs to Ru(t). This is true for any past history. This fact clearly suggests that the first term in the right-hand side of (3.160) describes an irreversible processes. Thus the expression (3.160) gives the decomposition of hysteresis nonlinearity described by the classical Preisach model into irreversible and reversible components. It is stressed in the above discussion that the expression (3.160) is mathematically equivalent to the classical definition (3.149). However, it is apparent that this equivalence holds only for input and output variations confined to the region enclosed by a major hysteresis loop. Outside this region, the classical Preisach model prescribes flat saturation values for output, while the moving model (3.160) prescribes the actual experimentally observed values.fu+(t) and f~t) for the states of negative and positive saturation, respectively. This is the case because for these states ascending

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

(~,P) s-

365

/,

,

(a,13o)

FIGURE 3.52

and descending branches merge together and consequently fult) "-- f~t)" As far as the first term in (3.160) is concerned, it is clear from (3.151) that this term vanishes. Thus, the moving model (3.160) can be regarded as a generalization of the classical model as far as the description of hysteresis nonlinearities beyond the limits of major loops is concerned. It is instructive to consider directly the identification problem for the model (3.160) without invoking our previous discussion of this problem for the classical model (3.149). The essence of this problem is in determining the function ~ by fitting the model (3.160) to some experimental data. Suppose that, starting from the state of negative saturation, the input u(t) is monotonically increased until it reaches some value e. Then the input is monotonically decreased until it reaches some value ft. The corresponding diagram is shown in Fig. 3.52. By using (3.160) and this diagram, we find that the output valuef~fl corresponding to the final value of output is given by

#(~', fl') da' dfl' - ffR(~o,~o,~,~) #(cr fi') do~'dfl' + f; +fff.

(3.162)

2

Here R(a0, rio, a, fi) is the rectangle whose opposite vertices are the points (c~0, fl0) and (c~,fi); the same meaning as the notation R(c~, fl0, fl, fl). Now suppose that, starting from the state of positive saturation, the input u(t) is monotonically decreased until it reaches the value ft. The corresponding c~-fl diagram is shown in Fig. 3.53. From this diagram and

CHAPTER 3 ScalarPreisach Models of Hysteresis

366

FIGURE 3.53

(3.160) we find the following expression for the resulting value of output:

f~- =

f•R

lt(e', fl') de' dfl' + f ; + f~

(=o,~o,~,~)

---T~"

(3.163)

Next we introduce the function T(e, fl) =f~- -f~/~,

(3.164)

which is equal to output increments between the limiting descending branch and first-order transition curves. From (3.162), (3.163) and (3.164) we find

T(e, fl) = 2 f f

,IJR (~o,&,~,/~)

#(e', fl') de' dfl'

- 2 f~~ (f~i #(0r fl') d]/t de.

(3.165)

By differentiating the last expression twice, we obtain #(e, fl) =

1 c32T(e, fl) ~ . 2 cgec3fl

(3.166)

By recalling the definition (3.164) of T(e, fl), from (3.166) we derive

#(~' fl) = 2 8e 8fl'

(3.167)

which coincides with the expression (3.29) obtained for the classical Preisach model.

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

367

0{ s-(t)

I I I

I

Rk

FIGURE 3.54

Next, we show that the integration in (3.160) can be avoided and that the explicit expression forf(t) in terms of experimentally measured function T(~, fl) can be derived. To start the derivation, consider the cr diagram shown in Fig. 3.54. Then, according to (3.160), we have

fit) -

IL

+(t) la(a, fl) da dfl -

=2iL

SL

-(t)

u(t)

l~(cr

#({z, fl)d{zdfl-i/R +(t)

#(~, fl) da dfl +

u(t)

+

q2 f~t) +f~t) 2 "

(3.168) To evaluate the integral over Ru(t) in (3.168), we consider a monotonic increase of input from the state of negative saturation to some value u(t) Then from (3.160) we find

fu+(t) = - ifI~ u(t)

P(~' ~) d(x dfl + fu(+) + f~t) 2 "

(3.169)

By substituting (3.169) into (3.168), we conclude

fit) = 2 i L

+(t)

/z(oc, fl) dcz df1-4- fu+(t).

(3.170)

It is clear that the moving model (3.160) is equivalent to the classical Preisach model as far as description of purely hysteretic behavior is concerned. For this reason, the wiping-out property and congruency property of minor loops are valid for the moving model. According to the wipingout propertj6 the staircase interface L(t) in Fig. 3.54 has vertices whose

CHAPTER 3 Scalar Preisach Models of Hysteresis

368

a and fl coordinates are equal to Mk and mk, respectively, and, as before, numbers Mk and mk form the alternating series of past dominant input extrema. Now we subdivide S+(t) into rectangles Rk and represent the integral in (3.170) as n(t)

/fs +(t)

#(a, fl)de dfl = ~ / / R k=l

fl(~'/J) de dfl.

(3.171)

k

It is easy to see from the expression (3.165) for the function T(a, fl) that the integrals over Rk can be represented in the form

f/R #(a, fl) da dfl = T(Mk+I, ink) -- T(Mk, ink).

(3.172)

k

From (3.170), (3.171) and (3.172) we obtain n(t)

f(t) = 2 ~[T(Mk+I, mk) - T(Mk, mk)] + fu+(t).

(3.173)

k-1

This is the final result which expresses explicitly the outputf(t) in terms of experimentally measured function T. Up to this point, we have discussed the modeling of 'counterclockwise' hysteresis. For this type of hysteresis, the positive saturation state is achieved for a larger input value than the negative saturation state, and for any two consecutive descending and ascending branches the descending branch is above the corresponding ascending branch (see, for instance, Fig. 3.50). This type of hysteresis is typical for magnetic materials and it can be modeled by using the classical Preisach model with positive measure #(a, fl). It is apparent that the classical Preisach model with negative measure can be used for the modeling of 'clockwise' hysteresis when the positive saturation state is achieved for a smaller input value than the negative saturation state. This type of hysteresis occurs in superconductors due to their diamagnetic nature. There exists, however, the 'clockwise' hysteresis with the property that the positive saturation state is achieved for a larger input value than the negative saturation state. This is typical for hysteresis that occurs for certain front propagation problems such as nonlinear diffusion of electromagnetic fields in magnetically nonlinear conductors. The modeling of this type of hysteresis is not readily available within the framework of the classical Preisach model, however it can be easily accomplished by using the 'moving' Preisach model (3.160). Indeed,

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

369

the proper saturation states can be modeled by the appropriate choice of the reversible terms 1 (fu+(t)ff-f~t))" while the 'clockwise' nature of hysteresis is imposed by choosing a negative measure #(a, fl) in the irreversible component. In other words, the decomposition (3.160) of hysteresis nonlinearity on reversible and irreversible components makes it possible to control separately the (clockwise or counterclockwise) orientation of hysteresis and the location of saturation states. 3.2.2 PREISACH MODEL OF HYSTERESIS WITH INPUT-DEPENDENT MEASURE

The Preisach model with input-dependent measure is discussed in this section. It has the following advantages over the classical model. First, the congruency property of minor loops is relaxed for this model. This results in a broader area of applicability of this model as compared with the classical model. Second, the model with input-dependent measure allows one to fit experimentally measured first- and second-order reversal curves. Since higher-order reversal curves are' sandwiched' between firstand second-order reversal curves, it is natural to expect that this model will be more accurate than the classical one. The Preisach model with input-dependent measure can be mathematically defined as

fit) - ffR

u(t)

#(a' fl'

u(t))~flu(t) d~ dfl +

du+(t)if-f ~t)

(3.174)

It is clear that a new feature of this model in comparison with the 'moving' model (3.160) is the dependence of the distribution function # on the current value of input, u(t). For this reason, the model (3.174) is also termed as the 'nonlinear' Preisach model. Owing to the new feature mentioned above, the first term in the right-hand side of (3.174) can be construed as a partially reversible component of hysteresis nonlinearity. Indeed, for each pair (~, fl), the integrand #(~, fl, u(t))~u(t) is reversible for input variations between ~ and ft. The 'nonlinear' Preisach model admits a geometric interpretation, which is similar to that for the classical model. In other words, it could be easily demonstrated that at any instant of time t the rectangle Ru(t) is subdivided into two sets (see Fig. 3.55): S+ (t) and S- (t) consisting of points (~, fi) for which ~aflu(t) = 1 and ~ u ( t ) - - 1 , respectively. The interface L(t) between S+ (t) and S-(t) is a staircase line whose vertices have a and fl coordinates equal (respectively) to past dominant extrema Mk and mk. Using the above geometric interpretation, the model (3.174) can be represented

CHAPTER 3 Scalar Preisach Models of Hysteresis

370

,

/

u(t)

/ FIGURE 3.55

in the form

fit) = f~S+(t ) #(a, fl, u(t)) da dfl - / ~ s - ( t ) 11(0r fl' u(t)) da dfl +

f u+((t) -Jr-f ~ t)

. (3.175) 2 It can be shown that the following two properties are valid for the 'nonlinear' model. PROPERTYA (WIPING-OUTPROPERTY). Only the alternating series of subsequent global extrema Mk and mk are stored by the "nonlinear" Preisach model. The proof of this property is identical to that for the classical model.

PROPERTYB (PROPERTYOF EQUAL VERTICALCHORDS). All minor loops resulting from back-and-forth input variations between the same two consecutive extrema have equal vertical chords (output increments)for the same input values (Fig. 3.56). PROOF. Consider a minor loop formed as a result of back-and-forth input variations between u+ and u_ (Fig. 3.57). Letfu f andfu be output values on descending and ascending branches of this loop, respectively. These values correspond to the same value of input u from the interval u_ < u < u+. By using the expression (3.175),fu I andfu can be computed as f ~ = ffs~

~(~, fl, u) d~ dfl - f/s" l~(~,fl, u)dadfl

+ ~fu+ ,+ f u 2

(3.176)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

U_

U

U+

371

--U

F I G U R E 3.56

fu'

U_

U

U+

F I G U R E 3.57

fu' - / ~ s p(~, fl, u) &z dfl - f f s #(a, fl, u) d~ dfl ~_ + fu+ + f u , (3.177) 2 where the sets S~_, St__, S~_ and S'_~ are shown on the diagrams presented in Fig. 3.58. It is clear from these diagrams that S+ - S+ ' - R(u+, u , u _ ) ,

S ~_ - S"_ - R(u+, u, u_).

(3.178)

From (3.176), (3.177) and (3.178), we derive fu' - f u - 2 I I p(~, fl, u) da dfl. J JR (u+,u,u_)

(3.179)

According to (3.179), we find that for any u ~ (u+, u_) the corresponding vertical chord does not d e p e n d on a particular past history preceding the formation of the minor loop. This proves that all comparable minor loops, that is, the loops with the same reversal input values u+ and u_, have equal vertical chords. It is left to the reader as a useful exercise to prove that comparable minor loops described by the nonlinear model (3.174) are not necessarily congruent.

CHAPTER 3 Scalar Preisach Models of Hysteresis

372 (X

/

/

I

8"

I S'+

' I

R,u_

~

u

R(u_,u,u+)

I

u

u

u+

(a)

u+

(b) F I G U R E 3.58

Now we turn to the identification problem of determining the distribution function #(a,/}, u) by fitting the model (3.174) to some experimental data. It turns out that for the solution of this identification problem the sets of first- and second-order reversal curves are required. These curves can be measured experimentally as follows. We first decrease the input, u(t), to such a negative value that the outputs of all operators ~a~ are equal to - 1 (state of negative saturation). Then we monotonically increase the input until it reaches some value a. As we do this, we will follow the ascending branch of the major loop (see Fig. 3.59). As we already know, the first-order reversal curves are attached to this ascending branch and they are formed when the above monotonic increase of u(t) is followed by a subsequent monotonic decrease. The notation f + will be used for the output values on the first-order reversal curve attached to the ascending branch of the major loop at the pointf +. The second-order reversal curves are attached to the first-order reversal curves, and they are formed when the above monotonic decrease is followed by a monotonic increase. The notation f~u will be used for the output values on the second-order reversal curve attached to the first-order reversal curve f~u at the point f ~ (Fig. 3.59). Consider the function

P(~, ~,

u) - f~u - f ~ , u ,

(3.180)

which has the physical meaning of output increments between the firstand second-order reversal curves.

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

373

I

u

FIGURE 3.59

It is clear from the definition of this function that

P(o~, u, u) = P(u, fi, u) = P(u, u, u) = O.

(3.181)

This property will be employed later in our discussion. Now we will try to relate the function, P(a, fl, u), to the distribution function, #(a, fl, u). To this end, we will use the diagrams shown in Fig. 3.60. From these diagrams and (3.175) we conclude

f~u = ff~+ l~(~x', fi', u) da' dfl' - ff~ la(a', fl', u) d~x' dfl' + f : , , u - ff~ ~(~', fl', u)d~' a f t ' - f J i #(~'' fl'' u)d~' dfl +

fu+ +f; (3.182)

-

f++fu 2 S+ - S+ - R(~, fl, u),

S_ - S_ - R(~, fl, u).

(3.183)

By using (3.182) and (3.183), we derive

P(~, fl, u)

-

2 ffR

= 2

~,fl,u)

#(~" fl'' u) d~' aft'

fu (f, u#(e', fl', u)dfl' ) d~'.

(3.184)

374

i

CHAPTER 3 Scalar Preisach Models of Hysteresis

i

d

d

/

I

H~ u L

U

(a)

~,

(b)

FIGURE 3.60

By differentiating the last expression twice, we obtain 1r p(a,//, u) -

2

]~, u) r

.

(3.185)

By recalling the definition (3.180) of P(a, fl, u), from (3.185) we find 1 02fa~u # ( ~, ~6, u) = 2 &z cgfl

(3.186)

It is clear from the above derivation, that the expressions (3.185) and (3.186) are valid if ]/< u < a. If uK~fl or u~>a, then we define p(a, fl, u) = 0,

(3.187)

which is consistent with (3.181). Thus, if the distribution function, p(a, fl, u), is determined from (3.184) or (3.186), then the nonlinear model (3.174) will match the increments between first- and second-order reversal curves. Next we shall show that the limiting ascending branch will be matched by this model as well. Indeed, it is easy to see that the output values, fu, on the limiting ascending branch predicted by the nonlinear model (3.174) are equal to

fu = -

IfR

#(a" fl'' u) da' dfl' + f+ + f u (~0,&,u) -----2---"

(3.188)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

375

I

1

foq3u

FIGURE 3.61

According to (3.184) and (3.180), we have

ffR

(~o,&,u) #(a'' fl'' u) d~dfl' - ~P(ao, rio, u) = -~(f~ou -f~o&u) 1 1 1

= i 0 u-fu+). By substituting (3.189) into (3.188), we find

(3.189)

fu -- fu+. Thus, if the function,/~(a, fl, u), is determined according to (3.184) or (3.186), then the nonlinear model (3.174) fits: (a) the output increments between the first- and second-order reversal curves, and (b) the ascending branch of the major loop. Since the ascending branch of the major loop can be construed as a second-order reversal curve, we conclude that the nonlinear model (3.174) fits the sets of first- and second-order reversal curves. In the above discussion we have used second-order increasing reversal curves (see Fig. 3.61) in order to determine the distribution function, #(~, fl, u). However, by almost literally repeating the previous line of reasoning, a similar expression can be found for #(a, fl, u) by using secondorder decreasing reversal curves f ~ . One of these curves is shown in Fig. 3.62. By using this figure, we can introduce the function (3.190)

CHAPTER 3 Scalar Preisach Models of Hysteresis

376

I

I I

i

i

I

I I

,/

',

o,,

FIGURE 3.62

In the same way as before, we can show that

P ( -fl, ~ , fi ) = 2 f f # ( o{ , fl' , fi ) d o{ d fl' 3JR ~,~,~

(3.191)

and

1 (92P(~, ~, U) #(~' fl' u) =

2

O~Ofl

"

(3.192)

If --a,

&=-fl,

fi--u,

(3.193)

then due to the symmetry between increasing and decreasing second-order reversal curves (see Fig. 3.62) we find (3.194) By using (3.193) and (3.194) in (3.192), we derive 1 02P(a, fl, u) /~(-fl, -c~, - u ) =

2

0c~cgfl

.

(3.195)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

377

From (3.195) and (3.185), we conclude #(~, fl, u) = # ( - / / , - ~ , - u ) .

(3.196)

The last formula can be regarded as a generalization of the mirror symmetry (3.46) previously established for the classical Preisach model. [] We next proceed to the proof of the following important result. REPRESENTATION THEOREM. The wiping-out property and the property of equal

vertical chords for minor loops constitute the necessary and sufficient conditions for the representation of a hysteresis nonlinearity by the nonlinear Preisach model on the set of piecewise monotonic inputs. PROOF. Necessity: If a hysteresis nonlinearity is representable by the nonlinear Preisach model, then this nonlinearity should have the same properties as the model. This means that this nonlinearity should exhibit the wipingout property and the property of equal vertical chords for comparable minor loops. Sufficiency: Consider a hysteresis nonlinearity that has both the wipingout property and the property of equal vertical chords. For this nonlinearity we find the distribution function #(a, fl, u) by using formula (3.185) (or (3.38)). Then the nonlinear Preisach model (3.174) will match exactly the sets of all first- and second-order reversal curves. We intend to prove that this model will match all possible higher-order reversal curves as well. The proof is based on the induction argument. Let us assume that the above statement is true for all possible reversal curves up to the order k - 1. Then, for the induction inference to take place, we need to prove that the same statement holds for any reversal curve of order k. Let a be a point at which a reversal curve of order k is attached to a reversal curve of order k - 1 (see Fig. 3.63). According to the induction assumption, the output values for the actual hysteresis nonlinearity and for the nonlinear Preisach model coincide at each point of the reversal curve of order k - 1. Thus, it remains to be proven that the output increments between the reversal curves of orders k and k - 1 are the same for the actual hysteresis nonlinearity and for the nonlinear model (3.174). It is here that the wiping-out property and the property of equal vertical chords will be used. According to the wiping-out property, the kth reversal curve should meet the (k - 1)th reversal curve at the point b, which is the point of inception of the latter curve. As a result, a minor loop is formed. Consider a comparable minor loop (loop with the same reversal input values), which is attached to the limiting ascending branch (Fig. 3.63). This loop is formed by some first- and second-order transition curves. According to the property of equal vertical chords, this

378

CHAPTER 3 Scalar Preisach Models of Hysteresis

U

FIGURE 3.63 o~

u

FIGURE 3.64

loop has the same vertical chords as the loop formed by (k - 1)th and kth reversal curves. Consider an arbitrary value u of input such that fl < u < a. Using the diagram shown in Fig. 3.64, it is easy to derive the following expression for the output increment

f(k) _ F(k- 1) _ 2 ffR

#(a', fl', u ) d e ' dfl',

(3.197)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

379

wherefu(k) andfu(k-l) are the output values on the kth and (k - 1)th reversal curves, respectively, corresponding to the input value u and predicted by the model (3.174). From (3.184) we find that the right-hand side of (3.197) is equal to the output increment el between the first- and second-order transition curves. Consequently, f(u k) -- f(u k-l) -- el.

(3.198)

According to the property of equal vertical chords we have

el = cd.

(3.199)

From (3.198) and (3.199), we conclude f(u k) -- ju'c(k-1) _ c d .

(3.200)

Thus the nonlinear model predicts the correct output increments between the kth and (k - 1)th reversal curves. From this fact and the above induction assumption we conclude that the nonlinear model predicts the correct output values on the reversal curve of order k. This concludes the proof of the theorem. [] The proved theorem establishes the exact bounds of applicability of the nonlinear Preisach model. It is apparent that the property of equal vertical chords is more general than the congruency property. Indeed, if comparable minor loops are congruent, then they have equal vertical chords. However, if comparable minor loops have equal vertical chords, they are not necessarily congruent. This clearly shows that the nonlinear Preisach model (3.174) has a broader area of applicability than the classical Preisach model. Next we shall show that the nonlinear model (3.174) contains the classical Preisach model as a particular case. The exact statement of this fact is given by the following theorem. REDUCTIONTHEOREM. If all comparable minor loops of hysteresis nonlinearity are congruent, then the nonlinear Preisach model for this nonlinearity coincides with the classical Preisach model as far as purely hysteretic behavior of this nonlinearity is concerned. PROOF. Since all comparable minor loops are congruent, these loops have equal vertical chords. Assuming also that the wiping-out property holds, we can represent the mentioned hysteresis nonlinearity by the nonlinear Preisach model (3.174). We next show that, because of the congruency property, the distribution function # in (3.174) does not depend on u. It is

CHAPTER 3 Scalar Preisach Models of Hysteresis

380

f

II I I I

I I I

0~

0~

P

U

FIGURE 3.65 clear from Fig. 3.65 that the congruency property results in the congruency of second-order reversal curves fa'Bu, fa~u and fa"~u. This means that the derivative -N--u G~u does not depend on cr Consequently,

c~f~u - O.

(3.201)

&~~//Ou From (3.201) and (3.186) we find

~(~,/~, u) 0u

--- 0.

(3.202)

Thus/~ does not depend on u and is only a function of ~ and fl: /~(~, ]3, u) - v(c~, fl).

(3.203)

By substituting (3.203) into (3.174), we obtain

fit) = ffR

v(a, fl)~flu(t) dec dfl +

du~t) q- du~t)

(3.204)

u(t)

It remains to be proved that v(ce, fl) in (3.204) coincides with #(a, ]3) in (3.160). The proof is straightforward. By using (3.160) and (3.204) we find

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

381

that for any a, fl and u such that flKu~a we have

P(~, fl, u) = f~u -f~flu - 2

ff. (~,~,u),(~,, fl,) da' dfl',

(3.205)

F/"

P(:r fl, u) =f~u - faflu - 2

__JJR(a,fl, u) v(a', fl') da' dfl'.

(3.206)

From (3.205) and (3.206), as before, we derive #(~, fl) = v(~, fl) =

1 02P(a, fl, u) 2

#~afl

.

(3.207)

Thus, it is proved that under the congruency condition, the nonlinear Preisach model (3.174) coincides with the moving Preisach model (3.160). On the other hand, the moving model coincides with the classical Preisach model as far as the description of purely hysteretic behavior is concerned. From here we conclude that under the congruency condition the nonlinear model (3.174) coincides with the classical Preisach model, and this concludes the proof of the theorem. We next turn to the discussion of numerical implementation of the nonlinear model (3.174). We intend to show that double integration in (3.174) can be completely avoided and that explicit expressions for the outputflt) in terms of the experimentally measured function P(a, fl, u) can be derived. The starting point of our derivation is the expression (3.175) that can be modified as follows:

f(t) -

ffRu(t)

/~(~, fl, u(t)) da dfl - 2

-(t) Ia(a, fl, u(t)) doc dfl

+ fu+(t) -}-f ~t) 2

(3.208)

According to (3.189), we have

~fRu(t ) [.1(0~,~, u(t)) da dfl -

f ~t) -- f u+(t)

(3.209)

By substituting (3.209) into (3.208), we obtain

fit) -- f ~ t ) - 2 fL-(t) #(~, fl, u(t)) da dfl.

(3.210)

We next subdivide S-(t) into rectangles Rk (see Fig. 3.66) and represent the integral in (3.210) as

n(t) Ia(:r fl, u(t)) d~dfl - E ffR IJ(~, fl, u(t)) d~dfl.

fL -(t)

k=l

k

(3.211)

CHAPTER 3 Scalar Preisach Models of Hysteresis

382

s-(t)

o~

W___]_', ~-R~

/

FIGURE 3.66

It is easy to see that

Rk = R(Mk, mk, u(t)) -- R(Mk+I, mk, u(t)).

(3.212)

From (3.212) and (3.184), we conclude

fRk ~(~' fl' u(t)) de aft = ff

ddR (Mk,mk,u(t))

lz(or fl, u(t))dadfl - f f

#(~,

dd R(Mk+l,mk,u(t))

-- l [p(Mk, ink, u(t)) -- P(Mk+I, ink, u(t))]. 2

u(t)) dadfl

(3.213)

By substituting (3.213) into (3.211) and then (3.211) into (3.210), we obtain n(t)

fit) = f~t) + ~-'~.[P(Mk+I, mk, u(t)) -- P(Mk, mk, u(t))].

(3.214)

k=l

This is the final formula which expresses explicitly the output f(t) in terms of the experimentally measured function P. This formula has been used to develop a digital code which numerically implements the nonlinear Preisach model. This code computes output values by using input values, a set of second-order reversal curves and an input history which are all specified by the user. The structure of the code is very similar to the structure of the code which implements the classical Preisach model. For this reason, the detailed discussion of this code is omitted. We next proceed to the discussion of hysteretic energy losses for hysteresis nonlinearities described by the nonlinear Preisach model (3.174). We

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

383

A

~ ( a, [~ u ) T~, u

.(a,~,u) b

v

~

h

~

~

U

\ a

FIGURE 3.67

begin with the case of elementary hysteresis nonlinearity represented by the expression #(a, fl, u(t))~u(t). It follows from (3.187) that the input-output relationship for this nonlinearity is described by a loop shown in Fig. 3.67. It is clear that branch I of this loop is fully reversible except for its part a along which the irreversible switching-up occurs. Similarly, branch 2 of the loop is fully reversible except its part b along which the irreversible switching-down occurs. If a periodic variation of input is such that the whole loop is traced, then the hysteretic energy loss for one cycle, Qcycle, is equal to the area enclosed by the loop Qcycle - 2

p(a, ]/, u) du.

(3.215)

It is clear that this energy loss occurs as a result of irreversible switchings along the vertical parts of the above loop. Consequently Qcycle -- q~ + q~,

(3.216)

where q~ and q~ are losses occurred as a result of irreversible 'up' and 'down' switchings, respectively. Owing to the symmetrybetweenbranches I and 2, it can be concluded that q~ - q~.

(3.217)

CHAPTER 3 Scalar Preisach Models of Hysteresis

384

J

FIGURE 3.68

From (3.215), (3.216) and (3.217) we find q= - q/~ =

/~(a, fl, u) du.

(3.218)

In the nonlinear Preisach model, any input variation is associated with irreversible switchings of some elementary loops #(a, ]/, u(t))~=~u(t). It is natural to equate the hysteretic energy loss occurring for some input variation to the sum of energy losses resulting from the irreversible switchings of the above elementary loops. Since in the nonlinear Preisach model we deal with continuous ensembles of the elementary hysteresis loops, the above summation should be replaced by integration. Thus, if f~ denotes the region of points on the ~-//diagram for which the elementary hysteresis loops are irreversibly switched during some monotonic input variation, then the hysteretic loss, Q, for this input variation is given by Q=

/~(~, ]/, u) du d= d]~.

(3.219)

In the particular case when the distribution function # does not depend on u, from (3.219) we obtain the expression

Q -//n #(~'fl)(a-//)da dfl,

(3.220)

which has been previously derived for the classical Preisach model (see Section 3.1). A typical shape of the region f2 is shown in Fig. 3.68. The region f2 can be subdivided into a triangle and some trapezoids which in turn can

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

385

{2

(u,,u_)

+,

_

FIGURE 3.69

"8 u

(X

FIGURE 3.70

be represented as differences of triangles. Thus, if the integral (3.219) can be evaluated for any triangle region, then it will be easy to determine this integral for any possible shape of f2. In the case when f2 is a triangle, a very simple expression for the integral (3.219) in terms of experimentally measured function P can be derived. The derivation proceeds as follows. Let be a triangle (see Fig. 3.69) swept during a monotonic increase of input from some local m i n i m u m u_ to some value u+. According to (3.219), this input variation results in the hysteretic loss:

T(u+,u_)

Q(u_, u+) -

(u+,u_)

/a(~,fl, u) du d~ dfl

=/ui+(/ui(/flaP(cx'fl'u)du)dfl) d~"

(3.221)

By using Fig. 3.70, we can change the order of integration in the two interior integrals as follows:

/ui (/fl~P(cx,fl, u)du)dfl- /ui (/ulP(~' fl'u)dfl) du"

(3.222)

CHAPTER 3 ScalarPreisach Models of Hysteresis

386

FIGURE 3.71

Now, by invoking the formula (3.185), from (3.222) we find

fu~(ffl ~

#(o~,fl, u) du

)

l fu~(fuU~ap(~x'[3'u)

d~ = --~

O~c]fl

d]/) du

lfu~[aP(~,u,u) 2

_

a~

]

c~P(a, u_, u) du. l

(3.223)

By substituting (3.223) into (3.221) and by changing this time the order of integration according to Fig. 3.71, we derive

lfuU+(fu~[C]P(cx'u'u) Q(u_, u+) = --~ _ _ O~

OP(cx,u_,u)]3~ du)doe

l fuU+(fuU+[cqP(cx'u'u) _ c]P(cx'u-'u)]&x)du 2

-~

_

~

l f u+[P(u+, u, u) -

~

P(u, u, u) - P(u+, u_, u)

+ P(u, u_, u)] du.

(3.224)

By using (3.181) in (3.224), we finally obtain

l fun+P(u+, u_, u) du. Q(u_, u+) - -~

(3.225)

The formula (3.225) is remarkably simple if compared with a similar formula (3.115) derived for the classical Preisach model. The formula (3.225) allows one to evaluate hysteretic energy losses for arbitrary input variations. Indeed, consider some input variation for which the region f] has

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

387

the shape shown in Fig. 3.68. Then, the corresponding hysteretic energy loss is given by (3.219), which can be written as k Q - Q(Mn, mn) + ~-~[Q(Mn-i, mn-i) - Q(Mn-i, mn-i-1)].

(3.226)

i=1

Each term in (3.226) can be evaluated by employing the formula (3.225). The formula (3.225) admits the following geometric interpretation. Consider a minor loop formed as a result of back-and-forth input variations between two consecutive extrema u_ and u+. If a hysteresis nonlinearity is representable by the nonlinear Preisach model, then all comparable minor loops of this nonlinearity have equal vertical chords. Consequently, P(u+, u_, u) can be construed as a vertical chord of the above minor loop. This implies that the integral in (3.225) is equal to the area A of this loop. Thus, the expression (3.225) can be written as A Q(u_, u+) - ~.

(3.227)

From (3.227) we immediately infer the following result.

For any loop of hysteresis nonlinearity representable by the nonlinear Preisach model, hysteretic losses occurring along ascending and descending branches are the same. By using this result we can relate hysteretic losses occurring for arbitrary input variations to certain cycle losses. This can be accomplished in exactly the same way as was done for the classical Preisach model. In particular, the expressions (3.122) and (3.123) can be derived. We conclude this section by the discussion of two facts that may facilitate further appreciation of the advantages of the nonlinear Preisach model over the classical one. Consider the hysteresis nonlinearities exhibited by the Stoner-Wohlfarth magnetic particles (see Fig. 3.72). It is pointed out in Section 3.1 that these nonlinearities cannot be described by the classical Preisach model. The reason is that degenerate minor loops traced as the input varying back and forth between ar and fl' are not congruent. It is easy to see that the hysteresis nonlinearities exhibited by Stoner-Wohlfarth magnetic particles can be described by the nonlinear Preisach model. First, this is clear from the fact that the property of equal vertical chords is satisfied because all minor loops are degenerate and, consequently, they have equal (zero) vertical chords. Second, the explicit representation (3.174) for the Stoner-Wohlfarth hysteresis nonlinearities can be found. Indeed, it can be

CHAPTER 3 ScalarPreisach Models of Hysteresis

388

fu

,,, ,,, ~N~,~

-hrn

I

h m U

I

II 1I I

I

I I

FIGURE 3.72

shown that

[a(a, fl, u(t))

= f~t)

- fu+(t) c~(o~ -

2

hm, fl q- hm)

(3.228)

where ~ is the Dirac function. By substituting (3.228) into (3.174), we find

f(t) = f~t) - fu+(t) ~hm,_hmU(t ) q_ f~t) + fu+(t) 2 2 "

(3.229)

It can be easily checked that the expression (3.229) indeed represents the hysteresis nonlinearities depicted in Fig. 3.72. Finally, the following observation may help the reader to appreciate the extent to which the nonlinear Preisach model is more general than the classical one. The classical Preisach model represents hysteresis nonlinearities that, for any reversal point, have (regardless of past history) only one branch starting from this point (see Fig. 3.73(a)). This branch is congruent to one of the first-order reversal curves. This is an apparent consequence of the congruency property of the classical model. On the other hand, the nonlinear Preisach model (3.174) describes hysteresis nonlinearities with the property that for any reversal point there are infinite possible branches starting at this point (see Fig. 3.73(b)). A particular realization of these branches is determined by a particular past history. This property follows, for instance, from the expression (3.214) where all terms are input-dependent. For the classical model, the similar expressions (3.71)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

389

f

Ji ///

/1 I'

I

- -

/

U

/ U

max

max

(b)

(a)

FIGURE 3.73

and (3.74) have only the last terms which are input-dependent. The above observation shows that the nonlinear model is endowed with a much more general mechanism of branching than the classical model. 3.2.3 'RESTRICTED' PREISACH MODELS OF HYSTERESIS

In the previous section we have discussed the nonlinear Preisach model of hysteresis which is a far-reaching generalization of the classical Preisach model. This generalization has been achieved by assuming that the distribution function y is dependent on the current value of input u(t). Another approach to the generalization of the classical Preisach model is to assume that the function y depends on stored past extremum values of input, {Mk, mk}. This approach was briefly explored in [47]. Here, we shall follow this approach as well, however, our treatment of the model itself and the identification problem for this model will deviate appreciably from the discussion in [47]. We begin with the simplest case when the function y depends only on the first global maximum, M1. Thus we define a new Preisach-type model as

fit) - l I T

#(~' fl' M1)~aflu(t) da dfl + C M 1

,

(3.230)

,~ ,J l M 1

where M1 is the largest input maximum since the departure from the state of negative saturation, and the support of #(a, fl, M1) is the triangle TM1 defined by inequalities flo~flKa~M1 (see Fig. 3.74). We shall use the following interpretation of the above model. For any fixed extremum M1, the model (3.230) will be used to describe hysteresis behavior in the region confined between an ascending branchfu~t ) and a first-order reversal curve fMlfl (see Fig. 3.75). For this reason, the model (3.230) can be termed a

CHAPTER 3 Scalar Preisach Models of Hysteresis

390

TM1

FIGURE 3.74

'restricted' Preisach model of hysteresis. However, the above'restriction' does not diminish at all the region of applicability of the model. This is because M1 may assume any value between fl0 and a0. The constant CMI in (3.230) can be easily determined from the condition that the model matches the output values f+l for u(t) - M1 and f_ for u(t) = flo. Indeed, from (3.230) we easily derive

ft~l = f f ~ , It(a, ]~, M1) da dfl +

CM1 ,

(3.231)

9,,,1 M 1

f

fl, M1) da d ] / + CM1.

= - ff.

(3.232)

.J,J I M1

From (3.231) and (3.233) we find

CM1 =

fM1 -'~"f -

2

'

(3.233)

which leads to the following representation for the model (3.230):

fit) - ffTM It(a, fl, M1)~&~u(t) da d~ +

f~l +f-

(3.234)

1

Next, we shall be concerned with the solution of the identification problem. The essence of this problem is in determining the function #(a, fl, M1) by fitting the model (3.234) to some experimental data. It turns out that It(a, fl, M1) can be found by matching second-order reversal curves f M l ~ shown in Fig. 3.75. To do this, we introduce the function

F(a, fl, M1) "-

1

-~(fMlfl ~ --&l/J).

(3.235)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

391

f

we

~/

/

L//.'///

~

--.

M1

fMI~OC fMll3 /

~__~........~---~:~

,L_______second-.order reversal curve

FIGURE 3.75

d

d

-

(a)

(b) FIGURE 3.76

This function is equal to one-half of the output increments along the secondorder reversal curves. To relate this function to #(~, fl, M1), consider the diagrams shown in Fig. 3.76.

CHAPTER 3 Scalar Preisach Models of Hysteresis

392

By using exactly the same line of reasoning as in the case of the classical Preisach model (see Section 3.1.3), from these diagrams and (3.234) we derive F(~, fl, M1) --~

,liT (~,fl)

=

#(0r fl', M1) da' aft'

, #(a', fl', M 1 ) d s

aft'.

(3.236)

By differentiating (3.236) twice, we obtain #(a, fl, M1) = -

~2F(a, fl, M1)

.

(3.237)

By recalling the definition (3.235) of F(a, fl, M1), from (3.237) we find #(a, fl, M1) =

1 ~.02fM~fl~ 2 Oa c~fl

(3.238)

Thus, if the distribution function # is determined from (3.237) or (3.238), then the restricted Preisach model will match the output increments along second-order reversal curvesfM1/~a attached to the first-order reversal curve fM~fl. This implies that the limiting ascending branch f + will be matched because this branch coincides with the second-order reversal curvesfMlfloa and because the value f_ is matched by the model. From (3.235) we also find that the output increments along the first-order reversal curve fMlfl will be matched because these increments are twice F(M1, fl, M1). Since the limiting ascending branch, output increments along the first-order transition curvesfMlfl and the output increment along the second-order reversal curves fM~fla will all be matched, we conclude that the second-order transition curves fM~fl~ themselves will be matched. Up to this point it has been tacitly assumed that input variations are started from the state of negative saturation (u(t)<,flo). This is a natural assumption because the state of negative saturation is a well-defined state. However, another well-defined state is the state of positive saturation (u(t)>~ao). If input variations start from this state, then it is natural to assume that the distribution function depends on the first global m i n i m u m ml. This leads us to the following restricted Preisach model:

f(t)

-

ff_ fa(a, fl, ml)~/~flu(t) da dfl + Cml,

(3.239)

,I ,l l ml

where the support of ~(~, fl, ml) is the triangle ml~
Zml

defined by inequalities

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

393

/

Tm1

/

FIGURE 3.77

The model (3.239) can be regarded as a counterpart of the model (3.230) and our immediate goal is to establish some connections between distribution functions # and ~. To this end, we first find the constant Cm~ in (3.239) by matching the output values fml and f+:

fml -- - []~ ddl f+ - dff~ dl

~(:<,fl, ml) d~ dfl + Cm~,

(3.240)

ml

~(~, fl, ml) da dfl + Cml.

(3.241)

ml

From (3.240) and (3.241) we find Cml

fml 2+f+ '

=

(3.242)

and the restricted model (3.239) can be represented as

fit) -- fiT

~(~' fl' ml)7~3u(t) d:r dfl +

fml+f+

(3.243)

mI

To determine the distribution function ~(~, fl, ml), we shall use the decreasing second-order reversal curvesfm 1~ which are attached to the increasing first-order reversal c u r v e s f m 1~ (see Fig. 3.78). As before we can introduce the function 1 F(~, fl, ml) (3.244) -

-

-2(fml~--fmlSc-fl)

and show that F(~, fl, ml) - [ [

JJT

~(a', fl')d~' dfl'.

(3.245)

CHAPTER 3 Scalar Preisach Models of Hysteresis

394

f

mj

T2

T//q/

/

"~

FIGURE 3.78 From (3.245), we derive 632F(&, fl, ml) ~(~, fl, ml) = -

(3.246)

By invoking (3.244) in (3.246), we obtain 2

1 c~

~(~,fl, m l ) -

~ r

7ml~fl "

(3.247)

Next, we will utilize the symmetry between the ascending and descending second-order reversal curves. In mathematical terms, this symmetry means that if ml---M1, ~---fl, fl - -0r (3.248) then F(~, fl, rex) = F(~, fl, MI).

(3.249)

The last formula can also be expressed as F ( - f l , - ~ , - M 1 ) - F(~x, fl, M1).

(3.250)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

395

By substituting (3.249) and (3.248) into (3.246) and taking into account (3.237), we derive ~t(~, fl, ml) = --02F(~'/~, M1) __ #(0r 8~8//

M1).

(3.251)

From (3.248) and (3.251) we finally obtain U(a, fl, M1) = ~(-fl, - a , -M1).

(3.252)

The last expression can be regarded as a generalization of the mirror symmetry property (3.46) previously derived for the classical Preisach model. Indeed, if M1 = a0 and ml = fl0, (/30 = -a0), then the models (3.234) and (3.243), respectively, coincide with the classical Preisach model. Consequently, we have U(a, fl, ~0) = U(a, fl),

ft(a, fl,-a0) = U(a, fl),

(3.253)

and the equality (3.252) can be rewritten as ~(a, fl) = ~ ( - f l , - a ) ,

(3.254)

which is the property of mirror symmetry for the classical Preisach model. The formula (3.252) also shows that, having determined the function /,t(a, fl, M1), we have solved the identification problem for the model (3.234) and its counterpart (3.243). This suggests that the model (3.243) is not independent of the model (3.234), but rather complements the latter model. In a way, we deal here with the same model written in two different forms. These two different forms correspond to two different initial states (states of positive or negative saturation). For this reason, in our subsequent discussion we will not draw any distinction between the models (3.234) and (3.243). In the sequel, all results will be discussed for the model (3.234), and it will be tacitly implied that they are also valid for the model (3.243). It is clear that the model (3.234) has an almost identical structure to the classical Preisach model. For this reason, it is apparent that the wipingout property holds for the restricted model (3.234). However, the congruency property of minor hysteresis loops undergoes some modification. This modification can be described as follows. We call minor hysteresis loops comparable if they are formed as a result of back-and-forth input variations between the same consecutive reversal values and these input variations take place some time after the same largest input maximum was achieved. The given definition of comparable minor loops implies the possibility of different past input histories between the time when the largest maximum M1 was achieved and the time when the above-mentioned backand-forth input variations commence. By using the same reasoning as in

CHAPTER 3 Scalar Preisach Models of Hysteresis

396

Section 3.1.2, we can prove that all comparable minor loops described by the restricted model (3.234) are congruent. It is clear from the above statement that the congruency of minor hysteresis loops prescribed by the model (3.234) for the same input reversal values are to a certain extent history dependent. This is not the case for the classical Preisach model. It turns out that the wiping-out property and the modified congruency property are characteristic of the restricted model (3.234) in the sense that the following result is valid. REPRESENTATIONTHEOREM. The wiping-out property and the congruency property

of comparable minor loops constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the restricted Preisach model (3.234). The proof of this theorem is very similar to the proof of the representation theorem for the classical Preisach model. For this reason, this proof is omitted. It is clear that the modified congruency property is more general than the congruency property of the classical Preisach model. Thus the restricted model (3.234) is more general than the classical one. In addition, the restricted model (3.234) allows one to fit experimentally measured firstand second-order reversal curves, while the classical Preisach model is able to fit only first-order reversal curves. Since higher-order reversal curves are 'sandwiched' between first- and second-order reversal curves, it is natural to expect that the restricted model will be more accurate than the classical Preisach model. We next turn to the discussion of numerical implementation of the restricted model (3.234). As before, we will show that double integration in (3.234) can be completely avoided and that an explicit expression for the output, fit), in terms of the experimentally measured function, F(~, fl, M1), can be derived. The derivation proceeds as follows. From (3.234) by means of simple transformations we find

fit)- ffs ~l ~t~~(a' fl' M1) d~ dfl - ffs ;~ ~t~la(a, fl, M1) do~dfl + f~l + f 2

=2f sM1(t) q_ f/~l

+f-.

2

fl, M1) d~xdfl -

fWM 1

fl, M1) d~x dfl (3.255)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

397

F I G U R E 3.79

By invoking (3.231) and (3.232), we derive

f/rM l~(a, fl, M1) dc~ dfl

-- f~ll

1

-f-

(3.256)

2

By substituting (3.256) into (3.255), we obtain

fit) - 2 H /a(c~,fl, M1) dc~ dfl + f _ . dds ~VI1(t) The s e t S~dI (t) can be subdivided into n trapezoids As a result, we have

(3.257)

Qk shown in Fig. 3.79.

n(t) /fs+- (t) #(~X,fl, Ml)d~xdfl - y~ f/Q la(~x,tiM1) dc~ dfl. M1

k=l

(3.258)

k

Each integral in the right-hand side of (3.258) can be represented as

ffQ k

fi, M1)

d, - ff r (Mk,mk-1) l~(a, fl, M1) da dfl - [[ #(~, fl, M1) dc~ dfl. Jdr (Mk,mk)

(3.259)

By recalling (3.236), from (3.259) we find

ffQ #(~, fi, M1) dc~dfl - F(Mk, mk-1, M1) - F(Mk, mk, M1). k

(3.260)

CHAPTER 3 ScalarPreisach Models of Hysteresis

398

By substituting (3.260) into (3.258) and then (3.258) into (3.257), we finally obtain n(t)

fit) - 2 E[F(Mk, mk-1, M1) - F(Mk, mk, M1)] + f - .

(3.261)

k=l

Thus, the formula (3.261) expresses the output, fit), explicitly in terms of experimentally measured second-order reversal curves which are related to the function F by Eqn. (3.235). This formula has a two-fold advantage over the expression (3.234). First, double integration is avoided. Second, the determination of ~(~, fl, M1) through differentiation of experimentally obtained data (see (3.237) or (3.238)) is completely circumvented. This is a welcome feature because the above differentiation may amplify noise inherently present in any experimental data. So far we have discussed the restricted Preisach model for which the distribution function/~ depends only on the first global extremum M1 or ml. It is natural to call it a first-order restricted Preisach model. However, further generalizations in this direction are possible. In these generalizations, the function # is assumed to be dependent on some finite sequence of past dominant extrema: la(~, fl, M1, ml, M2, m2 . . . . . Mk, mk). It is natural to call these models high-order restricted Preisach models. To make the general idea of these models clear, consider a second-order restricted Preisach model. In this model, the function/~ is assumed to be dependent of M1 and ml, and the model itself is defined as

fit) - ffTM

#(~' fl' M1, ml)~;aflu(t) da dfl + CMlml,

(3.262)

lml

where the support of/~ is the triangle ZMlml specified by inequalities ml<<,fl<<,o~M1 (see Fig. 3.80). For any fixed extrema M1 and ml, the model (3.262) will be used to describe hysteresis behavior in the region confined between the firstorder reversal curve fMl2 and the second-order reversal c u r v e fMlmlOr (see Fig. 3.81). This explains the origin of the term 'restricted', used for the name of this model. However, the described restriction does not narrow at all the region of applicability of this model. This is because M1 and ml are free parameters that may assume any values between fl0 and a0. These values depend on a particular past input history and may even change with time when the previous extrema M1 and ml are wiped out by subsequent input variations.

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

399

1,

FIGURE 3.80 f

fM1m10~p

/ m,

/

,m,o /

/ I "i-;f~f/J/',

~U

FIGURE 3.81 The c o n s t a n t CMlml in (3.262) can be determined from the condition that the model matches the output values fM1 and fMlml"

fM1 -- iirM

la(Sr fl, M1, ml) de dfl + CMlml,

(3.263)

lml

fMlml -- -- ffTM

/./(0~,fl, M1, lml

ml) de dfl + CMlml.

(3.264)

CHAPTER 3 Scalar Preisach Models of Hysteresis

400

From (3.263) and (3.264), we find

CMlml --

f~I1 if- fM1 ml

(3.265)

which leads to the following expression for the model (3.262)"

fit) -

ffr

lml

~(~' ~' M1, ml)~aflu(t) d~ dfl +

f~I 1 -[-fM1 ml 2

(3.266)

To determine the distribution function #(~, fl, M1, ml), the experimentally measured third-order reversal curves fMl,ml~x~ (see Fig. 3.81) should be employed. By using these curves, we can introduce the function F(or ]~, M1, ml)

--

1 ~(fMlmlO~ --&lml~/J).

(3.267)

Exactly in the same way as before it can be shown that the second-order restricted Preisach model will fit the third-order reversal curves if the function F(a, ]/, M1, ml) is related to the function p(a, fl, M1, ml) by the equation

F(0r

M1, ml) - ] / ddT

#(0r fl', M1, ml) da' dfl'.

(3.268)

From the last equation we easily derive p(~,//, M1, ml) - -

02F(0r

M1, ml)

(3.269)

and

1 ~2fM 1ml or /2(cz, fl, M1, ml) - ~ c]~z~gfl

(3.270)

By its design, the model (3.262) is valid under the condition that the initial state of the hysteresis transducer is the state of negative saturation. If the initial state is the state of positive saturation, then the following counterpart of the model (3.262) can be used:

f(t) = ffTK 4

[_l(O~,fl, 17111,~/I 1)~a~u(t) d~ dfl +

ft~1 "q-fffl1~I1

(3.271)

lrhl

where rhl is a global m i n i m u m and M1 is a subsequent global m a x i m u m of the input.

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

401

As before, we can establish the following relationship between the function # and ~: (3.272)

/~(~, fl, M1, ml) - ~(-fl, -~x, -M1, - m l ) ,

which shows that (3.266) and (3.271) are just different forms of the same model. These two different forms correspond to two different well-defined initial states of the hysteresis transducer. It is apparent that the congruency property of minor hysteresis loops undergoes further modification (further relaxation) in the case of the secondorder restricted Preisach model (3.266). The essence of this modification is that for minor loops to be congruent the corresponding inputs must vary back and forth between the same consecutive reversal values and also assume in the past the same values of M1 and ml. It can be proven that the wiping-out property and the above-mentioned modified congruency property of minor loops constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the model (3.266). This is the so-called representation theorem for the second-order restricted Preisach model. As before, it can be shown that the following expression can be used for the numerical implementation of the model (3.266): n(t)

fit)

-

2 E[F(Mk, mk_ 1, M1, ml)

-

-

f(Mk, mk, M1, ml)]

-~&lm

I9

(3.273)

k

The main advantage of this expression over the formula (3.266) is that it represents the output,fit), directly in terms of the experimentally measured third-order transition curves. It is clear from the previous discussion that higher-order restricted Preisach models of hysteresis can be defined as

fit) -- f (_

[a(~, fl, M1, ml ..... Mk)~u(t) d~ dfl

J J l M lml""Mk

q'- CMlml...Mk.

(3.274)

It is also apparent now how higher-order transition curves can be used for the determination of the function #(~, fl, M1, ml . . . . . Mk). It goes without saying that by increasing the order of the restricted Preisach model we can increase the accuracy of this model. However, this increase in accuracy is amply paid for by the increase in the amount of experimental data required for the identification of higher-order models. For this reason, the use of restricted Preisach models of order higher than two does not seem to be practically feasible or attractive. It is instructive to note that another

402

CHAPTER 3 Scalar Preisach Models of Hysteresis

o~

X

R(Z,~.u) FIGURE

3.82

approach to the construction of Preisach models that can match any desired number of higher-order reversal curves has been recently proposed in [48-50]. We conclude this section with a brief discussion of the 'restricted nonlinear' Preisach model of hysteresis. It is clear from the name of this model that it combines the main features of the 'restricted' and 'nonlinear' models. The first-order restricted nonlinear (input-dependent) Preisach model can be defined as

fit) = /R M1U

#(:e, fl, u(t), M1)4&flu(t) d~ dfl + fMlu(t) q-fu+((t) 2

(3.275)

where the moving support of #(a, fl, u(t), M1) is the rectangle RMIu specified by inequalities flo~flKu(t), u(t)KaKM1 (see Fig. 3.82), and the meaning offMlu(t) is clear from Fig. 3.83. By introducing the function (3.276)

P(CX, ]~, U, M1) = fMlflOe u -- f M l f l u

and by using the same reasoning as in the previous section, we can establish that

P(0~, ]~, u, M1) -- 2 [ ] #(a', fl', u, M1) doe' dfl'. ddR (o4fl,u) From (3.277) we can derive the following expressions: /~(0e, ]3, u, M1) =

1 c~2P(0e, ]3, u, M1) 2

0a3fl

,

(3.277)

(3.278)

and

/~(0e, j~, U, M1) --

1 ~.c92fMlfl~u 2 c~e aft

(3.279)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

403

I

61

FIGURE 3.83

We can also derive the following explicit expression for the output, fit), of the model (3.275): n(t) f ( t ) -- fMlu(t) -Jr- ~[P(Mk+I, k

mk, u(t),M1) - P(Mk, mk, u(t),M1)]. (3.280)

It is worthwhile to remember here that the function P(~, fl, u, M1) is directly related to experimental data and has the meaning of output increments between the second- and third-order reversal curves. It is easy to formulate and to prove the representation theorem for the model (3.275). This is left to the reader as a useful exercise. 3.2.4 DYNAMIC PREISACH MODELS OF HYSTERESIS

In all the previous sections our discussion has been centered around the classical and generalized Preisach models of hysteresis that are rateindependent in nature. The term 'rate-independent' implies that in these models only past input extrema leave their mark upon the future values of output, while the speed of input and output variations has no influence on branching. The intent of this section is to relax the 'rate-independence

404

CHAPTER 3 Scalar Preisach Models of Hysteresis

property' of Preisach-type models. For this reason, new Preisach-type models that are applicable to the description of dynamic hysteresis are introduced. The identification problem of fitting these models to some experimental data is then studied. Finally, some discussion is presented concerning numerical implementation of the dynamic Preisach models of hysteresis. The main idea behind the dynamic Preisach-type models of hysteresis is to introduce the dependence of/x-functions on the speed of output variations, df This leads to the following dynamic Preisach models of hysteresis:

f(t) - ff~>>.3;(a, fl, d--fdt)9=3u(t)da dfl, fit) -

l/R( /x u(t)

df)

a, fl, u(t), -~ ~;~flu(t) da dfl +

(3.281)

f~t)-I-fu+(t). (3.282) 2

The above models are 'dynamic' generalizations of the classical Preisach model and the nonlinear Preisach model, respectively. Similar generalizations for the restricted Preisach models are apparent. They will not be discussed in this section because their treatment mostly parallels that of the model (3.281). The direct utilization of the models (3.281) and (3.282) is associated with some untractable difficulties. First,/x-functions depend on the unknown quantity df and this complicates numerical implementations of the models (3.281) and (3.282). Second, it is not clear how to pose the identification problems for these models. The above difficulties can be completely circumvented by using the power series expansions for/x-functions with respect to ~df.. (

d f ) -- //0 (~, fl) + ~-~//1(0r df ]~) q - . . . ,

(3.283)

/x ~, fl, ~

(

dr)

/X a, fl, u(t), ~

df (~, fl, u(t))+

-/X0 (a, fl, u(t))+ ~--~/Xl

"''.

(3.284)

By retaining only the first two terms of the above expansion, we arrive at the following dynamic models:

fl t ) - / f ~>>,~ #o(~, fl)9~3u(t) da dfl

-}--~dfffa)fl/xl(~,fl)~a3u(t) do~dfl,

(3.285)

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

fit) = ffI~

t~~176fl' utt))~a~u(t) da dfl +

405

f~t) q-fu+(t)

u(t)

q- --~df//Ru(t) #1 (or flu(t))~a~u(t)

da

dfl.

(3.286)

It is clear that in the case of very slow output variations, the above models are reduced to the corresponding rate-independent hysteresis models. This means that #0-functions in (3.285) and (3.286) should coincide with the/~functions of rate-independent models (3.149) and (3.174), respectively. In other words, the #0-functions in (3.285) and (3.286) can be determined by matching first-order and second-order transition curves, respectively. The above reasoning suggests that the models (3.285) and (3.286) can be represented in the following equivalent forms:

df

fit) =)~(t) + ~-~ JJ~>~/~fll (or f l ) ~ u ( t ) d~ dfl,

(3.287)

fit) =)~(t)+ ~df //R u(t) fl l (:r fl , u ( t ) ) ~a~ u ( t ) d :r d fl ,

(3.288)

where the f-terms stand for the 'static' components of hysteresis nonlinearities:

f (t) = ff~ >>-fl#o(~, fl)}~flu(t) d:c dfl, J:(t) - [[_ JJl<

u(t)

#o(O~,fl, u(t))}a~u(t) d~ dfl + fu+(t)q-Juft) f- . 2

(3.289) (3.290)

The expressions (3.287) and (3.288) are transparent from the physical point of view. They show that the instant speeds of output variations are directly proportional to the differences between instant and 'static' output values. We next turn to the identification problems of determining the #lfunctions by fitting the models (3.287) and (3.288) to some experimental data. We first consider the identification problem for the model (3.287). The following experiments can be used to solve this problem. Starting from the state of negative saturation, the input u(t) is monotonically increased until it reaches some value a at t = to and is kept constant for t ~>to. As the input is being kept constant, the output relaxes from its value fa at t = to to its static value )?a. According to the model (3.287), this relaxation process is described by the differential equation df + f =)~

(3.291)

406

CHAPTER 3 ScalarPreisach Models of Hysteresis

FIGURE 3.84

where

"ca = f f s ; #l (~X',fl') do~' dfl' - f f s + #l(Or

(3.292)

and S~- and S+ are negative and positive sets on the Preisach diagram (see Fig. 3.84) corresponding to the above input variation. The solution to Eqn. (3.291) is given by

fit) = (f~ - L ) e - G

t

'~

+ f~.

(3.293)

Thus % has the meaning of relaxation time and can be experimentally measured. Next, the hysteresis nonlinearity is brought back to the state of negative saturation. Starting from this state, the input is again monotonically increased until it reaches the value ~. Then the input is monotonically decreased until it reaches some value fl at time t - t or and is kept constant for t > t~. As the input is being kept constant, the output relaxes from its value f ~ at t = t~ to its static value)?~/~. The model (3.287) yields the following differential equation for the above relaxation process: -ca/~df ~ + f - f~/~,

(3.294)

where according to the model (3.287) and the Preisach diagram shown in Fig. 3.85 the coefficient za/~ is given by the expression za/~ --/fS~-~ #l(a" fl') d~162 dfl' -/fs+8 #1 (a', fl') d~' dfl'.

(3.295)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

407

c~

s~ {~.~1

FIGURE 3.85

By solving (3.294), we find t

fit) = (f~ - f~a)e ~ + f~a.

(3.296)

Thus, ra~ also has the meaning of relaxation time and can be experimentally measured. It is apparent that the relaxation time ra can be construed as the relaxation time ~aa, and, consequently, it belongs to the set {raft}. We next show that by knowing these relaxation times, we can find the function /A1(0~, ]~). To this end, we introduce the function q(~, ]/) = ~a - za/~.

(3.297)

From (3.292), (3.295) and (3.297), we find q(~, fl) = -2 ff_

/~1(~', fl') d~' dfl'.

(3.298)

J Jl

where Taft is a triangle shown in Fig. 3.85. From (3.298), we can easily derive the expression 1 02qa]~ #1(a,//) - 2 ~ ~fl"

(3.299)

By recalling (3.297), from (3.299) we find

/A1 ( a , fl) - -

1 02raft 2 ~ ~fl"

(3.300)

The solution to the identification problem for the input-dependent (nonlinear) model (3.288) can be found in a similar way. For this reason, our comments will be concise. We consider two types of relaxation

CHAPTER 3 Scalar Preisach Models of Hysteresis

408

processes. A first type process occurs after a monotonic input increases to some value a and a subsequent monotonic decrease of input to some value u. According to the model (3.288), the first type relaxation processes are described by the differential equation df + f =f~u

(3.301)

where

l~u = f f

JJs a

#1 (0r fl', U)da' d f l ' - / /

JJs+~

~1 (0r ]~', U)da' dfl'.

(3.302)

From the previous differential equation, we find fit) = (f~u-f~u)e - t

+f~u.

(3.303)

A second type relaxation process occurs when the input is first monotonically increased to a, then decreased to fl, again monotonically increased to u, and kept constant afterwards. These relaxation processes are described by the equation q

-

r~flu - ~ + f = f ~ u ,

(3.304)

where ~ c
(3.305)

As before, we have t

fit) = (f=flu - J:=flu)e "~'" + J:aflu.

(3.306)

It is clear from (3.302) and (3.305) that Z=u and z~/~uhave the physical meaning of relaxation times and can be measured experimentally. Knowing these relaxation times, we can define the function Q(~, fl, u) = ~u - r=/~u.

(3.307)

From (3.301), (3.304) and (3.306), we derive Q(~, fl, u) - - 2 / /

J JR (o~,fl,u)

fll (~', fl', u) d~' dfl'.

(3.308)

From (3.307), we obtain

I ~2Q(0r fl, u) ~0~cg/Y "

fll (0~, fl, U) -- ~

(3.309)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

409

By invoking (3.306), from (3.308) we have the alternative expression for/'1: 1 (?2ra/~_______2u" 2 {?~c~fl

/21 ((X,/J, U) --

(3.310)

Consider the relaxation processes that are symmetric to those discussed above. Suppose that we start from the state of positive saturation and reduce the input to some value fl, then increase the input to some value and keep it constant thereafter. Suppose also that the subsequent output variations are characterized by the relaxation times r ~ . If fl--~,

~--fl,

(3.311)

then due to the symmetry we have r ~ - r~/j.

(3.312)

1 02r~ /11 (~' ~) -- 2 {?&{?~"

(3.313)

As before, we can derive that

By substituting (3.311) and (3.310) into the right-hand side of (3.312) and by recalling (3.300), we obtain /11(~, fl) =

1 ~2"cafl __ -/11 (0{, fl). 2 0fl {?~

(3.314)

From (3.313) and (3.310), we conclude /11 (--/J, --0~) -- --/11(0r fl).

(3.315)

Similarly, we can introduce the relaxation times r~aa and prove that ~2 1 r~ /11(~, fl, U) -- 2 {?~0fi"

(3.316)

~=-fl,

(3.317)

If ~=-u,

then due to symmetry we have r.~

- "C~lJu.

(3.318)

CHAPTER 3 ScalarPreisach Models of Hysteresis

410

By using the same line of reasoning as before, from (3.309), (3.315), (3.316) and (3.317) we derive #i (-/~, -~, -u) - -#i (a,/~, u).

(3.319)

We next proceed to the numerical implementation of the dynamic models. The models (3.287) and (3.288) can be represented as the following differential equation:

df +fit)=;~(t), ?l(u(t))-~

(3.320)

where the hysteretic coefficient a(t) is defined by the formulas

?~(u(t)) = - [ [

#1(a, f l ) ~ u ( t ) d~ dfl,

(3.321)

?t(u(t)) = - ]]..

#1 (~, fl, u ( t ) ) ~ u ( t ) d a dfl

(3.322)

dJ~

J J l~u(t)

for the models (3.287) and (3.288), respectively. The explicit solution to Eqn. (3.319) is well known and can be expressed

as f(~) d~] ' fit) -- b(t) If0 + f0t a(~)b(~)

(3.323)

where f0 is the initial output value, and b(t) is given by the formula

b ( t ) - e x p ( - f0 t ad~ -~ ) "

(3.324)

The expressions (3.322) and (3.323) can be efficiently used for computing fit), iff(t) and a(t) are known. The calculation of static f-components in the case of the classical Preisach model as well as the nonlinear Preisach model has been discussed in detail in the previous sections. Since the mathematical structure of a(t) is similar to the mathematical structure of the rate-independent Preisach models, similar explicit formulas can be derived for the numerical evaluation of a(t). In particular, it can be shown that the following formulas are valid for a(t) in the case of models (3.287) and (3.288), respectively: n(t)

?z(u(t)) -- 2q(~0, fl0) - 4 y~[q(Mk, mk-X) -- q(Mk, mk)],

(3.325)

k=l 1

?l(u(t)) = --~Q(ao, flo, u(t)) n(t)

- E [ Q ( M k + I , mk, u(t)) -- Q(Mk, mk, u(t))].

k=l

(3.326)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

411

The above formulas are convenient not only because they give explicit expressions for integrals in (3.320) and (3.321), but also because these expressions are presented in terms of experimentally measured data. It has been tacitly assumed in the foregoing discussion that the relaxation processes used in the identification procedures are well characterized by single relaxation times. If this is not the case and several relaxation times have to be employed to describe the above relaxation processes, then the discussed dynamic models must be generalized. The natural way to generalize these models is to use higher-order differential equations with hysteretic coefficients in order to account for several relaxation times. We demonstrate such a generalization for the second-order dynamic model: d2f fl(1) (u(t)) df ~(2)(u(t))h- ~ + +fit)-f(t)

(3.327)

where

37(t) = ff~

po(a, fl)~u(t)dadfl,

~(1) (u(t)) -- l~

111 (:r

fl)~=flu(t) d:r dfl,

~(2)(u(t))= ff~~>~ #2(~,fl)~=~u(t)da dfl.

(3.328) (3.329) (3.330)

To find #1 (0r fl) and/A 2(0r fl), we shall use the same two relaxation processes as for the identification of the model (3.287). In the first process, we start from the state of negative saturation and monotonically increase input to some value a and keep it constant thereafter. As the input is being kept constant, the output relaxes. According to (3.326), this relaxation is described by the differential equation a(2) d2f

(1)df

~

--~t~ + a~ --~ + f - f~.

(3.331)

A solution to this equation has the form t

t

fit) = c(~l)e ~(1~+ C(2)e ~(2~+ ~ ,

(3.332)

where r(~1) and r(~2) are the roots of the characteristic equation a (2)"C2 Jr- a(~l)r + 1 - O.

(3.333)

CHAPTER 3 Scalar Preisach Models of Hysteresis

412

Consequently, a(2) =

1 (1) (2)'

(3.334)

a(~1 ) - _,c~1) _}_1:(2)

(3.335)

The constants r(1) and z~2) have the physical meaning of relaxation times and can be measured experimentally. This leads to the experimental determination of a(~2) and a(~1) according to (3.333) and (3.334). N o w we consider the second relaxation process. In this case we start from the state of negative saturation, increase input to some value a, then decrease input to some value fl and keep it constant thereafter. As the input is being kept constant, the output relaxes and this relaxation is governed by the equation a(2) d2f _(1) df ~fl ~ + u ~ ~ + d - f ~ ,

(3.336)

whose solution is given by t

c~(1)e fit)-..~

t

,~(2) ~ + t. ~e

3(1)

r (2)

~

~ +f~.

(3.337)

The constants 1:(1)a/~and f~/~-(2)have the meaning of relaxation times and can be measured experimentally. As soon as this is done, we can find a (2) and a(1). a(2) __ 1 a// 1:(1)1:(2)'

a(1)-~]/

.L.(1) .C(2) ~]~ + ~// 1:(1)1:(2) "

(3.338)

(3.339)

Next we introduce the functions (1) _ a(1) _ a(1)

(3.340)

(2) _. a(2) _ a(1) qafl aft'

(3.341)

which are directly related to the above-mentioned experimental data. Using the same reasoning that we have used many times before, we can

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

413

establish the formulas (1)-2fjf

(2) _ 2

ffT

111 (~', fl') d~' dfl',

(3.342)

112(0r fl') da' dfl',

(3.343)

from which we derive

111(0{, fl) --

,,2 (1) 1 v q~/~ 2 0~ Off'

#2(~, fl) -

lvq~-----~fl.

(3.344)

,,2 (2) (3.345) 2 0~ 0fl Thus, the identification problem for the second-order model (3.326)-(3.329) is solved. Extensions to higher-order dynamic models are straightforward. 3.2.5 PREISACH MODEL OF HYSTERESIS WITH ACCOMMODATION

The Preisach-type models described in previous sections exhibit the wipingout property. This property results in an immediate formation of minor hysteresis loops after only one cycle of back-and-forth input variations between any two consecutive extremum values. However, experiments show that in some cases hysteresis loop formations may be preceded by some stabilization process that may require an appreciable number of cycles before a stable minor loop is achieved. This stabilization process is often called in the literature the 'accommodation' or 'reptation' process. Sometimes this accommodation process can be appreciable and then it is important to model it. In this section we discuss a certain modification of the moving Preisach model which allows one to account for the accommodation process. Another approach to the Preisach modeling of accommodation is described in [51]. We define the Preisach model with accommodation as

fit) -

fro

fl,d(m))5;aflu(t)do~ dfl + r(u(t)).

(3.346)

J J l~ u(t)

In the previous formula r(u(t)) stands for a fully reversible component which is represented by some single-valued function of u(t), while

d (m) - Mfk

if

(a, fl), Rk(t),

where Mkf are local extremum values of o u t p u t f and

(3.347)

Rk(t)

are rectangular

regions formed after the extremum M f has been achieved (see Fig. 3.86).

CHAPTER 3 Scalar Preisach Models of Hysteresis

414

a 0

R2k+l

/

R2k

_ _ _

FIGURE 3.86 ao

I

i

v'

/

FIGURE 3.87

It is easy to see that the regions Rk(t) are reduced with time and can even be completely wiped out. This fact is reflected in (3.346) by indicating that Rk are dependent of t. It is also clear from Fig. 3.86 that MkI are local output minima for even k and local maxima for odd k. We next prove that the model (3.345) does account for the accommodation process. For the sake of clarity, we will show this for the simplest case when the initial state of a hysteresis transducer is the state of negative saturation, and the input u(t) is monotonically increased to some value u+ and is varied back and forth between u+ and u_ thereafter. More general cases can be treated in a similar way. As the input is increased to the value u+, the output reaches a maximum value M( which, according to (3.345) and Fig. 3.87, is equal to

M( = -//~R #(~' fl' Mr~ da dfl + r(u+), 0

(3.348)

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

415

(X Ro i

FI

FIGURE 3.88

where M0f =f_,

(3.349)

and f_ is the output value in the state of negative saturation. As the input is monotonically reduced to the value u_, the output reaches minimum value M2f. According to (3.345) and Fig. 3.88, this minimum value is given by

ffR1

Mr2 =

fl' M() dadfl -

ffRo

fl'MfO) dadfl

+ r(u_).

(3.350)

It is clear that M2f ~=M0f - f_.

(3.351)

As the input is monotonically increased again to the value u+, the output reaches some maximum value M3f. According to (3.345) and Fig. 3.89, for this value we find

Mf = -

ffRo

fl' MfO) d e d f l - / / a 2 #(a, fl, M f) dadfl

+ r(u+).

(3.352)

By comparing Figs 3.87 and 3.89 as well as formulas (3.347) and (3.351) and by taking the inequality (3.350) into account, we conclude M3f # M(.

(3.353)

As the input is monotonically decreased again to the value u_, the output reaches some minimum value M f, which, according to Fig. 3.90, is

416

CHAPTER 3 ScalarPreisach Models of Hysteresis (~

Ro

u

i

1

Ro

FIGURE 3.89 (x

/

R

FIGURE 3.90 given by

Mf = ffe3 /~(a' fl'Mf3) d a d f l -

ffRo

fl'MfO) d~dfl

+ r(u_).

(3.354) By comparing formulas (3.349) and (3.353) and by taking into account that R1 = R3 and the inequality (3.352), we find M f ~: M f.

(3.355)

By continuing the same line of reasoning, we can establish that

M /2k+l # Mf2k-1 '

(3.356)

Mf2k 7~ Mf2k_2.

(3.357)

and

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

417

f I

i

i

U+

U

F I G U R E 3.91

The last two inequalities show that some stabilization (accommodation) process is going on before a stable minor loop is asymptotically reached. This accommodation process is illustrated by Fig. 3.91. We next proceed to the discussion of the identification problem. It turns out that the model (3.345)-(3.346) is very general in nature and it is not clear at this point how to solve the identification problem for this model. However, the identification problem becomes tractable if the dependence of the #-function on f(m) is factored out. In this way, we arrive at the following model:

v(f(m))#(O~,fl)~&~u(t) dadfl

fit) - ffR

+

r(u(t)),

(3.358)

u(t)

where f(m) are defined in the same way as in (3.346), and v-function is assumed to be known. Thus the identification problem consists of determining the functions r(u) and #(0r fl) by fitting the model (3.357) to some experimental data. We begin with the determination of the function r(u). Suppose that starting from the state of negative saturation the input is monotonically increased to some value u. As a result of this monotonic increase, the output will reach some value f + on the limiting ascending branch. According to (3.357), for this output value we have

fu+ - -v(f_) ffR(~o,#o,U)

#(~, fl) d~ dfl + r(u).

(3.359)

Now consider the state of positive saturation and suppose that the input is monotonically decreased to the same value u. In this case, the limiting descending branch will be followed and some output value f u on this branch will be reached. According to (3.357), for the above-mentioned

CHAPTER 3 Scalar Preisach Models of Hysteresis

418

output value we find

fu

v(f+) ffR(~o,flo,U) #(a, fl)da dfl 4- r(u),

-

(3.360)

where as before f+ is the output value in the state of positive saturation. From (3.358) and (3.359), we derive

r(u) =

v (y+ ) f u+ + v (f _ ) f ;

vq+) + vif_)

.

(3.361)

Consider a particular case when v-function is even. In this case, since f_ = -f+, we find v(f+) = v(f_), (3.362) and from (3.260) we conclude 1

r(u) = -~ (f+ 4- fu).

(3.363)

This coincides with the previous result for the moving Preisach model. We next turn to the determination of the #-function in (3.357). Consider first-order reversal curves f~fl which are attached to the limiting ascending branch. By fitting these curves to the model (3.357), we arrive at the following equation:

f~fl = v(f~)

-

(~,flo,fl,fl)

d s dfl'

+ v(f_) ffR (~o,flo,~,fl)~(~" fl') d~' dfl'

F ( fl) .

(3.364)

vq_) ffR(~o,flo,~,fl) "(~" fl') da' dfl. v (f~)

(3.365)

From (3.363), we obtain

f~fl - r(fl)

v(f~)

[[

= JJR (~,flo,fl,fl) #(a',fl')da'dfl'

On the other hand, by fitting the limiting descending branch ffl- to the model (3.357), we find

f ~ = v(f+) ffR(~o,flo,fl, fl)

#(s fl') da' dfl' + r(fl).

(3.366)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

419

The expression (3.365) can be transformed as follows:

J;

f[ ~(~,, fl')da' aa R(~o,flo,~,fl)

r(fl)

v(f+)

_-

+ ffR(o~,flo,fl,fl)

dfl

#(a', fl') da' dfl'.

(3.367)

By introducing the function T(o~, fl)

- ffR(~o,flo,~,fl)#(a', fl')d~' dfl'

(3.368)

and by using (3.364) and (3.367), we derive the following expression for T(c~, fl) in terms of experimentally measured data: T(~, fl) =

v(f~)[f~-

r(fl)]-

v(f+)[fafl -

r(fl)]

v(f+)[vq;) + v(f-)l

.

(3.369)

By knowing the function T(a, fl) and by employing the same reasoning as usual, we can derive from (3.367) the following expression for/2: #(~, fl) = -~92T(a' fl). & Sfl

(3.370)

For the purpose of numerical implementation of the model (3.357) we do not need to know the function/2. This is because an explicit expression for the output in terms of the function T(c~, fl) can be derived. The derivation proceeds as follows. From (3.357) and Fig. 3.86, we find n(t)

f(t)- Ev(Mfk_l)ffR k=l

fl) dadfl 2k-1

n(t)

P l,

v(M~k) [[.. k=0

/~(~, fl) de aft.

(3.371)

,J,J l,(2k

Owing to the definition of T(a, fl), we have

fiR

#(0~,fl) da dfl

- T(Mk+I, ink) -- T(Mk, ink).

(3.372)

2k-1

Similarly, we find

ffR #(~' fl) d~ dfl 2k

- T(Mk+I, m k + l ) -- T(Mk+I, mk),

(3.373)

CHAPTER 3 Scalar Preisach Models of Hysteresis

420

where it is assumed that

T(M1, m0) = 0.

(3.374)

By substituting (3.371) and (3.372) into (3.370), we derive n(t)

f(t) = y~ v(Mfk_l)[T(Mk+l, mk) -- T(Mk, mk)] k=l

n(t)

- ~ v(Md2k)[T(Mk+l, mk+l) -- T(Mk+I, mk)].

(3.375)

k=0

This is the final expression for the output in terms of the function T which is directly related to the experimental data by the formula (3.368). We will conclude this section by the discussion of one sufficient condition which guarantees the convergence of the stabilization (accommodation) process. This condition is given by the following inequality:

maxlv'(Md)l .mffx ffR MY

#(a, fl) da dill - q < 1,

(3.376)

where vI stands for the derivative of v-function, and R is any rectangle within the limiting triangle. The proof that the condition (3.375) guarantees the convergence of the stabilization process is given below for the simplest case. This is the same case that was discussed in the beginning of this section when we proved that the model (3.345) did account for accommodation. More general cases can be treated similarly. From (3.357), we find 2k+1 -- --v(f_)

#(~, fl) de dfl 0

- v(M;)//R

l~(e, fl)dedfl + r(u+),

(3.377)

+

where the notation R+ is used for R2k because they do not change from one cycle to another. Similarly we have M2fk = v ( M f - 1 ) f f R _

#(~' fl) dadfl

- v(f_) [[_ #(a, fl)dadfl + r(u_), d,I R 0

where R_ stands for R2k+l.

(3.378)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

421

From (3.376) and (3.377), we derive Mf2k+l--Mf2k_l----[v(Mf2k)--v(Mf2k_2)]

x [[_

/~(0r fl) dc~dfl,

(3.379)

JJN +

Mf2k --Mf2k_2--[v(Mf2k_l) - v(M2fk_3)]

x fro

fl) de dfl.

(3.380)

J,JN

It is apparent that

]v(Mf2k) -- v(Mf2k_2) l<~max v' (Mf ) ]lMf2k - Mf2k_21, Mf

(3.381)

]v(Mf2k_l) -- v(Mf2k_3) l~ maxlv'(MJ) llM~2k_l -

(3 382)

Mf'

M 2k_31

From (3.375) and (3.378)-(3.381), we derive

[Mf2k+1 -- M ; _ 11~ q[Mf2k -- M;_21

(3.383)

]M; - Mf2k_21 <~qlMf2k_l - Mf2k_31.

(3.384)

By combining (3.382) and (3.383), we obtain ]M2fk+l _

Mf2k_l 1<~q2 IMf2k_l _ Mf2k_3 ],

IM;+2 - M ; I ~

q2[M; - Mfk-2

["

(3.385) (3.386)

Inequalities (3.384) and (3.385)mean that extrema {Mfk+l } and {Mfk } form two contracting sequences. For this reason, there exist the limits: lim M f = Mf+ k--~cx~ 2k+1

(3.387)

lim Mf2k = Mfu_. (3.388) k--~ Consequently, the condition (3.375) guarantees the convergence of the accommodation process. 3.2.6 MAGNETOSTRICTIVE HYSTERESIS AND PREISACH MODELS WITH TWO INPUTS

The essence of the magnetostrictive phenomenon is the dependence of strain on magnetization. Since the magnetization can be varied by applied magnetic fields, this opens the opportunity to control the strain of

CHAPTER 3 ScalarPreisach Models of Hysteresis

422

magnetostrictive rods by magnetic fields. For this reason, magnetostrictive materials (especially Terfenol-type materials with giant magnetostriction) are very attractive as actuator materials for many applications, such as robotics, active vibration damping, micromotors, etc. However, magnetostrictive materials exhibit hysteresis, which represents a problem for fine-positioning applications. If hysteretic effects of magnetostrictive materials could be predicted, then actuator controllers could be designed to correct for these effects. This would result in high precision actuators powered by magnetostrictive materials. It is clear that mathematical models for magnetostrictive hysteresis could facilitate the design of the above controllers. The Preisach modeling of magnetic hysteresis of Terfenol-type materials was first attempted in [52]. It is known that magnetostrictive materials exhibit hysteretic behavior with respect to variations of two variables: magnetic field and stress. Thus, the problem of developing the Preisach-type models with two inputs presents itself. For the sake of generality, we consider a hysteresis nonlinearity that can be characterized by two inputs u(t) and v(t) and an output fit). In magnetostriction applications u(t) is the magnetic field, v(t) is the stress, whilef(t) is the strain. We shall discuss the following Preisach-type model with two inputs:

fit) = ['[" la(~, fl, v(t))~aflu(t)d~ dfl ,Ida

+ f~

v(a, fl, u(t))~aflv(t)dadfl.

(3.389)

In the above model, the dependence of the functions # and v on v(t) and u(t), respectively, reflects the cross-coupling between two inputs. The investigation of the model (3.388) is greatly facilitated by its equivalent representation as a model with moving supports for [a(a, fl, v(t)) and v(cr fl, u(t)). To arrive at this representation, we introduce the sets + + . . Su( t) , Ru(t), Su(t), Sv(t), Rv(t) and Sv(t), which can be defined by the same formulas as (3.150)-(3.152). By using these sets, the expression (3.388) can be transformed as follows:

fit)- ffR

u(t)

+ JJs II

#(~' fl' v(t))ya~u(t) dec dfl + ffI~

u+(t)

JJs v+(t)

II

J J S-u(t)

J J Sv(t)

v(t)

v(~, fl, u(t))~&~v(t) da dfl

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

423

We next find a simple expression for the last four terms in (3.389). Consider first a state of negative saturation and a subsequent monotonic increase of inputs until they reach some values u(t) and v(t) 9 Letfu(t)v(t + ) be the resulting output value. Then, according to (3.389), we find:

--

f

ff~

,Idle, u(t)

+ dds II

#(~, fl, v(t))d~

aft- [1~

e I Kv(t)

,, v(t))dadfl

-

u+(t)

II

v(~, fl, u(t))da

dfl

vt , ,, u(t))dadfl

J,]Su(t)

+ ff v(a, fl, u(t)) dadfl dds v~t)

-

ff

JJSv(t)

v(a, fl, u(t)) dad//. (3.391)

Similarly, starting from the state of positive saturation, we can definef~t)v(t) and derive the following expression: --

f~t)v(t)

If l~(a, fl, v(t)) dadfl ddRu(t)

+

fl v(a, fl, u(t)) dadfl d,I Rv(t)

+ ff l~(a,fl, v(t))dad//- ff l~(a,fl, v(t))dadfl dis u+(t) JJS~(t) + ff v(a, fl, u(t)) dadfl - ff v(a, fl, u(t)) dad//. JJS v(t) + ,],Y gu(t) (3.392) From (3.390) and (3.391) we derive

-21 +(fu(t)v(t) + f~t)v(t)) -- ~

Su+(t)

#(~, fl, v(t)) dadfl -

/s u(t)

~(~, fl, v(t)) dadfl

+ f v(~, fl, u(t))dadfl- f v(a, fl, u(t))d~dfl. d.qSv+(t) J Sv(t) (3.393) By substituting (3.292) into (3.389), we arrive at the following representation of the model (3.388):

fit)

-

ff

fl. v(t))~aflu(t)dadfl

JJ l
+

fro v(=.fl. u(t))~,flv(t)dadfl

d d tKv(t)

+

if- -2 (fu(t)v(t) "[- f~t)v(t))"

The last expression can be regarded as a generalization of (3.160).

(3.394)

424

CHAPTER 3 Scalar Preisach Models of Hysteresis

(U2,V2)

(u~ ,v~) ,-v FIGURE 3.92

By using the same line of reasoning as in previous sections, it can be established that the model (3.393) has the following two characteristic properties. WIPING-OUT PROPERTY. Only the alternating series of past dominant extrema of

u(t) and v(t) are stored by the model (3.393), while all other past extrema of u(t) and v(t) are wiped out. PROPERTYOF EQUALVERTICALCHORDS. All

minor hysteretic loops corresponding to the same consecutive extremum values of u(t) for the same fixed value of v have equal vertical chords regardless of the past history of variations of u(t) and v(t). The same is true for minor hysteretic loops formed as a result of back-and-forth variations of v(t) for any fixed value of u. It turns out that the model (3.393) has also the following distinct property. PATHINDEPENDENCEPROPERTY. Consider two points (Ul, Vl) and (u2, v2) on the u-

-v plane and a set of paths connecting these points and corresponding to monotonic variations of both u(t) and v(t) (see Fig. 3.92). Then, the output increment predicted by the model (3.393) does not depend on a particular monotonic path between the points (Ul, Vl) and (u2, v2). PROOF. For any instant of time the model (3.393) can be represented as

fit) =/~S+(t ) p(a, fl, v(t)) da d f l - /~S-(t) #(a, fl, v(t)) dadfl

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

q- fff2 + (t)

v(oe, fl, u(t)) doe dfl -

~ff2-(t)

425

v(oe, fi, u(t)) doedfl

1 + q- -2 G(t)v(t) if- fff((t)v(t))"

(3.395)

Here S+ (t), S- (t), f2+ (t) and f2- (t) are positive and negative sets on which Ru(t) and Rv(t) are subdivided, respectively. It is clear that the above subdivision on positive and negative sets will be the same at the point (u2, v2) for all monotonic paths connecting this point with (Ul, Vl). This is because the monotonic variations of u(t) and v(t) will result in the same modification of negative and positive sets upon achieving the values u2 and v2, respectively. From here it is easy to conclude the validity of the above property. We next proceed to the discussion of the identification problem for the model (3.393). In order to determine the functions ~(oe, fl, v) and v(oe, fl, u) we shall use the first-order transition curves fa~v and fua/~. As before, these curves are measured for piecewise monotonic input variations started from the state of negative saturation. Namely, f~v is the output resulting from monotonic increases of two inputs to the values oeand v, respectively, and subsequent monotonic decrease of u(t) to the value ft. The output value fu~ can be determined in a similar way. We shall define the functions 1

F(oe, fl, v ) - ~ (f~/~v -ffl+), 1

p, u)=

-&).

(3.396) (3.397)

By using (3.393) and the diagram technique, it is easy to show that ,

v) -

(~,fl)

fl', v)doe' dfl'

= ffla(fflila(oe',fl',v) dfl')doe',

(3.398)

G(oe, fl, u) - ffR (~,~) v(oe', fl', u)doe' dfl' - ffla (ffl/~ v(oe', fl', u)dfl')doe',

(3.399)

0

where R(oe, fl) is the rectangle shown in Fig. 3.93. From (3.397) and (3.398) we derive 02F(oe, fl, V) ~(~,/~, v)=

a a,

(3.400)

CHAPTER 3 Scalar Preisach Models of Hysteresis

426

(a,~)

/,

R(O~,I~) I

FIGURE 3.93

v(a, t, u) =

c~2G(0r t, u)

.

(3.401)

As far as the numerical implementation of the model (3.393) is concerned, it can be accomplished without the formulas (3.398) and (3.400) but by employing explicit expressions for the integrals in (3.393) in terms of the functions F(~, t, v) and G(a, t, u). The derivation of these expressions proceeds as follows. For the first integral in (3.393) we find

([ #(a, t, v(t))~u(t) dadfl ddRu(t)

= 2 [[

JJS+(t)

#(a, t, v(t)) dadfl

-//R u(t)

/a ( a , fl, v ( t ) ) d a d fl.

(3.402) Similarly,

//R

u(t)

v(~, fl, u(t))4&~v(t) d~ dfl = 2 //~

+(t)

v(a, fl, u(t)) dadfl

-- /fav(t, V(~, j~, u(t)) da dfl. (3.403) From (3.390) and (3.391), we derive

1+ -~(l:u(t)v(t)- f ~t)v(t)) = -

u(t)

#(o~, [3, v(t)) do~aft

- ffRv(t~ v(o~,fl, u(t))de aft.

(3.404)

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

427

(X

s-It)

I

1

,

FIGURE

3.94

By substituting (3.401) and (3.402) into (3.393) and taking (3.403) into account, we obtain

f(t) -- 2[ f~s+(t) #(e, fl, v(t))

dedfl +

ffn

+(t) v(e,

fl, u(t)) de dfl]

+

(3.405)

ff- fu(t)v(t)

A typical geometry for the set S+ (t) is shown in Fig. 3.94. From this figure follows

n(t)

ffs

l~(e, fl, v(t)) dedfl +(t)

= ~ k=l

ffR fl'v(t)) dedfl.

(3.406)

k

By using (3.397), we find

ffa #(e, fl, v(t)) de dfl = F(M~u), m~u), v(t)) - F[M(u)v,,k+l,m~u)v(t))'

(3.407)

k

where M~u) and m~u) form the alternating series of past dominant extrema of u(t). From (3.405) and (3.406) we conclude

f +(t)f #(e,sfl, v(t)) de dfl n(t)

= z..,LF,M, u) , m k(u), v(t))- F(M~U)_F1 , m~u) , v(t))]. k=l

(3.408)

CHAPTER 3

428

Scalar Preisach Models of Hysteresis

By using the same line of reasoning as before, we can derive the expression

fff~

+(t)

v(~, fl, u(t)) d~dfl

l(t)

= ~ [ G ( M k (v), m~v) , u(t)) - Gt^/t(V)v,.k+X,m~v) , u(t))] .

(3.409)

k=l

By substituting (3.408) and (3.407) into (3.404), we obtain (t)

+ 2 ~ [ F ( M ~ u), mp ) , v(t)) - F(M~U)+I,m~u) , v(t))] fit) --fu(t)v(t) + k=l

t(t)

+ 2 y~[G(M~ v) , m k(v) ,u ( t ) ) - G ( M 2 1 , m k(v) ,u (t))].

(3.410)

k=l

This is the final expression for the output in terms of experimentally measured functions F and G defined by (3.395) and (3.396), respectively. We next formulate the following result. REPRESENTATIONTHEOREM. The wiping-out property, the property of equal vertical

chords and the path independence property constitute the necessary and sufficient conditions for the representation of actual two-input hysteresis nonlinearities by the model (3.393) on the set of piecewise monotonic inputs. The proof of this theorem closely parallels the proof of the representation theorem from Section 3.2.2. The only new element is the utilization of the path-independence property. By using this property, it can be easily shown that an arbitrary simultaneous piecewise monotonic variation of u(t) and v(t) can be reduced to two consecutive piecewise monotonic single-input variations during which the other input remains constant. Further details of the proof of the theorem are left to the reader. In conclusion of this section, we consider one peculiar property of strain hysteresis in comparison with magnetic hysteresis. This peculiarity stems from even symmetry of strain hysteresis which exhibits butterflyshaped major hysteretic loops with respect to variations of magnetic field (see Fig. 3.95). It is clear that the first-order transition curvesf~v will also exhibit even symmetry. This can be expressed mathematically as

f~'~'v - fa~v

and f~v - f~+

(3.411)

if fl'--a

and

a'--fl.

(3.412)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

429

f

'--.

.

.

.

.

FIGURE 3.95

By using (3.410) and (3.411), it can be easily proven that

~(a,/~, v) = - ~ ( - f l , - ~ ,

v).

(3.413)

Thus, the function #(a, fl, v) is of odd symmetry with respect to the line = -ft. In the case of magnetic hysteresis, the/2-function has an even symmetry with respect to the same line. It follows from (3.412) that the function/2(a, fl, v) cannot be positive everywhere on the a-fl plane. Actually, this function changes its sign across the line ~ = -ft. As a result, first-order and higher-order transition curves are not enclosed by major hysteretic loops. This property has been experimentally observed and it also distinguishes strain hysteresis from magnetic hysteresis. 3.2.7 EXPERIMENTAL TESTING OF PREISACH-TYPE MODELS OF HYSTERESIS

In our previous discussions of Preisach-type models of hysteresis we consistently tried to establish the necessary and sufficient conditions under which actual hysteresis nonlinearities can be represented by these models. These conditions establish the limits of exact applicability of Preisach-type models. As a result, these conditions allow one to judge to what extent one or another Preisach-type model is applicable. These conditions also clearly reveal the phenomenological nature of Preisach-type models and

430

CHAPTER 3 Scalar Preisach Models of Hysteresis

their physical universality. However, it is unrealistic to expect that actual hysteresis nonlinearities will satisfy these conditions exactly. For instance, the experimental results presented in [53] show that neither the congruency property nor the property of equal vertical chords of comparable minor loops are satisfied for some typical materials used in magnetic recording. For this reason, the classical and 'nonlinear' Preisach models, as well as some other Preisach-type models, cannot be expected to be absolutely accurate. However, we may try to use these models as approximate ones. Then the question arises how to judge the accuracy of these models. It has been emphasized throughout our discussion that branching is the essence of hysteresis. Consequently, any mathematical model of hysteresis is expected to predict this branching. In other words, models of hysteresis are expected to predict higher-order reversal curves (branches) which occur after several (or many) reversals of input. This prediction should be based on some limited experimental data used for the identification of hysteresis models. Thus, the accuracy of the classical, 'nonlinear' and 'restricted' Preisach models can be measured by their abilities to predict higher-order reversal curves on the basis of information provided by the first- and second-order reversal curves. This is exactly the criterion that has been chosen for the experimental testing of the accuracy of the Preisach-type models. This testing was first performed for two typical particulate materials: Co-coated ~;-Fe203 and iron magnetic-tape materials. The experimental work consisted of two main stages. During the first stage, the first- and second-order transition curves were measured by using a vibrating sample magnetometer. Experimentally measured first-order transition curves for the above two materials are shown in Figs 3.96 and 3.97, respectively. For each first-order transition curve there is a set of second-order transition curves which are attached to the above first-order curve. Experimentally measured sets of second-order transition curves are shown in Figs 3.98 and 3.99, respectively. These first- and second-order transition curves were used for the identification of the classical, 'nonlinear', and 'restricted' Preisach models. During the second stage of the performed experiments, several sets of higher-order (with the order up to six) transition curves were measured. The sequences of field reversal for these higher-order transition curves were chosen in such a way that these higher-order transition curves were more or less uniformly (and densely) distributed over the area enclosed by major hysteresis loops. This was important in order to get the information about uniform accuracy of Preisach-type models. Then, by using the developed computer software, which implements the Preisach-type models, various sets of higher-order transition curves predicted by the

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

431

.04

.02

-.02

-.04

-2000

-1500

-1000

-500

0

500

1000

1500

2000

FIGURE 3.96

Preisach-type models were computed for the same sequences of field reversals as in the experiment. The comparison of the higher-order transition curves predicted by the classical Preisach model (3.149) with the experimentally measured higher-order transition curves is demonstrated in Figs 3.100 and 3.101. Similar comparisons for the nonlinear Preisach model (3.174) and the restricted Preisach model (3.234) are shown in Figs 3.102 and 3.103. It is evident from these figures that the actual (experimentally measured) transition curves are 'sandwiched' between the predictions of the nonlinear and restricted Preisach models. This observation has prompted the idea to use an 'average' (superposition) model

1//

fit) -- -~

4-

[#(~, fl, u(t)) 4- #(a, fl, M1)]~a]~(t) d~ d]~

4

"

(3.414)

CHAPTER 3 Scalar Preisach Models of Hysteresis

432

1L

0.-

-.05-

-Z

: -,!

-3000

-2000

-tO00

0

1000

2000

3000

FIGURE 3.97

The numerical implementation of this model can be carried out by using the formula 1 n(t) fit) - f - + f~t) + ~.[P(Mk+I mk u(t)) -- P(Mk mk, u(t))] 2 -2 ' ' ' k=l

n(t)

+ ~ [ F ( M k , mk-1, M1) - F(Mk, ink, M1)],

(3.415)

k=l

which follows from (3.214) and (3.261). It is clear that the 'average' model inherits the common wiping-out property of the 'nonlinear' and 'restricted' models. Functions/~(~, fl, u) and ~(~, fl, M1) can be individually determined by matching first- and second-order reversal curves exactly as in the case of those models. On the other hand, the 'average' model further relaxes the congruency property of the 'restricted' Preisach model and the property of equal vertical chords of the nonlinear model. Namely, the 'average' model describes hysteresis nonlinearities whose minor loops have

,;,,,~

,~' o.

3.2

.

I

N o

.

~,,,~

GENERALIZED

~,

o

.

(nuIa) UOl",,~z.t:laul~l~

oN

,

~

SCALAR

i

~

9

,

PREISACH

~

o.

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MODELS

.o I"

o.

I

O 0

O O

I

-o

e2

o

OF HYSTERESIS

I"

o,,. I

~o

I

?,

9

(nuaa) uo!~z!lau~eR

"

o~o

q

-'-T-

o

(nu,te) uo~.~,~zr~au~l~

~.

433

434

,,-:

~ q (nu,.m)

(n~e)

o

uo1"*,mzr

uol~ez;'~uhR

i

,~ o

i

9

,-, ,"

i

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CHAPTER 3

I

o

,-:

-

.

i|i|

o

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~

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w|l!

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Scalar Preisach Models of Hysteresis

ill|

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(nt.u~) uol1,~zr~u3ux

~n

i i

9

(nuze) uo!'~zp,~uleR

q

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9

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I' I

I

I

l

I

NoLDrZLT~ND'C'R

I 'C

t"

I

i

l

1

I

o

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!

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c~

iI

o

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i

'

f 0

''

I

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(n=a)

"

~ (nuaa)

9

g

I

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NOU.Y'ZLL,3NOYR

o

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g

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"o NOLI,V'ZLT~N DV'R

'

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oi

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

I

' i ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 '

~

. 1

.

, (nru=~) NOU,Ir"ZU,3NDYR

435

436

....

I ....

~i

(ntua) NOIJ.V21J~N~VM

q

' ' l ' ' ~ ' ' l " ' ' " l

_

9

''1

CHAPTER 3

'

tli

I ' ' t ~

t"

I ....

'I ....

I ....

o

I ~'

f

i

"

i

Scalar Preisach Models of Hysteresis

....

.

(nma) NOIJ,VZI.~NgYI~

'_

.

I

.

.

NOIJ.Y"LLL3NDYI'I

| f

I

. ~

!t

.....

!-' 9

1 ~ 1 7 ,

'

,

'

,

I

;

,

, i"

1

NOIJ.Y'ZI.L.qNOYI~

I

( n m ~ ) NOIJ.V'ZLI.3.qDYN

(nmo)

"

1'1',

,

,

,

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

t-'

x

(nm~)

I

( n m a ) NOIJ.Y'ZIJ,,.qNDYR

I i

I

,

o

- o

437

438

1

, ~,

i

"7

I

~

,

, ~l

i

,,,,gl,Ill

,o o.

, t , ,

(ntn~)

9

(nm~)

o

! ....

, I ....

,i

CHAPTER

o

5'

~'

~

t

I,

!

.-.

I ,

I ....

|

I ....

9 I"

o NOI.L'rZLI~i~I~

3

L

Scalar

-

z

~ I

~.

Models

NOl~ll~tl~'CM

I'

~

NOI.LWZII,~NDWI

Preisach

ul

"

(nma)

~

o

..................

(nma)

~

~~ \\~

i o

~

of

Hysteresis

7

o

o o

o

oi I

/t~

~

~\\\ ~o~

.

,

~ 5

.

~,,,

(nmo)

(nine)

1,,,

.

t

1,,,

0 I

N OI.l.'krz LL;3 INIDVIW

.

NOI.I,YZLI.~NDYI~

.

i

.

~ ,

I

r~-

2

o

i

:I

i

IF'' i ....

gl

Ir

I ~ I ' P , , I ,

l~.

i Li~ . (nuae)

,

I,

~

,

i

i

~

439

0 .-.

o~

'~I ~

O. I

I , , , 1 , | I INIOLI,Y'ZI,I,..~NDY]I~

~

3.2 GENERALIZED S C A L A R PREISACH MODELS OF HYSTERESIS

i , ~ , , ,

R ~

E ~ FR L~

L;'

.

C~

[.,T.1

I,-,nq

440

i

''1''''1''''1''"'!''''1''

i

I" (nu~a) HOLLYZIJ.3NDV~

'

z

''l''''!

-

-

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o

8

.r

I''''I''~'IT~-I~

..,

~

!

P.

~

~=I

~,

Scalar Preisach Models of Hysteresis

....

i

i" NOM.~LI~.~NDYI~

(.u~a) ~otLvzu~xov~

"

i

(nuar

.., 9

i

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3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS 0.06

'

1

'

'

'

'

I

'

'

'

'

I

'

441

-

0.04 0.02 o

-0.02 -0.04 -0.06

-0.6

0 H [KOe]

.6

FIGURE 3.106

equal vertical chords only if they are formed after the same largest input maximum M1 was achieved. In other words, the property of equal vertical chords becomes history-dependent. It is clear that the average model relaxes much further the congruency property of the classical Preisach model than the 'nonlinear' or 'restricted' models. At the same time, the average model requires the same experimental data for its identification as the two aforementioned models. The 'average' model turned out to be remarkably accurate for all used (and densely distributed) sequences of reversal values of magnetic fields. The comparison of the higher-order transition curves predicted by this model with the experimentally measured higher-order transition curves is demonstrated in Figs 3.104 and 3.105. The above comparison suggests a higher accuracy of the 'average' Preisach model. This prompted the attempt to test this model for other magnetic materials. This testing was performed for six different magnetic-tape materials: Ampex641, Ampex-797, Ampex-D1, TDK-YHS, Maxell-Beta, and Memorex-3480. Sample results of the comparison of the higher-order transition curves predicted by the 'average' model with the experimentally measured higherorder transition curves are demonstrated for the above materials in Figs 3.106-3.111, respectively. Thus, it can be concluded that the extensive experimental testing presented above has revealed a remarkable accuracy of the 'average' Preisach model of hysteresis. It is believed that this accuracy is related to the fact that the average model provides the far-reaching generalization of the congruency property of the classical Preisach model. Actually, this generalization goes much further than in the cases of 'nonlinear' and 'restricted' Preisach

442

CHAPTER 3

.a.

,

,

0.02

I

'

'

'

S c a l a r P r e i s a c h M o d e l s of H y s t e r e s i s

'

I

'

'

'

'

I

Exp. -

Model

0.01 ~E

0

-0.01 -0.02 -0.5

0 H [KOe]

FIGURE 3.107 '

0.01 _- -

;

'

'

I

'

;

;

;

I

;

;

'

'

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o

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-2

-1

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0.02

Exp. Model

0.01 o

0

-0.01 -0.02 -1

0 H [KOe] F I G U R E 3.109

1

'

'

;-

REFERENCES

443 __

'

'

I

'

'

'

'

I

'

'

'

'

I

'

'

4

0.02 0.01 o

-0.01 -0.02 -1

0 H [KOe]

1

F I G U R E 3.110 '

0.02

'

'

I

'

'

'

'

I

'

'

'

'

I

I

I

I

'

'

'

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I

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0.01 ,~, o

-0.01

-0.02 -1

0 H [Koe]

1

F I G U R E 3.111

models, although the average model needs the same experimental data for its identification as the two aforementioned models.

References 1. H. Poincar6, Science and Hypothesis, Dover, New York (1952). 2. F. Z. Preisach, Ober die magnetische nachwirkung: Zeitschrift f~ir Physik, 94, 277-302 (1935). 3. L. N6el, Sur les effects d'un couplage entre grains ferromagn6tiques dou6s d'hyst6r6sis, Comptes rendus hebdomadaires des seances de 1 academie des sciences, 246(16), 2313-2319 (1958). 4. G. Biorci and D. Pescetti, Analytical Theory of the behaviour of ferromagnetic materials, Nuovo Cimento 7(6), 829-842 (1958).

444

CHAPTER 3 ScalarPreisach Models of Hysteresis

5. G. Biorci and D. Pescetti, Some consequences of the analytical theory of the ferromagnetic hysteresis, Journal de Physique et le Radium 20(2-3), 233-236 (1959). 6. G. Biorci and D. Pescetti, Some remarks on hysteresis, J. Appl. Phys. 37(1), 425-& (1966). 7. W.F. Brown Jr, Failure of local-field concept for hysteresis calculations, J. Appl. Phys. 33(3), 1308 (1962). 8. G. Bate, Statistical Stability of preisach diagram for particles of gammaFE203, J. Appl. Phys. 33(7), 2263 (1962). 9. J. G. Woodward and E. Della Torre, Particle interaction in magnetic recording tapes, J. Appl. Phys. 31, 56-62 (1960). 10. E. Della Torre, Measurements of Interaction in an Assembly of ?-Iron Oxide Particles, J. App1. Phys. 36(2), 518-522 (1965). 11. J.A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and minor loops -models and experiments, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 386 (1791), 251-261 (1983). 12. A. Damlanian and A. Visintin, A Multidimensional generalization of the preisach model for hysteresis, Comptes rendus de 1 academie des Sciences serie I-Mathematique 297(7), 437-440 (1983). 13. D. H. Everett and W. I. Whitton, A General Approach to hysteresis, Trans. Faraday Soc. 48(8), 749 (1952). 14. D.H. Everett, A General approach to hysteresis. 3. A Formal treatment of the independent domain model of Hysteresis, Trans. Faraday Soc. 50(10), 1077-1096 (1954). 15. D. H. Everett, A General approach to hysteresis. 4. An Alternative formulation of the domain model, Trans. Faraday Soc. 51(11), 15511557 (1955). 16. J.A. Enderby, The domain model of hysteresis. 2. Interacting domains, Trans. Faraday Soc. 52(1), 106-120 (1956). 17. A. Friedman, Foundation of Modern Analysis, Dover, New York (1982). 18. M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis, Nauka, Moscow (1983). 19. MI Brokate, Some Mathematical properties of the preisach model for hysteresis, IEEE Trans. Magn. 25(4), 2922-2924 (1989). 20. M. Brokate and A. Visintin, Properties of the preisach model for hysteresis, J. Reine Angew. Math. 402, 1-40 (1989).

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445

21. M. Brokate and A. Friedman, Optimal-design for heat-conduction problems with hysteresis, SIAM J. Control Optim. 27(4), 697-717 (1989). 22. A. Visintin, On the preisach model for hysteresis, Nonlinear Analysistheory methods & Applications 8(9), 977-996 (1984). 23. A.A. Visintin, A model for hysteresis of distributed systems, Ann. Mat. Pura Appl. 131, 203-231 (1982). 24. A. Friedman and K.-H. Hoffmann, Control of free-boundary probles with hysteresis, SIAM J. Control Optim. 26(1), 42-55 (1988). 25. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proceedings of the National Academy of Sciences of the United States of America-Biological Sciences 79(8), 2554-2558 (1982). 26. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, Wiley, New York (1961). 27. R. deGroot and P. Mazur, Non-Equilibrium Thermodynamics, NorthHolland, Amsterdam (1963). 28. W.G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York (1965). 29. G. Bertotti, Energetic and thermodynamic aspects of hysteresis, Phys. Rev. Lett. 76, 1739-1742 (1996). 30. G. Bertotti, Hysteresis in Magnetism, Academic Press, Boston (1998). 31. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, New York (1991). 32. I.D. Mayergoyz, Mathematical-Models of Hysteresis, Phys. Rev. Lett. 56 (15), 1518-1521 (1986). 33. I. D. Mayergoyz, Hysteresis models from the mathematical and control-theory points of view, J. App1. Phys. 57(8), 3803-3805 (1985). 34. I. D. Mayergoyz, Mathematical-Models of hysteresis, IEEE Trans. Magn. 22(5), 603-608 (1986). 35. T. Doong and I. D. Mayergoyz, On numerical implementation of hysteresis models, IEEE Trans. Magn. 21(5), 1853-1855 (1985). 36. I.D. Mayergoyz and G. Friedman, The preisach model and hysteretic energy-losses, J. App1. Phys. 61(8), 3910-3912 Part 2B (1987). 37. I. D. Mayergoyz and G. Friedman, Generalized preisach model of hysteresis, IEEE Trans. Magn. 24(1), 212-217 (1988). 38. I. D. Mayergoyz, Dynamic preisach models of hysteresis, IEEE Trans. Magn. 24(6), 2925-2927 (1988).

446

CHAPTER 3 ScalarPreisach Models of Hysteresis

39. I. D. Mayergoyz, G. Friedman, and C. Salling, Comparison of the classical and generalized preisach hysteresis models with experiments, IEEE Trans. Magn. 25(5), 3925-3927 (1989). 40. I. D. Mayergoyz, G. Friedman, and A. A. Adly, Superconducting hysteresis and the preisach model, J. Appl. Phys. 67(9), 5466-5468 Part 2B (1990). 41. I. D. Mayergoyz, The Classical Preisach model of hysteresis and reversibility, J. Appl. Phys. 69(8), pp. 4602-4604 (1991). 42. A. A. Adly, I. D. Mayergoyz, and A. Bergqvist, Preisach modeling of magnetostrictive hysteresis, J. Appl. Phys. 69(8), 5777-5779 Part 2B (1991). 43. G. Friedman and I. D. Mayergoyz, Input-dependent preisach model and hysteretic energy-losses, J. Appl. Phys. 69(8), 4611-4613 Part 2A (1991). 44. A.A. Adly and I. D. Mayergoyz, Experimental testing of the Average preisach model of hysteresis, IEEE Trans. Magn. 28(5), 2268-2270 Part 2 (1992). 45. I. D. Mayergoyz and A. A. Adly, Numerical implementation of the feedback preisach model, IEEE Trans. Magn. 28(5), 2605-2607 Part 2 (1992). 46. I. D. Mayergoyz and P. Andrei, Preisach modeling of clockwise hysteresis and its application to front propagation problems, J. Appl. Phys. 91(10), 7645-7647 Part 2 (2002). 47. K. Wiesen and S. H. Charap, A better scalar preisach algorithm, IEEE Trans. Magn. 24(6), 2491-2493 (1988). 48. G. Friedman and K. Cha, Power series expansion of hysteretic processes with wipe-out memory, Journal of Material Processing and Manufacturing Science 9 (1), 70-78 (2000). 49. G. Friedman and K. Cha, Representation of hysteresis with wipe-out memory, J. Appl. Phys. 89(11), 7236-7238 Part 2 (2001). 50. G. Friedman, Second-order Preisach model of scalar hysteresis, Physica B 275 (1-3), 173-178 (2000). 51. E. Della Torre, Magnetic Hysteresis, IEEE Press, Piscataway (1999). 52. J. B. Restorff, T. H. Savage, A. E. Clark, and M. Wum-Fogle, Preisach modeling of hysteresis in terfenol, J. Appl. Phys. 67(9), 5016-5018 Part 2A (1990). 53. C. Salling and S. Schultz, Using minor loop behaviour to investigate the preisach distribution for particulate media, IEEE Trans. Magn. 24 (6), 2877-2879 (1988).

CHAPTER 4

Vector Preisach Models of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Classical Stoner-Wohlfarth Model Vector Hysteresis Definition of Vector Preisach Models of Hysteresis and their Numerical Implementation Some Basic Properties of Vector Preisach Hysteresis Models Identification Problem for Isotropic Vector Preisach Models Identification Problem for Anisotropic Vector Preisach Models Dynamic Vector Preisach Models of Hysteresis Generalized Vector Preisach Models of Hysteresis. Experimental Testing References

4.1 C L A S S I C A L S T O N E R - W O H L F A R T H V E C T O R HYSTERESIS

447 458 467 485 494 504 511 526

MODEL

In magnetics, research on the modeling of scalar and vector hysteresis has been pursued along two quite distinct lines. Modeling of scalar hysteresis has been dominated by the Preisach approach. This approach can be traced back to a landmark paper [1]; it has been proved to be very successful and has won many followers. On the other hand, phenomenological modeling of vector hysteresis has long been centered around the classical Stoner-Wohlfarth model (S-W) model [2]. As a result, this model has further been developed and used in the area of magnetic recording [3-5]. The The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

448

CHAPTER 4 Vector Preisach Models of Hysteresis

attractiveness of the S-W can be attributed to its strong appeal to physical intuition. This appeal is, in turn, based on the fact that the S-W model is designed as an ensemble of single-domain, uniaxial magnetic particles. Since these particles have some features of physical realities, the S-W model is usually regarded as a physical (not mathematical) model. Owing to its popularity in magnetics, the S-W model is a natural benchmark for comparison with other vector hysteresis models. This is the main reason w h y we precede our discussion of vector Preisach models by the discussion of the S-W model. Since single-domain, uniaxial magnetic particles are the main building blocks of the S-W model, we begin with the discussion of hysteresis of these particles. We consider a single-domain, uniaxial magnetic particle with magnetization (magnetic momentum) M which may change its orientation under the influence of an applied field but has a constant magnitude. Such a particle is now commonly called a Stoner'Wohlfarth (S-W) particle. It is clear from the symmetry consideration that the vector M of this particle lies in the plane formed by the easy axis x and the applied magnetic field H (see Fig. 4.1). The orientation-dependent part of free energy g of the S-W particle is given by g = K sin 2 0 -/VI./7/,

(4.1)

where K is the anisotropy constant and 0 is the angle between the easy axis and M. The first term in the right-hand side of (4.1) represents the anisotropy energy, while the second term in (4.1) is the energy of interaction of magnetic m o m e n t u m M with the applied magnetic field. By using the Cartesian coordinates shown in Fig. 4.1, the expression (4.1) can be represented as g = K sin 2 0 - MHx cos 0 - MHy sin 0.

(4.2)

Equilibrium orientations of/VI correspond to minima of s and they can be found from the equations ~g --

~0 (~2g

=0,

(4.3)

~0 2 ~ 0.

(4.4)

2K sin 0 cos 0 + MHx sin 0 - MHy cos 0 - 0.

(4.5)

From (4.2) and (4.3) we derive

4.1 CLASSICAL STONER-WOHLFARTH MODEL VECTOR HYSTERESIS

449

M

H

,,

•

F I G U R E 4.1

By introducing the so-called switching field a: 2K - M'

(4.6)

the expression (4.5) can be rewritten as sin 0 cos 0 + Hx sin 0 - Hy cos 0 = O,

(4.7)

which is equivalent to sin 0

cos 0

= a.

(4.8)

Equation (4.7) (as well as (4.8)) is a quartic equation with respect to cos 0. For this reason, this equation m a y have two or four real solutions. Which of these two cases is realized depends on the applied magnetic field, H. In the first case, we have only one m i n i m u m and, consequently, only one equilibrium orientation of M. In the second case, there are two minima and this results in two equilibrium orientations of M. Thus, on the H-plane there are two different regions where M has one and two equilibrium orientations, respectively. On the b o u n d a r y between the above two regions one m i n i m u m and one m a x i m u m merge together. In the case of a minim u m , the inequality (4.4) holds, while for a m a x i m u m we Consequently, on the above b o u n d a r y we have 0g 0---0 = 0

have (~2~/~02~0.

02g and

~02

= 0

9

(4.9)

The first condition in (4.9) leads to Eqn. (4.8). Now, let us derive the second equation by using both conditions in (4.9). From (4.2), by using the notation (4.6), we derive

1

egg

sin 0 cos 0 00

= a -t

Hx

Hy

cos 0

sin 0"

(4.10)

CHAPTER 4 VectorPreisach Models of Hysteresis

450

Hy (Hxl ,

H

FIGURE 4.2

By differentiating (4.10), we obtain c9( 1 )egg 1 cgagHxsinOHycosO 0---0 sin 0 cos 0 ~ + sin 0 cos 0 002 = cos 2 0 -ff ~ ' s i n 2 0

(4.11)

By using both conditions (4.9), from (4.11) we find

HI

cos3---~ +

Hy sin 3 0

= 0.

(4.12)

Thus, on the boundary which separates regions with one and two minima, Eqns (4.8) and (4.12) are satisfied. These are two linear equations with respect to Hx and Hy. By solving these equations, we find

Hx =

- a c o s 3 (9,

Hy

-- 0r s i n 3 0.

(4.13)

From (4.13) we find the equation for the boundary separating the above two regions: Hx2/3+--y /_/2/3 _. 0~2/3 (4.14) This equation represents the astroid curve shown in Fig. 4.2. This astroid curve helps to visualize the solution of the quartic equation (4.7). This solution can be found by using the following geometric construction [6,7]. For a given external magnetic field H with components Hx and Hy the directions of M that satisfy Eqn. (4.7) are parallel to those tangent lines to the astroid that pass through the point H (see Fig. 4.2). The proof of the above statement proceeds as follows. Let Hx and Hy be Cartesian components of f/, and let Hxl and Hy1 be Cartesian coordinates of the point on the astroid at which the abovementioned tangent line touches the astroid. For the slope of this tangent

4.1 CLASSICAL STONER-WOHLFARTH MODEL VECTOR HYSTERESIS 451 line we have m=

I-ly1 - I-Iy Hx~ - H ~ '

(4.15)

which is tantamount to Hy 1 - Hy = m(Hxl - Hx).

(4.16)

Since the point (Sx1 , S y 1) belongs to the astroid, we have 2/3

/_/2/3 __ ~2/3

(4.17)

By using implicit differentiation of (4.17) with respect to Hxl, we derive 2 H~11/3 + 2-H,,--11/3 dHyl = 0 -3 3-dHxl "

(4.18)

From (4.18) we obtain m ~-~

ally 1 Hy I 1/3 " dHxl = - - ( ~ )

(4.19)

L e t / / b e the angle formed by the above tangent line with the easy axis of the particle. Then, m = tan//, (4.20) and from (4.19) and (4.20) we find HYl -- - tan 3 ft.

(4.21)

Hx I

Expressions (4.17) and (4.21) can be construed as two simultaneous equations with respect to two unknowns: Hxl and Hy 1. By solving these equations, we obtain Hxl -- - ~ c o s 3 fl,

Hy 1 -- cxsin 3 ft.

(4.22)

By substituting (4.20) and (4.22) into (4.16), we find cxsin 3 f l - Hy = tan/J(-cx cos 3 f l - Hx).

(4.23)

A trivial transformation leads to 0r

3 fl cos/J + cos 3 fl sin fl) q- Hx sin fl - Hy cos fl - 0,

(4.24)

CHAPTER 4 VectorPreisach Models of Hysteresis

452

Hy 6

a

Hx

FIGURE 4.3

which can be simplified as sin fl cos fl 4- Hx sin fl - Hy cos fl = 0.

(4.25)

It is clear that Eqns (4.7) and (4.25) are identical. Consequently, fl = 0.

(4.26)

This proves the validity of the above-described geometric construction. It can be shown that equilibrium orientations of M correspond to the tangent lines with smallest slopes. It is clear that, when the point H is outside the astroid, only two tangent lines_,are possible, and therefore there is only one equilibrium orientation of M. When H is inside the astroid, there are four tangent lines. However, only two of these tangent lines represent equilibrium orientations of the magnetization M. Which one is realized depends on the previous history of the magnetization. The described geometric rules allow one to compute hysteresis loops of a S-W particle for the case when the applied magnetic field is restricted to vary along one arbitrary chosen direction. Suppose that this direction is specified by the line a-a' (see Fig. 4.3) and that the magnetic field is first monotonically increased from its value H_ corresponding to point I to the value H+ corresponding to point 6 and then is monotonically decreased back to H_. The dependence of the magnetization projection along the line a-a t on the value of the magnetic field H exhibits hysteresis that is shown in Fig. 4.4. This is clear from Fig. 4.3. Indeed, as we move up along the line a-a t, the equilibrium (stable) orientations of M coincide with directions of tangent lines to the right-hand side of the astroid until we reach point 5.

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 453 Ma_a,

H 8-8'

/

FIGURE 4.4

At this point a 'switch' from the right-hand side to the left-hand side of the astroid occurs and, after that point, stable orientations of M coincide with directions of tangent lines to the latter part of the astroid. As we move down along the line a - a I from point 6, we use the tangent lines to the lefthand side of the astroid to determine the stable direction of/~'i. However, at point 2 a 'switch' from the left-hand side to the right-hand side of the astroid occurs and, after that point, stable orientations of M coincide with directions of tangent lines to the latter part of the astroid. Thus for the points of the line a - d , which are inside the astroid, there are two stable orientations of M that result in two different branches of the hysteresis loop shown in Fig. 4.4. It is clear from the preceding discussion that if the applied field varies along the easy axis x then a S-W particle exhibits the rectangular hysteresis loop shown in Fig. 4.5. It is also clear that if the applied magnetic field is varied along the direction perpendicular to the easy axis, then due to symmetry there is no hysteresis effect and a S-W particle exhibits the single-valued magnetization curve shown in Fig. 4.6. Thus, the shapes of hysteresis loops depend on the direction along which the applied field is being varied. It has been shown in Chapter 3 (see formula (3.221)) that the loop shown in Fig. 4.4 can be represented in terms of the rectangular ~-loop. G. Friedman found a very interesting generalization of the formula (3.221) (see reference [8]). This generalization represents the magnetization of a S-W particle for arbitrary (not only collinear) variations of magnetic fields in terms of rectangular ~-loops. The basis for this representation is the notion that there are two distinct states (vector branches) for any S-W

454

CHAPTER 4 VectorPreisach Models of Hysteresis

MX

HX

F I G U R E 4.5

,'H

F I G U R E 4.6

particle. In the first state, stable orientations of the magnetization coincide with directions of tangent lines to the left-hand side of the astroid. The notation/vl + (r H(t)) will be used for the magnetization in the first state, where ~(t) is the angle formed by the applied magnetic field with the easy axis and H(t) is the magnitude of the magnetic field. In the second state, the stable orientations of M coincide with directions of tangent lines to the right-hand side of the astroid. The notation M-(q~(t), H(t)) will be used for the magnetization in the second state. It is clear that ~r + (q~(t), H(t)) and/~I- (~(t), H(t)) can be found geometrically by using the previously described 'astroid' rule or by solving the quartic equation (4.7).

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 455 The rectangular loop operator ~a,_~ will be employed to describe switching from the first state to the second state and vice versa. The input v(t) for this operator is given by cos q0(t) [Hx(t)2/3 + Hy(t)2/313/2 I cosq~(t)l

v(t)-

(4.27)

It is clear from (4.27) and (4.14) that v(t) reaches the value 0r as the tip of /7/(t) crosses the right-hand side of the astroid, and v(t) reaches the value -c~ as the tip of H(t) crosses the left-hand half of the astroid. This shows that the switching of the rectangular loop ~a,_~ occurs at the same time as the switching of the S-W particle from one state to another. By using this fact we can represent the magnetization M(t) of the S-W particle as

~r

= -~(t)~a,_~v(t) + ~t(t),

(4.28)

where 1

3(t) - ~[/~+ (~0(t), H(t)) - ~l-(qo(t), H(t))],

(4.29)

-. - ~[/~I 1 d(t) + (q0(t) , H(t)) + ~ i - (q0(t) , H ( t ) ) ] .

(4.30)

It is clear from (4.28), (4.29) and (4.30) that/~(t) =/VI + (q0(t), H(t)) when ~a _~v(t) - 1, and M(t) - M - (qo(t), H(t)) when ~a,_~v(t) - - 1 . Switchings of r from I to - 1 and vice versa occur at the times when the tip of H(t) crosses the astroid. This proves that formulas (4.27)-(4.30) give the right representation for the magnetization of the S-W particle. Having described the basic properties of a S-W particle, we can now proceed to the discussion of the S-W hysteresis model. This model is designed as an ensemble of S-W particles. Consider an infinite set of S-W particles with different orientations of their easy axes and different values of switching field c~. The notation So, a will be used for a S-W particle whose switching field is equal to c~and whose easy axis forms the angle 0 with the x-axis. By using this notation, the S-W model can be represented mathematically as

A/i(t) - f f ~(0, a)'So,~H(t) dO da,

(4.31)

where ~(0, ~) is a distribution function that should be determined by fitting the model to some experimental data.

456

CHAPTER 4 VectorPreisach Models of Hysteresis

The expression (4.31) defines the S-W model in terms of magnetic quantities such as magnetization M and magnetic field H(t). However, it is possible to interpret the S-W model as a general mathematical model of vector hysteresis by writing this model in the form

A f" (t) - fl ~(0, a)So,~(t) dO da,

(4.32)

wheref(t) is the vector output, while fi(t) is the vector input. The output of the operator So,~ can be formally determined by using the astroid rule or the quartic equation (4.7) in which Hx and Hy are replaced by ux(t) and uy(t), respectively. In other words, the S-W model can be defined in purely mathematical terms without using any connections of this model to some physical objects such as uniaxial, singledomain magnetic particles. Such a purely mathematical point of view of the S-W model may have two-fold advantages. First, it suggests some possibilities of using this model not only in the area of magnetics. Secondly, it may open some opportunities for further generalization of this model. By using representation (4.28)-(4.30) for the S-W particles, the S-W model can be written in terms of ~-operators. This suggests some connections between the S-W model and the Preisach-type models. In particular, Preisach-type diagrams can be used to keep track of switching of different S-W particles. The S-W model has been known and used in magnetics for a long time. Gradually, it has been realized that this model has certain limitations. The most important of them can be summarized as follows. Since the S-W model is designed as an ensemble (superposition) of particles (hysteresis nonlinearities) with symmetric loops, this model does not describe nonsymmetric minor loops. This limitation is often attributed to the fact that the S-W model does not account for 'particle interactions'. This limitation can be somewhat corrected by expanding the set of elementary hysteresis operators So, a and by introducing the operators So,~,~ with shifted astroids (Fig. 4.7). Then, the generalized S-W model can be represented as A

f(t) - f f / f r

~, ~)'So,~,~u(t) dO da d~.

(4.33)

However, this generalization will require much more computational work for the numerical implementation of the S-W model. Even without this

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 457 Y -(X \\

,

FIGURE 4.7

generalization, the S-W model is computationally slow. This is in partbecause the calculations of outputs of individual elementary operators So,~ require the solution of quartic equations associated with the astroid construction. In addition, individual outputs should be integrated over some distributions of S-W nonlinearities (particles) with respect to their easy axis directions and switching fields. This requires the evaluation of double integrals in the case of the classical two-dimensional S-W model (4.32). The last difficulty is magnified in the case of the generalized S-W model (4.33). Furthermore, the identification problem of finding the distribution function ~(0, a) by fitting the S-W model to some experimental data has not been adequately addressed yet. Solutions to this problem are usually achieved by some artwork rather than by using a well-established procedure. Our research has been motivated by the desire to circumvent the limitations of the S-W model described above. To achieve this goal, we have turned to the Preisach approach and tried to extend it to the vector case. The guiding idea in our efforts has been the notion that the scalar hysteresis is a particular case of vector hysteresis. As a result, many important and characteristic properties of vector hysteresis can be exhibited in the scalar case. By exploring this notion, new vector Preisach models of hysteresis have been developed. These models have many of the desirable features of the scalar Preisach hysteresis models, and they represent a viable alternative to the S-W model.

458

CHAPTER 4 VectorPreisach Models of Hysteresis

4.2 D E F I N I T I O N O F V E C T O R P R E I S A C H M O D E L S OF HYSTERESIS AND THEIR NUMERICAL IMPLEMENTATION For the sake of generality, a vector hysteresis nonlinearity will be characterized below by a vector input ~(t) and a vector outputf(t). In magnetic applications, ~(t) is the magnetic field, whilef(t) is the magnetization. The most immediate problem we face is how to define vector hysteresis in a mathematically rigorous as well as physically meaningful way. To do this, it is important to understand what constitutes in the case of vector hysteresis the essential part of the input history that affects the future variations of output. In the case of scalar rate-independent hysteresis, experiments show that only past input extrema (not the entire input variations) leave their mark upon future states of hysteresis nonlinearities. In order words, the memories of scalar hysteresis nonlinearities are discrete and quite selective. There is no experimental evidence that this is the case for vector hysteresis. As a result, we must resign ourselves to the fact that all past vector input variations may affect future output values. The past input variations can be characterized by an oriented curve L traced by the tip of the vector input ~(t). Such a curve can be called an input 'hodograph'. Vector rate-independent hysteresis can be defined as a vector nonlinearity with the property that the shape of curve L and the direction of its tracing (orientation) may affect future output variations, while the speed of the input hodograph tracing has no influence on future output variations. Next, we shall give another equivalent definition of rate-independent vector hysteresis in terms of input projections. This definition is very convenient in the design of mathematical models of vector hysteresis. Consider input projection along some arbitrary chosen direction. As the vector ~(t) traces the input hodograph, the input projection along the chosen direction may achieve extremum values at some points of this hodograph. In this sense, the extrema of input projections along the chosen direction sample certain points of the input hodograph. If the projection direction is continuously changed, then the extrema of input projections along the continuously changing direction will continuously sample all points on the input hodograph. In this way, the past extrema of input projections along all possible directions reflect the shape of the input hodograph and, consequently, the past history of input variations. Thus, we arrive at the definition of vector rate-independent hysteresis as a vector nonlinearity with the property that past extrema of input projections along all possible directions may affect future output values. It is clear that mathematical models of vector hysteresis are imperative for self-consistent descriptions of systems with

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

459

i1 iI i

s

s

s S

,'

I III

Lr

" J"

o ~. *" ~, ~

.~ ~ ' '

s

i I

. . o

i S

i

II

i i

s

II

s

i

t

%

I

% %

i

s 9

...."

i

s s s

s

s S

]-~r-~(-~.U-~(t)).. ,~, *

r

..

A

%

i

s

t

i

I

II

9

I

i i s i

I

e

I

FIGURE 4.8 vector hysteresis. These models should be able to detect and store past extrema of input projections along all possible directions and choose the appropriate value of vector outputs according to the accumulated history. To detect and accumulate the past extremum values of input projections along all possible directions, the scalar Preisach models (Preisach's particles) can be employed. These scalar models are continuously distributed along all possible directions (see Fig. 4.8). Thus scalar Preisach models are the main building blocks of the vector model, which is constructed as a superposition of scalar models. This can be expressed mathematically in two dimensions as )7(t) - ~~1=1 ~F~(~. ~(t)) ds

(4.34)

and the integration in (4.34) is performed over a unit circle. Similarly, a three-dimensional vector Preisach model can be written in the form f(t) - ~~1 =1 ~ FF(r. ~(t))dsr, where the integration is performed over a unit sphere.

(4.35)

460

CHAPTER 4 VectorPreisach Models of Hysteresis A

The scalar Preisach models F~ are defined by

F~(~ 9 fi(t)) = fl)]~ v(e, fl)~/~(~, fi(t))de dfl

(4.36)

for isotropic vector models, and

F~(~ 9 fi(t)) - ff~>/~ v(e, fl, ~)~//(? 9 fi(t)) de dfl

(4.37)

for anisotropic vector models. Ideas of the construction of vector Preisach models that are somewhat similar to those described above have been briefly mentioned (without any analytical details) in [9,10]. Some similarities can also be found between our definition of the vector Preisach models and a purely mathematical vector generalization of the scalar Preisach model discussed in [11]. The following proposition further elucidates the above definition of vector Preisach models. PROPOSITION. The integration in (4.35) over a unit sphere can be reduced to the

integration over a unit hemisphere. PROOF. Consider the partition of the unit sphere into two hemisphere C+ and C-. For every point ~ + ~ C + there is a corresponding opposite point ~- ~ C- (see Fig. 4.9) ~+ = - ~ - . (4.38) From (4.35) we find

f(t) = ffc+ ?+F~+ (? +" fi(t)) ds+ + fJc- ?-F~- (?-" fi(t)) ds_,

(4.39)

where F~+ (~ +. fi(t)) - ff~

v(e, fl, ~ +)~a/~(~+. fi(t)) de dfl,

(4.40)

F ~ - ( ~ - . fi(t)) = ff~

v(e, fl, ~ - ) ~ ( ~ - .

(4.41)

fi(t))dedfl.

The following identity can be verified for any continuous piecewise monotonic function v(t)" ~ v ( t ) = -~_~,_~ (-v(t) ). (4.42)

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

461

+

FIGURE 4.9

Indeed, if v(t) < fi, t h e n - v ( t ) > -fl and

~aflv(t)- -1,

while

~_~,_a(-v(t)) - +1.

(4.43)

If v(t) increases and exceeds ~, then -v(t) decreases and becomes smaller than-c~. Consequently,

~v(t)-

1,

while

~_~,_~(-v(t)) - -1.

(4.44)

Similarly, it can be shown that the identity (4.42) holds for any monotonic decrease of v(t). In the last integral in (4.39) we change the variable ~- to -~ +. As a result, we find

l / c - -f -F~- (~- " ~(t)) as_ - - f/c+ ~+F_~+ ( - ~ + . ~(t))ds+,

(4.45)

where F_~+ (-~ + 9~(t)) -

)/~ v(e, fl, -~ +)~c~(-~ +. ~(t)) de aft.

(4.46)

By using the identity (4.42), the last integral in (4.46) can be transformed as follows" ff~ ~>~v(~,fl,-~ - - ~ ddc~

+. ~(t)) d~ dfl

v(c~, fl, -~+)~_1~_~(~+. ~(t)) da dfi.

(4.47)

CHAPTER 4 VectorPreisach Models of Hysteresis

462

In the last integral we will change a to -]7 and ]7 to -a, then according to (4.46) and (4.47) we obtain F_~+ ( - 7 + . fi(t)) = - ~

dda >~

v(-fl, -~, -~+)~3(7+. fi(t)) d~ aft. (4.48)

By substituting (4.48) into (4.45), we find as_

fc

=

fZ >.3

+

ds+"

(4.49)

By substituting (4.40) into the first integral in (4.39), we have

f f + ~+?~+ (F+. fi(t)) ds+ ,tIC =ffc~+(fZ +

>.3

v(e, fl,-f+)~aB(-f+.fi(t))&zdfl)ds +.

(4.50)

From (4.39), (4.49) and (4.50) we obtain

f"(t) =

+ ff + 7[fZ tvt=, >>.

fl, ~ +) + v(-~, -~, --f +))

x9~3(~ +. fi(t)) dad//] ds+.

(4.51)

By introducing a new function

v' (o~,fl,-f +) = v(o~, fl,-f +) + v(-fl, -a, -~ +),

(4.52)

from (4.51) we derive

f(t) -

ffc 7 +"F~+ (~+. fi(t)) ds+, +

(4.53)

where

~'~+(~+ 9~(t)) - f L ~ v'(~, fl, ~+)~3(~+. fi(t)) da dfl. This completes the proof.

(4.54) []

The proven proposition suggests that (4.53) and (4.54) can be regarded as an equivalent definition of the vector Preisach model. This definition will

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

463

be used in our subsequent discussions and the superscript ' will be omitted. It is also clear that a similar proposition is valid for the two-dimensional vector Preisach model (4.34) as well. Consequently, this model can be represented as )7(t) = f

~ +F~+ (~ +. fi(t)) dl+,

(4.55)

,/L +

where L+ is a semicircle and F'~+ is defined by (4.54). It is apparent from the above proof that particular choices of semispheres and semicircles in (4.53) and (4.55), respectively, are unimportant; all these choices will lead to equivalent vector Preisach models. From the above proof we can also find some interesting symmetry properties of the function v'. Indeed, if in (4.52) we change a to - f l , / / t o - a , and ~ + to - ~ +, then we find =

(4.56)

+

From (4.56) and (4.52), we obtain (4.57) This expresses the property of mirror symmetry of the function vf with respect to the line a - -//. This also extends the definition of the function vI from a unit semisphere or unit semicircle to the entire unit sphere or unit circle, respectively. In the case of isotropic vector models, the expression (4.57) can be simplified as follows: v' (a, fl) = v'(-fl, - a ) , (4.58) which is similar to the symmetry property of the #-function for the classical scalar Preisach model. Up to this point, the vector Preisach models have been defined in coordinate invariant forms (4.34), (4.35), (4.53) and (4.55). However, in many applications it is more convenient to use the expressions for the vector Preisach models in spherical and polar coordinates for three and two dimensions, respectively. In the case of spherical coordinates we have ds+

-

sin 0 dO d~o,

~+

- e 0 ,-~ r

-~ r +

.fi(t)

-

uo,~o

(t) ,

(4.59)

where e0,~0 is a unit vector along the direction specified by angles q~ and 0, and uo,~o(t) is the projection of fi(t) along the direction of e0,~0. By using (4.59), the three-dimensional vector Preisach model (4.53)(4.54) can be written as

f (t) =

f0270

A euo, e(t) sin 0 dO dq~, -~o,eFo,

(4.60)

CHAPTER 4 VectorPreisach Models of Hysteresis

464

where A

Fo, q)UO,q)(t) =

v(e, fl, O, q))4&flUO,q)(t) dedfl. >>.~

(4.61)

The last two formulas can be combined into one expression:

( ff~>~ v(e,fl, O,q));&~uo,~o(t)dedfl)

f(t) =

-do,~o

x sin 0 dO dq~.

(4.62)

Similarly, the two-dimensional vector Preisach model (4.55) can be represented in polar coordinates as A

f(t) =

~FCu~(t) dq~,

(4.63)

where A

Fq~uq~(t) -

v(e, fl, q))~flUq~(t) de dfl.

(4.64)

In the previous formulas, ~r is a unit vector along the direction specified by a polar angle q~, Fr is the scalar Preisach model (operator) for this direction, and uq~(t) is the projection of fi(t) along the direction of ~q~. By combining (4.63) and (4.64), we have

f (t) =

-do

v(e, fl, qo)~flu~o(t) de dfl dq).

(4.65)

The expressions (4.62) and (4.65) are written for anisotropic models. In the isotropic case, the function v should be independent of 0 and q~ (or of q~in the two-dimensional case). This leads to the following three-dimensional and two-dimensional isotropic vector Preisach models:

-"eo,~o( ff~ >~ v(e, fl)4&~uo,~o(t)dedfl) sinOdOdqo, (4.66)

f(t) = f (t) -

-@

v(a, fl)~fluq~(t) doe dfl dq~.

(4.67)

In the models described above, the functions v have not yet been specified. These functions should be determined by fitting the vector models to some experimental data. This is an identification problem. It is apparent that

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

465

d

(~, ~)

T| (0, ,~)

FIGURE 4.10

the identification problem is the central one as far as practical applications of the above vector hysteresis models are concerned. This problem will be discussed in sufficient detail in the subsequent sections. However, it is appropriate to comment already here that the solution of the identification problem is significantly simplified by the introduction of the auxiliary function P(~, fl, O, q~). For any fixed 0 and q~, consider a triangle T(~, fl) shown in Fig. 4.10. Then by definition, we have

P ( o~, fl, O, r ) -//T{~,fl) v(a', fl', O, q~)da' dfl'.

(4.68)

By using (4.68), it can easily be shown that P is related to v by the formula

v(c~, fl, O, qo) = -

a2P(~, fl, O, q~)

a~afl

.

(4.69)

Thus, if the function P is somehow determined, then the function v can be easily retrieved. However, from the computational point of view, it is more convenient to use the function P than v. This is because the double integrals with respect to a and fl in expressions (4.62), (4.65), (4.66) and (4.67) can be explicitly expressed in terms of P, and in this way the above double integration can be completely avoided. Indeed, for any fixed direction e0, we can consider the corresponding a-fl diagram. A typical example of such a diagram is shown in Fig. 4.11, where Mo, q~,kand mo, q~,kform an alternating series of dominant maxima and minima of input projections along the direction specified by e0, ~. By using these maxima and minima, the output

CHAPTER 4 VectorPreisach Models of Hysteresis

466

0'~'k)

FIGURE 4.11

of the scalar Preisach model Fo, q~uo,q~(t) associated with the direction e0,q~ can be evaluated as follows A

Fo, q~uo,r

= - P ( a o , flo) no,r +2 ~ [P(Mo, q~,k, mo, q,,k-1, O, r -- P(Mo, q~,k, mo, q~,k, O, r k=l

(4.70) The proof of (4.70) literally repeats the proof of the similar expression for the classical Preisach model and, for this reason, it is omitted. In the case of the two-dimensional vector Preisach model (4.63)-(4.65), a similar expression is valid for Fq~ur A

Feud(t) = - P ( a 0 , rio) %(t) +2 y~ [P(M~,k, mq~,k-1, q~) -- P(Mq~,k, mq~,k, q~)].

(4.71)

k=l

By using formulas (4.70) and (4.71) the numerical implementation of the three-dimensional vector model (4.60)-(4.62) and the two-dimensional vector model (4.63)-(4.65) can be reduced to the evaluation of double and single integrals, respectively. We note here that the numerical implementation of the classical two-dimensional Stoner-Wohlfarth model (4.32) requires the evaluation of double integrals. In this respect, numerical implementation of the two-dimensional vector Preisach model can be accomplished more efficiently than the numerical implementation of the

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

467

two-dimensional Stoner-Wohlfarth model. Another advantage of using the formulas (4.70) and (4.71) is that the function P can be directly related to experimental data. This will be demonstrated when we study the identification problems for the vector Preisach models. Using the expressions (4.70) and (4.71), digital codes that implement the vector Preisach models (4.60)-(4.62) and (4.63)-(4.65) have been developed. In these codes finite meshes of directions e0,~ and ~ are used to evaluate double and single integrals in (4.60) and (4.63), respectively. For each mesh direction the integrands in (4.60) and (4.63) are computed by using (4.70) and (4.71). Some numerical examples computed by using the developed digital codes will be given in the next section.

4.3 S O M E B A S I C PROPERTIES OF V E C T O R P R E I S A C H HYSTERESIS MODELS In the previous section we have defined the vector Preisach models of hysteresis and discussed their numerical implementation. The purpose of this section is to study some basic properties of these models and to show that these properties are qualitatively similar to those observed in experiments. We begin with the property of reduction of vector hysteresis to scalar hysteresis. This property is experimentally observed when an input is restricted to vary along an arbitrary chosen direction. We shall show below that a similar reduction property holds for the vector Preisach models; this property is stated more precisely below. For the sake of notational simplicity it is formulated and proven only for the two-dimensional Preisach model (4.63)-(4.65), although it holds for the three-dimensional Preisach model (4.60)-(4.62) as well. REDUCTION PROPERTYOF THE VECTOR PREISACH MODEL TO THE SCALAR PREISACH

MODEL.Consider an input fi(t) restricted to vary along some direction -~ofor times t>~to. Suppose that during t>~to, Ur (t) = u(t) consecutively reaches values u+ and u_ (with u+ > u_) and remains thereafter within these bounds. Then,for the Preisach vector model (4.63)-(4.65), the relationship between the output projection f~o(t) along the direction -e~o and the input u(t) exhibits the wiping-out and congruency properties. Since these properties constitute necessary and sufficient conditions for the representation of hysteresis nonlinearities by the classical scalar Preisach model, we conclude that the vector Preisach model is reduced to the scalar Preisach model.

CHAPTER 4 Vector Preisach Models of Hysteresis

468

(z+cos~ ~mj 'ucos~o FIGURE 4.12

PROOF. Without impairing the generality of our discussion, we can assume that q~0 = 0. Then, for any r the input projection u~(t) varies between u+ cos ~ and u_ cos ~. This means that for any ~ input variations m a y affect a - ] / d i a g r a m s only within the triangle T(u+ cos ~, u_ cos r (see Fig. 4.12). It is also clear that all input projections ur reach m a x i m u m or minim u m values at the same time and that these extremum values are 'cos qamultiples' of the corresponding extremum values of uo(t) = u(t). Consequently, if {Mk} and {ink} constitute an alternating series of dominant extrema of u(t), then {M~,k} and {m~,k} defined as

Mr

= Mk cos q~,

m~,k = mk cos ~

(4.72)

constitute the corresponding alternating series of dominant extrema of

u~(t). From (4.63) we find A

fx(t) -

cos ~FCu~(t) de.

(4.73)

2

From (4.71), (4.72) and (4.73) we conclude that only the alternating series of dominant extrema Mk and mk of u(t) affect the value offx(t). All other input extrema are wiped out. This is tantamount to the wiping-out property of the hysteretic relation betweenfx(t) and u(t). We shall next prove that the above hysteretic relation also exhibits congruency of minor loops. Let u (1) (t) and u (2) (t) be two inputs which vary between u+ and u_ for t~to and which m a y have different past histories I these inputs vary for to~t~t~o . However, starting from instant of time t 0,

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

469

(x

u~cos~o .

.

.

.

u cosq~

FIGURE 4.13

u+cosq~

/ ulcos "13 u_cosq~

FIGURE 4.14

back and forth between the same two consecutive extrema, u+I and u t_. As a result of these back-and-forth input variations, some minor loops are formed. We intend to show that these minor loops are congruent. The proof of the congruency of the above loop is equivalent to showing that any equal increments of inputs u (1) (t) and u (2) (t) result in equal increments of outputs fx(1) (t) andfx(2) (t). To this end, let us assume that both inputs after achieving the same value u'_ are increased by the same amount: Au (1) = Au (2) - Au. As a result of these increases, the identical triangles T~1) and T~2) are added to positive sets S + qo,1 (t) and S + qo,2 (t) and subtracted from the negative sets S~, 1(t) and S~, 2 (t) (see Figs 4.13 and 4.14). Since

F~ou~o(t) -

v(o~, fl, qo)de d f l +(t)

;(t)

v(~, fl, ~o)de dfl,

(4.74)

CHAPTER 4

470

Vector Preisach Models of Hysteresis

from (4.73) we derive

Af(x1) _ 2 f _ ~ (cos / / T q~ ( l V ( ~ f l ),

,

qo)dadfl)dq ~,

(4.75)

2

Afx(2) = 2

f

_~COS q)

(2)

)

v(~, fl, q~) da dfl dq~.

2

(4.76)

Since T (1) = T (2) for any q~, we conclude that Af(1) -- Af(2).

(4.77)

The equality (4.77) has been proven for the case when inputs U(1) (t) and u (2) (t) are monotonically increased by the same amount after achieving the same m i n i m u m value u'_. Thus, this equality shows the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (4.77) holds when the inputs u O) (t) and u (2) (t) are monotonically decreased by the same amount Au after achieving the same maximum value u~. This implies the congruency of descending branches of the above minor loops. Thus, the congruency property for minor loops is established. This completes the proof of the validity of the reduction property. [] In our discussion of the reduction property, we have proved the congruency of 'scalar' minor loops described by the vector Preisach models. The last result admits the following generalization. CONGRUENCYPROPERTYOFVECTORMINORLOOPS. Let the tips of two inputs Ft(1)(t) and ~(2)(t) trace the same closed curve for t>~to (see Fig. 4.15). Then the tips of the corresponding outputs ](1)(t) and/(2)(t) of the vector Preisach models trace congruent closed curves for t>~to (see Fig. 4.16). These curves may be noncollocated in space because of possibly different past input histories prior to to. PROOF. Consider the three-dimensional Preisach model (4.60)-(4.62). Since the tips of both inputs ~(1)(t) and ~(2)(t) trace the same closed curve, we conclude that for any direction eo,~0 the corresponding a-fl diagrams are modified with time in an identical way within the same triangles T(u+,o,~o, u-,o,~o). For instance, as the tips of the inputs ~ (1) (t) a n d v~(2) (t ) move from the point a to the point b (see Fig. 4.15), the same regions ftl,0,~0 and f22,0,~0m a y b e added to the positive sets S + 0,(p,1 and S + 0,q~,2 and subtracted

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

471

Uz

1

, Uy

Ux

FIGURE 4.15

from the negative

sets So,(pI and So, q~,2. Since

-Fo, q~UO,q~(t) - I/s ~,+~o~(t) v(c~,fl, O, qo)de d[3 -

[[ JJs ~,r

v(c~, fl, 0, ~o)d~ dfl,

(4.78)

we conclude that AFo, q~A u(ll0,q~(t) -- 2

ffolo

AFo, q ~ u(2)O,q~(t) = 2 [[_ J d.s

v(or fl, 0, q~) de dfl,

(4.79)

v(c~, fl, 0, qo) de dfl.

(4.80)

2,0,tO

By using (4.60), (4.79) and (4.80), we find that the corresponding output increments which connect the points A (1), B (1) and A (2), B (2) (see Fig. 4.16) are given by

af(l) -2 f2~f~o,~( ff n JO

Af (2) = 2

JO

fOarCfo rce0,~o(ff

v(a, fl, O,q))do~dfl)sinOdOdq),

(4.81)

1,0,rp

)

v(c~, fl, 0, qo) d~ dfl sin 0 dO dqo. (4.82) 2,0,(p

Since f21,o,~o - f22,o,~o for any 0 and q~, from (4.81) and (4.82) we conclude that A f ( l ) -- A f (2) . (4.83) The equality (4.83) holds for arbitrary chosen points a and b, and this proves that the vector minor loops shown in Fig. 4.16 are congruent. []

CHAPTER 4

472

Vector Preisach Models of Hysteresis

B2

2~

A2 " fy

fx FIGURE 4.16

We next proceed to the discussion of one remarkable property which is valid for the two-dimensional isotropic vector Preisach model (4.67). ROTATIONALSYMMETRYPROPERTY. Consider a uniformly rotating input (that is, one of constant magnitude and angular velocity)

Ft(t) - {ux(t) - UmCOscot, Uy - - Um Sin cot}.

(4.84)

Then the output of the two-dimensional isotropic Preisach model (4.67) can be represented as N

f (t) =f0 + f (t),

(4.85)

where fo does not change with time, while f (t) is a uniformly rotating vector. PROOF. It is clear from the very definition of the uniformly rotating input ~(t) and Fig. 4.17 that

u~o(t) -

U m

cos(cot - (p).

(4.86)

According to the proposition proved in Section 4.2, the integration over any semicircle can be used in the definition of the two-dimensional vector Preisach model. This fact allows one to modify the definition (4.67) of the two-dimensional isotropic Preisach model as follows: -

1

2re

f (t) - -~ f o

-G( ff~>~v(c~,fl)P~u~o(t)d~dfl) d~~

(4.87)

It is clear from (4.86) that for all directions e~0 the corresponding a-fl diagrams are modified with time within the same triangles T - T(um, -urn). Outside these triangles, the a-fl diagrams remain unchanged. These unchanged parts of c~-fl diagrams contribute to the term f0 in (4.85), while the

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

~

473

~(t)

FIGURE 4.17

time-varying parts of ~-fl diagrams result in the time-varying term By using the above comment as well as (4.86) and (4.87), we find: 1

f(t).

2~

f (t) - -i f ~ -dq,( f f v(a, fl)~t3Um COS(O~t- q)) da dfl) dq).

(4.88)

The expression (4.88) can be represented in terms of Cartesian components as

1 fO2re cos q)(ff v(~, fl)~Um COS(Ogt- qo)dadfl) dq~, fx(t)-~

(4.89)

1 fO2re sin q)(/f v(~, fl)~UmCOS(O~t- q~)d~dfl) dq~. fy(t) -- -i

(4.90)

Consider some instant of time t. For this instant of time, all directions ~ can be divided into two sets such that O~
(4.91)

~ < o 0 t - q~<2~.

(4.92)

and For the first set, all input projections u~o(t) are monotonically decreasing, and this results in the 0r diagram shown in Fig. 4.18. For the second set, all input projections uq,(t) are monotonically increasing, and this results in the ~-fl diagram shown in Fig. 4.19. Next we introduce new variable 0 - ogt- q~.

(4.93)

It is apparent that the double integral over T in (4.89) and (4.90) is the function of 0. This justifies the following notation

G(O) = ff~ v(e, fl)F~flUm A cos(o~t -

q~) d~ dfl.

(4.94)

CHAPTER 4 Vector Preisach Models of Hysteresis

474

(X

FIGURE 4.18

If 0~0~,

(4.95)

then the diagram shown in Fig. 4.18 is valid, and from this diagram and formulas (4.68) and (4.94) we find G(O) = P(um,

- - U m ) --

2P(um,

Um

COS 0).

(4.96)

If ~KOK2z~,

(4.97)

then the diagram shown in Fig. 4.19 is valid, and as before we find G(O) = 2P(um cos O, -urn) - P(um, -urn).

(4.98)

By using the change of variables (4.93), the notation (4.94), and by taking into account that dO - -d~o, (4.99) and c o t - 2z~O~cot,

(4.100)

we transform (4.89) as follows:

lj02

fx(t) - -~

where

cos(cot - O)G(O) dO - A cos cot + B sin cot,

lf02

(4.101)

cos OG(O) dO,

(4.102)

B -- ~1 f02~ sin OG(O) dO.

(4.103)

A- ~

475

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

/

FIGURE 4.19

Similarly, the expression (4.90) can be reduced to the form (4.104)

fy(t) - A sin cot - B cos cot. The formulas (4.101) and (4.104) can be modified as follows:

VIA2 q- B 2 cos(cot- ~),

(4.105)

y~y( t ) - v/A 2 + B2 s i n ( c o t - ~),

(4.106)

B tan ~ = A"

(4.107)

fx(t)

-

where

From (4.105) and (4.106), we conclude t h a t f is a uniformly rotating vector with the magnitude equal to v/A 2 + B2. We next express A and B in terms of the function P and prove that the angle ~ in (4.105) and (4.106) is acute. From (4.95)-(4.98) and (4.102) we find

1{/0 ~2~

cos O[P(um, -urn) - 2P(um, Um cos 0)] dO

A- ~

q-

}

cos O[2P(um cos 0, -urn) - P(um, -urn)] dO . (4.108)

In the second integral in (4.108), we will use the change of variables 0'= 0- ~

(4.109)

CHAPTER 4 Vector Preisach Models of Hysteresis

476

and take into account that

P(-um cos 0, -urn) ~- P(um, Um cos 0).

(4.110)

The last formula easily follows from the symmetry property (4.58) and the definition (4.68) of the function P. By using (4.109) and (4.110), we can transform (4.108) as follows: A = -2

~0 ~

cos OP(um, Um COS 0) d0.

(4.111)

From (4.95)-(4.98) and (4.103), we obtain

11/0

B= ~

+

sin O[P(um, -urn) - 2P(um, Um cos 0)] dO sin O[2P(um cos 0, -urn) - P(um, -urn)] d0 . (4.112)

To transform the second integral in (4.112), we will use the change of variables (4.109) and the identity (4.110). This eventually leads to the following expression for B:

/z

B - 2P(um, --Urn) -- 2

f0

sin OP(um, Um COS 0) d0.

(4.113)

Thus, we have found explicit expressions (4.111) and (4.113) for A and B in terms of the function P. In the next section, it will be shown that P can be related to some experimental data. In this way we can relate A and B to the experimental data, and, consequently, find the magnitude and phase of the uniformly rotating vectorf(t) in terms of these data. Now, we will use formulas (4.111) and (4.113) to prove that the phase angle ~ is acute under some general conditions. The first condition can be expressed mathematically as

P(a, fl)~P(a, fl')

if fl>~fl'.

(4.114)

The condition (4.114) means that P(a, fl) is a monotonically decreasing function of fl for any fixed ~. It is clear from the definition (4.68) of P that the condition (4.114) is satisfied if v(a, fl) is positive. From (4.111), we find

A 2[/0

cos OP(um, Um cos 0) dO q-

cos OP(um, Um cos 0) dO . (4.115)

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

477

By using the change of variables 0t = ~ - 0 in the second integral in (4.115), we derive /T

A

-

-2

f0

cos O[P(um, Um COS 0)

-- P ( u m , - u r n c o s 0)] dO.

(4.116)

From (4.114) and (4.116), we conclude A > 0.

(4.117)

O[P(um, -Um) -- 2P(um, Umcos 0)] d0 > 0.

(4.118)

The second condition is f0 ~rsin

From (4.114) and (4.118), we find B > 0.

(4.119)

From (4.107), (4.117) and (4.119), we conclude that the phase angle ~ is acute. This completes the proof of the rotational symmetry property. [] It is worthwhile noting that in spite of the nonlinear structure of the vector Preisach model (4.67) the time harmonic input (4.84) produces (up to a history-dependent constant term f0) a time harmonic output of the same frequency. In other words, no generation of higher-order harmonics is caused by the nonlinear structure of the vector Preisach model. This remarkable fact admits the following physical explanation. The isotropic vector Preisach model (4.67) has a mathematical form which is invariant with respect to any rotation of Cartesian coordinates. The mathematical form of uniformly rotating input (4.84) is also invariant with respect to any rotation of Cartesian coordinates. Thus, on symmetry grounds we expect that the mathematical form of the resulting output should also be invariant with respect to any rotation of Cartesian coordinates. This is possible only if the output is a uniformly rotating vector. The above discussion clearly reveals the meaning of the term 'rotational symmetry property'. The rotational symmetry property has been confirmed by numerical computations. By using a digital code that implements the vector Preisach model (4.67) and that has been briefly described in the previous section, the output of the Preisach model has been computed for the input shown in Fig. 4.20. This input gradually approaches the regime of uniform rotation (4.84). The results of computations shown in Fig. 4.21 demonstrate that the output also gradually attains the regime of uniform rotation. The property of rotational symmetry has also been observed in the experiments. The --4

CHAPTER 4 VectorPreisach Models of Hysteresis

478

5.0

2.5

_

o.o_

t,,._j

,

,

,)/, i

-5.0 -5.0

-2.5

0.0

2.5

5.0

FIGURE 4.20

results of these experiments are shown in Fig. 4.22. In these experiments, the applied magnetic fields were stationary, while the magnetizable sample of Isomax material was uniformly rotated. Thus, in these experiments, the magnetization was measured in the coordinate frame uniformly rotating with respect to the sample. The component of magnetization measured along the applied field is called 'parallel magnetization' while the component perpendicular to the field is named 'transverse magnetization'. The results shown in Fig. 4.22 clearly suggest that for large applied fields the whole magnetization of the sample moves in synchronism with the uniformly rotating coordinate frame. This means that we are dealing with uniformly rotating magnetization. As the applied field is reduced, the time constant term of magnetization appears, which reveals itself (in the uniformly rotating frame) as sinusoidally changing components of magnetization. PROPERTY OF CORRELATION BETWEEN MUTUALLY ORTHOGONAL COMPONENTS OF

OUTPUTANDINPUT. Suppose that the input ~(t) is first restricted to vary along the y-axis. It is increased from an infinitely negative value to some positive value u+, and then it is decreased to zero. Some remnant value of output fr = -eyfr results from the above input variations. After reaching zero, the input is restricted to vary

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH 20

I

I

0

10

479

10

-10

-20

-10

20

F I G U R E 4.21

along the orthogonal x-direction (see Fig. 4.23). It is asserted that, by increasing the input in x-direction, it is possible to reduce the orthogonal remnant component of the output to any, however small, value. PROOF. All directions ~r can be subdivided into sets: 0Kq~K~

and

~q0K0.

(4.120)

The input variations along the y-axis result in the a-fl diagrams shown in Figs 4.24 and 4.25 for the first and second sets of directions, respectively. As the input is increased along the x-axis, the above diagrams are modified. These modifications are shown by dashed lines. They result in continuous expansions of positive sets S~ (t) at the expense of negative sets S~ (t). From (4.67), we find

fy(t) = f_~ sinq~( / L 2

v(~, fl)~[~Ux(t)cosq~d~dfl) dq~.

(4.121)

~fl

By introducing the notation

G(qo, t) - ~ v(c~,fl)~ux(t) cos (p d~ dfl, dJa>~

(4.122)

480

'

I'

'

'

'

I

'

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Vector Preisach Models of Hysteresis

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uop, uzp, a u g g t u

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CHAPTER 4

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4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

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Vector Preisach Models of Hysteresis

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CHAPTER 4

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e,i r

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

f~

483

/~(t) ~X

FIGURE

4.23

u cos~

(~o)

u.~in~o

FIGURE 4.24

•u

Usn'q l

cos~

(~o)

FIGURE 4.25

formula (4.121) can be represented as follows:

fy(t) =/o ~ sin ~G(~,

t)do +/0

sin qaG(~, t)do. 2

(4.123)

CHAPTER 4 Vector Preisach Models of Hysteresis

484

Changing q~ to -q~ in the second integral in (4.123), we obtain 2

fy(t) -

Jo

sin q~[G((p, t) - G(-q~, t)] dq~.

(4.124)

If Ux cos q~a0, then according to the diagrams shown in Figs 4.24 and 4.25 we have G(q~, t) - G(-q~, t). (4.125) From (4.124) and (4.125), we find

fy (t) =

rccos

=0

sin q~[G(q~, t)

-

G(-q~, t)] dq~.

(4.126)

ux(t)

Since arccos Ux(t) ~0 ~ ~ as ux(t) ~ cx~,we conclude that

fy(t) ~ O.

(4.127)

This completes the proof of the above property. This property has been confirmed by numerical computations performed by using a digital code that implements the model (4.67). The results of the computations are shown in Fig. 4.26 for different remnant values of the

5

0

--5

-lO

1

~,~1~1,~,~1,,,,i,,,,1,,,,

0

1000

2000

3000

F I G U R E 4.26

4000

-

5000

6000

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

485

o

-5

-lo

j L i i 1 , l , l I i l , , 1 l o

.05

.1

.15

FIGURE 4.27

output. The property of correlation between orthogonal components of input and o u t p u t has also been observed in the experiments. The results of these experiments are presented in Fig. 4.27. There is an apparent qualitative similarity between the computational and experimental results. It is worthwhile noting here that the above property of orthogonal correlation has long been regarded as an important 'testing' property for vector hysteresis models in magnetics.

4.4

IDENTIFICATION PROBLEM FOR VECTOR PREISACH MODELS

ISOTROPIC

The essence of the identification problem is in determining the function v(a, fl) or P(a, fl) from some experimental data. It turns out that this problem can be reduced to the solution of a special integral equation that relates the function P(~, fl) to some 'unidirectional' (scalar) hysteresis data. We first present the derivation of this equation for the two-dimensional model (4.67). The derivation proceeds as follows. Consider the projectionfx(t) off(t) along the direction q~ = 0. According to Eqn. (4.67), we have

fx(t)

=

I' (o(iL'>~' v(a', fl')~,~,u~o(t) d~' aft' )de. cos

(4.128)

CHAPTER 4 Vector Preisach Models of Hysteresis

486

d

s:,

FIGURE 4.28

Now we restrict ~(t) to vary along the direction r = 0, which means that Ft(t) = -~xu(t). First, we assume that u(t) is made 'infinitely negative'. Then for any a,I fl! and (p we have ~,/~,u,(t) = -1. Next, we assume that the input is monotonically increased until it reaches some value a. Let fa denote the corresponding value offx(t). As a result of the above input increase, we find that for any q~ we have ~,/~,u,(t) = +1 if a' < a cos q~ and ~a,/~,u,(t) = - 1 if a t > a cos q~. This is shown geometrically in Fig. 4.28. Consequently, for any r we have

ff~,>>.~,v(a', fl')~,ffUq~(t) da' dfl' = ffs+ * v(a', i f ) d a ' d f f - ffs;.,

v(a', fl') da' dff.

(4.129)

Finally, we assume that the above input increase is followed by a subsequent monotonic decrease of u(t) until it reaches some value ft. Let f~/~ denote the corresponding value offx(t). The above input decrease modifies the previous geometric diagram that now for any q~ assumes the form shown in Fig. 4.29. According to this figure, we have ff~ ,>>.~'v(a',

= ffsL

fl')~,~,ur

da' dfl'

v(a', fl')da' d f l ' - f f

dis ~.~

v(a', fl')da' dfl'.

(4.130)

Consider the function 1

F(a, t) = ~(f~ -f~/~).

(4.131)

By using formulas (4.68), (4.128)-(4.131), it is straightforward to show that

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

487

Td (~) / ,

13cos~ s~*~,,I-

FIGURE 4.29

P(a, fl) is related to F(a, fl) by the expression

f

~ cos q~P(a cos q~, fl cos q~) dq~ - F(a, fl).

(4.132)

2

It is apparent that f~ and f~/~ can be found by measuring the first-order transition curves. Of course, the input u(t) cannot be made infinitely negative as was assumed in the derivation. However, by making the input 'sufficiently negative', a reasonable accuracy can be secured. The formula (4.132) can be construed as the integral equation which relates P(~, fl) to the first-order transition curves. This integral equation has a peculiar structure that is revealed by the following result. THEOREM. Consider the operator

AP = / ~ cos qoP(a cos q~, fl cos q~) d~0.

(4.133)

2

Monomials ~k fls are eigenfunctions of the operator A. PROOF. From (4.133) we find -

~ cos Oa k cos k Oti s cos s r d o 2

~xkfls f ~ cos k+s+l q~dq~ - 2k+lak fls, d-

(4.134)

CHAPTER 4 VectorPreisach Models of Hysteresis

488

where

(2n-l)!!

ifk + s + 1 = 2n, (4.135)

2k+s --

2 (2n)!!

ifk + s + 1 - 2n + 1.

By using the above theorem, the solution of Eqn. (4.132) can be easily found when the right-hand side F(a, fl) is a polynomial M F(a, f l ) = ~ ~-(m)akfls aks m=0 k+s=m

(4.136)

Indeed, looking for the solution of the integral equation in the form M P(a, fl) -- y ~ ~-~__"(m)akfls,laks m=0 k+s=m

(4.137)

from the above theorem we find ~(m) p~m) = Uks 2m"

(4.138)

By using this fact, the following algorithm for the solution of Eqn. (4.132) can be suggested. We first extend the function F(a, fl) from the triangle -a0~
(4.139)

where the Chebyshev polynomials are defined as To(a) = 1,

1

Tt(a) - ~--;--ccos(s

(4.140)

By assuming for the sake of notational simplicity that a0 = 1, for the expansion coefficients agq in (4.139) we have the formula (see [12])

agq=

2e+qf; f/ F(a,fl)Tt(a)Tq(fl)dadfl. /z2

1

1

v/l_a2v/l_fl2

(4.141)

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

489

Let teq(O~,fl) be the solution of the integral equation

/

7[

~ cos (pts

cos (p, fl cos q~) dq~ - Te(~) Tq(fl),

(4.142)

then the solution of the integral equation (4.132) can be represented in the form P(a' fl) - E as163 fl). (4.143)

s Thus, there are two major steps in finding the solution (4.143) of the integral equation (4.132). The first step is to solve Eqn. (4.142). This can be accomplished by using the technique (4.136)-(4.138). The second step is to evaluate the expansion coefficients as This can be handled by using the formula (4.141). However, for the sake of computation it is desirable to transform this formula as follows. By introducing new variables 0 and 0I = cos 0,

fl = cos 0',

(4.144)

cos qO' 2q_1 ,

(4.145)

F(cos 0, cosO') cosf.OcosqO' d0d0'.

(4.146)

we find cos t~0 , 2s

Ts

4/070

as - -~

Tq(fl)

-

Thus the expansion coefficients as can be evaluated by using fast cosinetransform schemes. There is another approach to the solution of the integral equation (4.132) that leads to a closed-form expression for P(a, fl). This approach is based on the following change of variables: x = a cos q~,

2 = ilia,

(4.147)

which after simple transformation yields the following integral equation of the Abel type:

f0 40r x-- X2 P(x, 2x) dx = ~F(~, 0~ 2a).

(4.148)

Some simplification of the above equation is achieved by introducing new auxiliary functions:

N(x) - xP(x, 2x),

R(a) - 2F(a, 2a).

(4.149)

CHAPTER 4 Vector Preisach Models of Hysteresis

490

oc

x

FIGURE 4.30

Then, from (4.148) we obtain

fo ~ N(x) dx

= R(~).

(4.150)

/0r _ x 2

The solution of the above equation can be found by using the following trick. We multiply both sides of (4.150) by ~/s2_~2 and integrate with respect to e from 0 to s:

f0s

o~ x/S 2 __ CX2

(f0

N(x) dx

) d ~ = f0

da.

(4.151)

~zdcz ~ dx. V/(S2_~2)(-~2_x 2) ]

(4.152)

~/0r __ X 2

&2

__ 0r

By using Fig. 4.30, we find

fos

cx (fo~ N(x)dX)dcz V/S2 -- 0~2

X/C(2 -- X 2

[~ ~N(x)dx d~ JJ. v/(S2 -- 0r162 2 -- X2) s =f0

s N(x)(L

Next, by using simple transformations, we obtain fx s

0rdcz

v/(S2_ CX2)(CX2_X2)

_ 1 fx s 2 ~

d(cz2)

v/(s22x2_)2 _(cx 2 -

~

(4.153)

s2+x22)2

By introducing a new variable of integration S2 + X2 09 -- O~2

(4.154)

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

491

from (4.153) we derive s2_x 2

/x s

o~da v/(S2 -- X2)(0r2 -- X 2)

_ 1 ~x22s2 2

do~

~ ~/ (S2-X2) 2 __ 092

1 209 = - arcsin 2 s 2 -- 09 2

s2 _x 2 09----

--. x2 -s 2 09----

2

(4.155)

From (4.151), (4.152) and (4.155) we obtain f0 s N(x) dx - -2 f0 s

c~R(c0 da.

(4.156)

v/S2 _ ~2

By differentiating (4.156) with respect to s and then by replacing s by and a by s, we find N(~) -

2 d [~ d~ ~0

sR(s)

ds.

(4.157)

V/or2 -- S2

This is the closed-form solution to the integral Eqn. (4.150). Now, by substituting (4.149) into (4.157), we find the closed-form expression for P:

P(c~, ;ccO- 1 d fo ~ s2F(s, ,~s) ds. ~:o~d~x

(4.158)

V/0r _ s 2

The last expression is valid for any value of 2. By varying 2 from 0 to 1 and using (4.158), we can compute all required values of P. However, for actual computations, it is convenient to transform (4.158) by integration by parts. The final expression is then given by

P(a, 2a)

1~

F(s, 2s) + s dF(s, ,ls)

7C Y0

V/0r 2 -- S 2

ds.

(4.159)

Now, we turn to the three-dimensional isotropic model (4.66). Consider the projection fz(t) of f(t) along the direction 0 - 0. According to (4.66), we have

fz(t) -

cos 0 x sin 0 dO dcp.

'~>/~'v(a', fl )?a,~,Uo, q~(t) da' aft' (4.160)

We restrict ~(t) to vary along the direction 0 - 0, which means that ~(t) -dzu(t). As before, we first assume that u(t) is made infinitely negative. Then

CHAPTER 4 Vector Preisach Models of Hysteresis

492

d ..

+

l~coso

/ FIGURE 4.31

I d (o , , ) / , +

FIGURE 4.32

for any a, fl, 0 and q~ we have r q~(t) - - 1 . Next, we assume that the input is monotonically increased until it reaches some value ~. Letf~ denote the corresponding value of fz(t). As a result of the above input increase, we find that for any 0 and r we have ~a,~,UO,q~(t) = +1 if a' < c~cos 0 and r q~(t) - - 1 if a' > a cos 0. This is illustrated geometrically in Fig. 4.31. Consequently, for any 0 and (p we have

fL ,>>.~'v(a', ff

dis

do~'dfl'

v(a', fl')da' a f t ' - f f

JJs;,o

v(a', fl')da' aft'.

(4.161)

Finally, we assume that the above input increase is followed by a subsequent monotonic decrease of u(t) until it reaches some value ft. Let f~/~ denote the corresponding value of fz(t). The above-mentioned input decrease results in geometric diagrams shown in Fig. 4.32. According to this

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

493

figure, we have /f~ '>~]~'

- dffi s

fl')~,/~, , u 0,q~(t) da' dfl'

v(~x',fl')&x' aft'- [[ v(~x',fl')d~x' dfl'. JJs;, ,o

(4.162)

By substituting (4.161) and (4.162) into (4.160), then by subtracting one expression from another and by using (4.68) and (4.131), we derive

f0

2~0 ~ COS OP(a cos 0, fl cos 0) sin 0 dO d o = F(a, fl).

(4.163)

Since the integrand does not depend on q~, we obtain 7r

/0

cos 0 sin OP(a cos O, fl cos O) dO -

1

F(a, fl).

(4.164)

This is the integral equation that relates P(a, fl) to the experimentally measured first-order transition curves. It turns out that this equation is easily solvable. By using the change of variables x = c~cos O,

2 = fl/~,

(4.165)

Eqn. (4.164) can be represented in the form

fo

C~XP(x, 2x) dx - ~nnF(a, 200.

(4.166)

By differentiating the last equation with respect to c~, we find 1 d [~2F(~ ' ,~)] c~P(~, 2~) - 2~z dc~

(4.167)

which leads to the following final expression: P(c~, 2cr -

1 d [0r162 ' ~0r 2rcc~ dc~

(4.168)

It is remarkable that the solution of the identification problem for threedimensional isotropic Preisach models of vector hysteresis turns out to be m u c h simpler than the solution of the same problem for two-dimensional isotropic models.

CHAPTER 4 VectorPreisach Models of Hysteresis

494

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR PREISACH MODELS We shall first discuss the identification problem for the two-dimensional anisotropic model (4.65). As before, we shall use the function P(~, fl, q0) defined by (4.68). We shall relate this function to the experimental data represented by the sets of first-order transition curves measured along all directions e~0. These experimental data can be characterized by the function 1

F(or fl, r = ~(f~r -f~/~r

(4.169)

which will be assumed to be known in the subsequent discussion. Of course, it is not feasible to measure first-order transition curves for all directions e~0. However, it is possible to measure these curves for some finite meshes of directions e~0and to use subsequent interpolation for computing F(~, ]~, (p) for all 9). Consider local polar coordinates (p, ~) with the polar axis directed along the vector e~o'. The model (4.65) in local polar coordinates can be written as

f (t) =

-de/

v(~, fl, q~' + d/)~ue/(t) d~ dfl dO,

(4.170)

where the relationship between the angles q0, opt and ~ is illustrated by Fig. 4.33. By using the expression (4.170) and by repeating almost literally the same reasoning as in the derivation of the integral Eqn. (4.132), we find that the function P(a, fl, q0) is related to the function F(a, fl, q0) by the expression

f

~ _~cos OP(a cos ~, fl cos ~, q0' + O)dO - F(a, fl, (p'). 2

FIGURE 4.33

(4.171)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

495

Next we shall use the following Fourier series expansions: (x)

F ( ~ , fl, (p' ) -

E

Fn ( o~, fl) e in~~,

(4.172)

Pn((X, fl)e inq~

(4.173)

Vn ( Ot, fl) e inq).

(4.174)

tl-----r

(x)

P(r fl, q ) ) -

E tl'----(X)

oo

V( Ot, fl, q)) -

E tl-'--O0

It is clear from (4.68) and (4.69) that Pn and

Pn(ot, fl) - / f T Vn(O~, f l ) - -

(~,~)

Vn are related by the expressions:

Vn(O~',fl') da' dfl',

~2pn(~, fl)

.

(4.175)

(4.176)

By substituting (4.172) and (4.173) into (4.171), we find

E

einqr / ~~ einr

n----oo

I[IPn(c~cos ~, fl cos I]/) dO

2 oo

-- E

Fn(c~, fl) e inq~

(4.177)

tl----(X)

From (4.177) we obtain

f

2 einr/ COSI~Pn(or cos ~, fl cos ~) dO = Fn(or fl) (n -- O, +1, 4-2 . . . . ).

(4.178)

Since sin n~ cos 6Pn(a cos 6, fl cos ~) is an odd function of ~, from (4.178) we derive

/2

_n

cos $ cos n$Pn((Z cos $, fl cos $) d $ - Fn(O~, fl)

(n = 0, 4-1, +2 . . . . ).

(4.179)

Thus we have obtained the infinite set of decoupled (separate) integral equations for Pn. The right-hand sides Fn(~X, fl) of these equations can be computed by using fast Fourier transform (FFT) algorithms.

CHAPTER 4 Vector Preisach Models of Hysteresis

496

For the case n = 0, from (4.179) we find _n 2 COS ~/P0( 0~c o s ~/, ]~ c o s ~/)

d6

= F0(a,

fl),

(4.180)

which naturally coincides with the integral Eqn. (4.132) for the isotropic vector Preisach model. If n = 4-1, from (4.179) we obtain

f_

~ c o s 2 ~P+l(a cos ~, fl cos ~) d~ = F+l ((x, fl). _n 2

(4.181)

Closed-form solutions to integral Eqns (4.180) and (4.181) can be found by using the change of variables x - a cos ~,

2=

fl/~

(4.182)

and by reducing the above equations to the Abel-type integral equations. In the case of Eqn. (4.180), the result is readily available and can be expressed by the formula P0(~, 2~)

1 d fa~ s2Fo(s, 2s) ~a d~ vu ~--~2~i2s as.

(4.183)

In the case of Eqn. (4.181), the change of variables (4.182) leads to the integral equation f0 a

X 2 dx 0r2 P+I (x, 2x) ~/0r2 _ x 2 -- -~-F+I (0r 200.

(4.184)

These equations can be solved in exactly the same way as Eqn. (4.148). The final result is given by the following formula P+I (a, 2a)

1

d [ ~ s3F+l(S, 2s) ds.

~0~2 d~ J0

(4.185)

V/o~2 -- S 2

Thus, if w e are interested only in the firstthree terms of the Fourier expansion for P (this is a first-order approximation for anisotropic media),

then the solution of the identification problem can be found by using the closed-form expressions (4.183) and (4.185). In the case of higher-order terms, the change of variables (4.182) leads to the following Abel-type integral equations

fo ~ Pn(x, 2x) ~/O~ xTn(X) ~ 2 _ X2 d x - -i-dFn((X, 2a),

(4.186)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

497

where Tn are Chebyshev polynomials defined by (4.190). Unfortunately, we have not been able to find closed-form solutions to these equations. However, there are many efficient numerical techniques developed for the solution of integral equations of this type (see, for instance, [12]). The analytical solutions to integral Eqn. (4.179) can also be found by exploiting the following property of this equation. THEOREM. Consider operators lr

~4nP - / 2~ cos ~ cos n~P(~ cos Ifi, fl cos ~) d~.

(4.187)

2

Monomials ~kfls are eigenfunctions of these operators. PROOF. It is straightforward to check the validity of the above statement and to get the following expressions for the eigenvalues "~k+s" j(n) . 2k+s (n) - 2

fo

cos nO cos k+s+l 0 dO

(4.188)

The explicit formula for the integral in (4.188) can be found in the literature (see [13]). [] By using the above theorem, polynomial solutions to the integral Eqn. (4.179) can be easily found. Indeed, if the right-hand sides of these equations are polynomials in the form (4.136), then the solutions to these equations will be polynomials in the form (4.137). Polynomial coefficients of the right-hand sides and the solutions will be related by the expression (4.138) where instead of eigenvalues ~m the eigenvalues ~(n) should "~k+s be employed. By taking advantage of this fact, Chebyshev polynomial expansions of type (4.139) for Fn(~, fl) can be used and the solutions can be found in the form (4.143), where the expansion coefficients can be computed by using expressions similar to (4.146). Now, we proceed to the identification problem for the three-dimensional anisotropic vector Preisach model (4.62). This problem is technically more complicated than the corresponding problem for two-dimensional models and, for this reason, it requires special treatment. It turns out that some facts from the theory of irreducible representations of the group of rotations of three-dimensional Euclidean space are instrumental in the treatment of this identification problem. As before we shall use the function P(~, fl, 0, q~) defined by (4.68) and we shall relate this function to the first-order transition curves experimentally measured along all possible directions e0,~0. These curves can be used

CHAPTER 4

498

Vector Preisach Models of Hysteresis

to define the function 1

F(o~, fl, O, (p) = -~(f~or -f~oq,).

(4.189)

It is convenient to use spherical harmonic expansions for the functions v, P andF: oo

V(O~,fl, O, q))= E k=0

k

E Vkm(O~'fl)Ykm(O' q))' m=-k

(4.190)

k

P(~, fl, O, r

E

E Pkm(~ fl)Ykm(O, r k=O m=-k o0 k

F(~, ,8, O,q~)= E

E Fkm(a,fl)Ykm(O,q~),

k=O

(4.191)

(4.192)

m=-k

where Ykm(O, q~) are spherical harmonics, while Vkm, Pkm a n d Fkm are corresponding expansion coefficients. It is clear from (4.68) and (4.69) that Pkm and Vkm are related by the following expressions:

Pkm(O~,fl) = f f Vkm(Off, if) ,liT (~,~) Vkm(e, f l ) = -

a2pkm(O~,fl)

d e ' dfl',

9

(4.193)

(4.194)

It turns out that the identification problem can be reduced to the solution of special integral equations which relate Pkm(~, fl) to Fkm(Ct, fl). The derivation of these equations proceeds as follows. Consider an arbitrary direction specified by the angles ff and q~'. We shall use a local coordinate system xyz, which is obtained from the system XYZ by Euler rotations Rz(0), Rx(0') and Rz(g7~ + q~') (see Fig. 4.34). The Euler angles for the inverse rotation from xyz to XYZ are equal to grc - ~0', 0 f and 7r, respectively. It is clear that the direction of the z-axis coincides with the direction of-do,e,. By using local spherical coordinates ~ and ~, we can represent the model (4.62) as

(/I., x sin r d~ d~.

o,,

d.d,) (4.195)

By using the last expression and by repeating almost literally the reasoning that was used in the derivation of the integral Eqn. (4.163), we can show

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

499

z z

y

o! i

x

,u

X

FIGURE 4.34

that the model (4.195) matches first-order transition curves measured along the direction e0'~0' if the function P satisfies the integral equation

J0270: cos ~P(a cos ~, fl cos ~, ~, ~, 0t, q~') sin ~ de d6 = F(~, fl, 0', ~0').

(4.196)

Next, we shall relate the function P in local coordinates to the same function in spherical coordinates 0 and q0. To achieve this, we shall use the spherical harmonic expansion (4.191) and the following facts from the theory of irreducible representations of the group of rotations (see [14-16]). Linear combinations of spherical harmonics Wkmof the fixed order k form a linear space Hk of the irreducible representation of the group of rotations. This means that any spherical harmonic Ykm(O, q)) in coordinates 0 and ~0 can be represented as a linear combination of the spherical harmonics Ykm(~, ~) of the same order k: k

Ykm(O,,q),)

-

-

~ k atom, ( l'r. -- q)') Ykm'(~, 0), ~, 0', -~ ml'--k

(4.197)

where atom, k are the matrix elements of the irreducible representation of the rotation group in Hk. These matrix elements are functions of Euler angles that determine the rotation from the local xyz coordinate system to the original XYZ system. These functions are sometimes called generalized spherical harmonics because they are reduced to spherical harmonics when m or m t is equal to zero.

CHAPTER 4 Vector Preisach Models of Hysteresis

500

By using (4.197) and (4.191), we find the expression for P in local coordinates

P(~, ~, ~, r 0', ~') k

k

--

amm, ~z, , -~ - q;' Pkm(O~,fl)Ykm'(~, 6). (4.198) m=-k m'=-k By substituting (4.198) into (4.196), we find k=0

F(~, ~, 0', ~') c~ -

k

k amm,

-

k=0

~,

, -~ -- qJ

m=-k m'=-k

x ~O2~z~O~ cos~Pkm(O~cos~, ~cos~)Ykm,(~, 0 ) s i n ~ d ~ d 0 .

(4.199)

In (4.199) the integral with respect to 6 is equal to zero if m' r 0, and it is equal to 2Ir if m' = 0. Thus we obtain

F(~, ~, 0', ~') oo

k

k=0

m=-k

= 27r

x

)

amo re, 0', ~ - q~'

~0 arc

COS~Pkm(~Xcos ~, fl cos ~)Yko(~) sin ~ d~.

(4.200)

It is known from group theory (see [16]) that

k ( 7r ) m / 4re amo ~z, 0',~--q0' - - ( - - 1 ) ~ 2 k + 1

Ykm(O', ~o').

(4.201)

It is also known (see [12]) that ~/2k + 1 Lk(cos ~), 4~r

Yko(~) =

(4.202)

where Lk are Legendre polynomials. From (4.200), (4.201) and (4.202), we derive oo

F(~, fl, 0', ~0') - 2Ir ~ k=0

k

~ (-1)mykm(O ', q)') m=-k

/[

x

Pkm(O~cos ~, ~ cos ~)Lk(cOS ~) cos ~ sin ~ d~.

(4.203)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

501

By comparing (4.203) with (4.192), we find f0 ~ Pkm(~Xcos ~, fl COS ~)Lk(cos ~) cos ~ sin ~ d~ = (-1)m 2----~Fkm(O~,fl). (4.204) These are the final integral equations that relate Pkm to Fkm. These equations can be reduced to Volterra integral equations by using the following change of variables x - ~ cos ~,

2 = -fl.

(4.205)

Indeed, after simple transformations, we derive

f0 ~ Pkm(X, 2x)XLk (X) dx --

~2 (--1)m-~-~Fkm(C~, ~,~x).

(4.206)

Explicit solutions to Eqn. (4.206) can be found for the cases k = 0 and k - 1. If k = 0, from (4.206) we obtain

~0~xPoo(x, 2x) dx = ~-~F00(c~, ~ 2~).

(4.207)

By differentiating (4.207) with respect to ~, we find P00(a 2a) -

1

d [a2F00(~ ,2~)].

2 ~ d~

'

(4.208)

As expected, this result coincides with the one obtained for the isotropic model (see (4.168)). If k = 1, according to (4.206) we have

f0 c~X2plm(X,

0r3 Flm(~X, ,~00 )~x) dx - ( - 1 ) m -~--~

(m = 4-1, 0).

(4.209)

By differentiating (4.209), we derive Plm( 0~, ,~,~) _

(-1) m d 27r~ 2

d~

[~Z3Flm(0~' ~)]

(m - +1 0).

(4.210)

Thus, if we are interested only in the first four terms of spherical harmonic expansion for P (this is a first-order approximation for anisotropic media), then the explicit analytical solution for the identification problem is given by formulas (4.208) and (4.210). For k > 1, Eqn. (4.206) can be solved numerically. Discretization procedures can be applied directly to Eqn. (4.206), or the equation can first be

502

CHAPTER 4 VectorPreisach Models of Hysteresis

reduced by differentiation to the Volterra equation of the second kind

Pkm(~Z,,~0~)-- fo ~Pkm(X, ,~x)-~x Lk,(X)d x = (-1) m d [o~2Fkm(~,~)]. (4.211) 2z~a da In both cases we shall end up with simultaneous algebraic equations with triangular matrices which are easy to solve. However, the reduction to the second kind of integral equation may be desirable as far as computational stability is concerned. Finally, the integral Eqn. (4.204) can be solved analytically if polynomial approximations for the right-hand side are employed. These analytical solutions can be found in exactly the same manner as for twodimensional identification problems discussed before. Up to this point, we have used scalar hysteresis data (4.169) and (4.189) measured for unidirectional variations of input ~(t) in order to solve the identification problems. However, for anisotropic media these input variations result in vectorial data that for two- and three-dimensional problems can be represented in the following forms, respectively 1

F(~, fl, ~#)= ~ ( f ~ -)?~/~),

(4.212)

F(a, ]/, 0, q~)= ~ 07~0~ -)?~//0r

(4.213)

The above vectorial data account for the output components that are orthogonal to the directions of input variations. There is a natural desire to utilize these vectorial data in the identification of vector Preisach models. This can be achieved by generalizing the models themselves. The essence of generalization is in employing vectorial functions ~. This leads to the following vector Preisach model:

f(t) - ~lTl:l ( Sf~>>.~7(~'fl")~a~(-f "u(t))d~

dsr"

(4.214)

By repeating the same line of reasoning as in Section 4.2, we can show that by means of redefining ~ the integration over a unit sphere in (4.214) can be reduced to the integration over a unit hemisphere

f(t) - //c+ ( ff~>>.~v(~z'fl'-f )~;a~(-f"u(t))d~

dsr"

(4.215)

It can also be shown that the redefined function ~ has the following symmetry property ~(~, fl, ~) - -~(-fl, -~, - ~ ) . (4.216)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

503

By using spherical coordinates, the generalized model (4.215) can be represented as

-* - ~02rc~0~( f ~ f(t)

~(a, fl, O, q~)~a~uoq~(t) da dfl ) sin 0 dO dq~.

(4.217)

Similarly, the two-dimensional model can be expressed in the form /r

f (t) =

~(a, fl, q~)~13uo(t) dadfl

dq~.

(4.218)

To solve the identification problems for the models (4.217) and (4.218), we as before introduce the following auxiliary functions P(~, fl, O, q~) = P(a, fl, q~) =

(~,/~) (~,/~)

?(a', fl', O, q~) dor aft,

(4.219)

~(a', fl', q~) de' dfl'.

(4.220)

These functions can be related to the experimental data (4.213) and (4.212) by the following equations, respectively

fo2O

P(0r cos ~, fl cos ~, ~, ~, 0', q~') sin ~ d~ dO = F'(a, fl, 0', q/), (4.221) /r

f ~ P(acos$, flcos$, q~' + ~ ) d $ - F ( ~ , fl, q~').

2

(4.222)

By employing spherical harmonic expansions in the three-dimensional case and Fourier expansions in the two-dimensional case, we derive as before the following integral equations cos O, fl cos O)Lk(cos ~) sin ~ d~ = (-1) mF-km(~, [3) (4.223) f0~_Pkm(O~ . 2~r /r

2

COSnl~Pn(o~ cos ~, fl cos ~) d o = Fn(o~, fl).

(4.224)

By using the change of variables (4.182) and (4.205), the above integral equations can be reduced to the following forms, respectively

f0

()

(4.225)

~0 ~ -" Zn(X) d x - 1 F-n(O~,,~,o0 Pn(x, ~,x) V/0~2 _ x 2 n2 .

(4.226)

Pkm(X, 2x)Lk -dx d x - (--1) m a ~_km(a ' 2~x),

504

CHAPTER 4 Vector Preisach Models of Hysteresis

When k and n are equal to one, the following explicit solutions of Eqns (4.225) and (4.226) can be derived: Plm(~, ,~or = (--1)m d [o~2~71m(Or ~,or 2rc~ dc~

(m = O, +1),

P1 (a, 2a) = 1 d f0 ~ SaFl(S, 2s) ~ dc~ v/---2-2 82 ds.

(4.227)

(4.228)

For other values of k and n, polynomial expansion techniques or numerical techniques can be employed for the solution of the above integral equations. For isotropic media, the data (4.212) and (4.213) are reduced to (4.131). It can be shown that in this case ~(~, fl) = ~q~V(~,fl) and thus the generalized model (4.218) is reduced to the model (4.67). The proof of the above statement is left to the reader as a useful exercise.

4.6 D Y N A M I C

VECTOR PREISACH MODELS

OF

HYSTERESIS The vector Preisach models of hysteresis that have so far been discussed are rate-independent in nature; they do not account for dynamic properties of vector hysteresis nonlinearities. The purpose of this section is to develop dynamic vector Preisach models of hysteresis. We shall discuss only isotropic dynamic vector models of hysteresis. We begin with two-dimensional dynamic models; a straightforward extension to three dimensions will then follow. The main idea of the design of the dynamic vector hysteresis models is to introduce the dependence of the function v for scalar Preisach models for all directions q0 on the speed of output variations, dd~, along these directions. This can mathematically be expressed as

f (t)

-

f ~ -de ff~ >>-~v a, fl, c3t ,I ~ u ~ o ( t )

d~ dfl) dqg.

(4.229)

The direct utilization of the above model is associated with some untractable difficulties. These difficulties can be circumvented by using the power series expansion of the v-function with respect to df~. dt

v(~, B, --~ df~) = vo(~,~) + --~-Vl(~, df~ fl) q - . - . .

(4.230)

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

505

By retaining only the first two terms of the above expansion, we arrive at the following dynamic model: 7[

f(t) =fo(t)+ L~-dodf~ ( f L 2

"-~

Vl(a, fl)~;afluo(t)dlzdfl)dq~,

(4.231)

~fl

where

fo(t) -

-do

vo(cc,fl)~3uo(t) d~ dfl dq).

(4.232)

It is clear that in the case of very slow input variations the second term in the right-hand side of (4.231) becomes negligible. Thus fo(t) can be construed as a rate-independent component. This means that the function v0(~, fl) should coincide with the v-function of the rate-independent model (4.67). In other words, the function v0(a, fl) can be determined by matching rate-independent first-order transition curves measured for unidirectional variations of the input ~(t). We next represent the model (4.231) in Cartesian coordinates. To this end, we shall use the following expressions: eo - ex cos q~ + ~y sin q~, (4.233)

df 0 =G df dt

dfx

9 ~-~ = cos q~~

dfy

(4.234)

+ sin q~ dt "

By substituting (4.233) and (4.234) into (4.231), after simple transformations we arrive at the following form of the model (4.231): .

-

--~ -- f (t) - fo(t),

(4.235)

where the matrix,4 is given by

fii_(?Zxx(FZ(t)) ?Zxy(~(t))) ?Tyx(~(t)) ~yy(~(t)) '

(4.236)

and the matrix entries are specified by the expressions: --

f

COS 2 q9

(If

Vl(a,

)

fl)9~3uo(t) da dfl dq~,

(4.237)

/z

--f

sin2 r ( / L

Vl(~Z,fl)7afluo(t)d~zdfl)dcP,

(4.238)

?Zxy(F~(t))- ?Zyx(~(t)) 7[

cos O sin q~

/z

2

Vl

>/3

(0~,fl)~flUo(t) de dfl)

do.

(4.239)

CHAPTER 4 Vector Preisach Models of Hysteresis

506

Thus the dynamic model (4.231) can be interpreted as a set of two coupled ordinary differential equations (4.235) with hysteretic coefficients (4.237)-(4.239). The expression (4.235) also suggests that the instant speed of output variations is directly proportional to the difference between instant and rate-independent output values. The last fact is transparent from the physical point of view. We next turn to the identification problem of determining the function Vl(a, t) by fitting the model (4.235) to some experimental data. The following experiments are used to solve this problem. We restrict ~(t) to vary along the direction r = 0, which means that ~(t) = -~xu(t). First, we assume that u(t) is made 'infinitely negative' and then it is monotonically increased until it reaches some value a at t = to. Afterwards, the input is kept constant. As the input is being kept constant, the output relaxes from its valuef~ at t = to to its rate-independent value f0~. Owing to symmetry, we have fo(t) - -~xfo(t),

f (t) - -~xf(t).

(4.240)

Thus, according to the model (4.234)-(4.239), the above relaxation process is described by the differential equation ~T~ ~d f

= f i t ) - foa

(4.241)

where "~a =

COS2 (p

(ff

V1(a~, fl') da t dfl'

- ffs+ ~ vl (a', fl')de ~dfl') do,

(4.242)

and geometrical configurations of S+~,r and S;, r are the same as in Fig. 4.28. The solution to Eqn. (4.241) is given by t

fit) -- (f~ - f o ~ ) e - ~

+ fo~.

(4.243)

Thus z~ has the meaning of relaxation time and can be experimentally measured. Next the input u(t) is made again 'infinitely negative'. Then it is monotonically increased until it reaches the value a. Afterwards, the input is ! monotonically decreased until it reaches some value fl at time t = t o and it is kept constant for t > t~. As the input is being kept constant, the output relaxes from its value f~/~ at t - t~ to its 'static' value f0~3. Owing to

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

507

symmetry, expressions (4.240) hold and the model (4.235)-(4.239) yields the following differential equation for the above relaxation process:

df =fit) - fo~fl,

(4.244)

-r~-d-i where 7C

~ COS 2 q)

~/~ --

2

(ffs

Vl(~Z', fl') &z' dfl'

_

~,fl,~o

~,/~,~

and the geometry of regions S + 0~,(p and S~,~ is the same as in Fig. 4.29. By solving (4.244), we find: t

f(t) = (f~/~-f0~/~)e ~ + f 0 ~ .

(4.246)

Thus, %/~ has the meaning of relaxation time and can be experimentally measured. We next show that by knowing relaxation times % and %/~ for all possible a and fl we can determine the function Vl (~, fl). To this end, we introduce the functions 1

q(~, fl)= ~(% - %/~),

(4.247)

P1 (~, fl) - I f

Vl (~, if) d~ ~dff, JJT (~,fl) where T(~, fl) is the triangle shown in Fig. 4.10. It is clear as before that P1 and Vl are related by the formula Vl(~, fl) - - ~2P1 (~' fl).

(4.248)

(4.249)

8~ 8fl Thus, if the function P1 (~, fl) is found, then the function Vl (~, fl) can be retrieved. However, from the computational point of view, it is more convenient to deal with the function PI(~, fl) rather than with Vl(~, fl). This is because the double integral with respect to ~ and fl in expressions (4.237)(4.239) can be explicitly expressed in terms of PI(~, fl) by using formulas similar to (4.71). Another advantage of using PI(~, fl) is that this function can be directly related to the experimental data (4.247). Indeed, by using the expressions (4.242), (4.245), (4.247), (4.248) and Figs 4.28 and 4.29, we derive

f2

n COS2 (PP1 (a c o s r

fl c o s ( p ) d o

= q(a, fl).

(4.250)

CHAPTER 4 VectorPreisach Models of Hysteresis

508

The expression (4.250) is the integral equation that relates the function P1 (a, fl) to the experimental data q(a, ]/). This equation is similar to the integral Eqn. (4.132) and, consequently, the same techniques can be used for the solution of Eqn. (4.250) as for Eqn. (4.132). Namely, by using the change of variables x = ~ cos q), 2 = ]//~, (4.251) Eqn. (4.250) can be reduced to the following Abel-type integral equation: fo ~

x 2 dx 0r2 P1 (x, ,,Ix)V/(x2_ x2 -- --~-q(~x,,~).

(4.252)

By using the technique discussed in Section 4.4, the following closed-form solution of the above equation can be found:

PI (O~,,~o0 -

1 d fo ~ s3q(s, ,~s) ds. ~0r2 d ~

(4.253)

~/0r _ s2

We next turn to the discussion of three-dimensional dynamic vector Preisach models of hysteresis. Similar to (4.231), these models can be represented in the following mathematical form

df~176 ( fl >~ vl(a,fl)%~uo~o(t)dadfl)

f_.(t) = fo(t) _. + foaZ~fO~ -dOq)-~-[x sin 0 dO do,

(4.254)

where f0 (t) represents a'static' component of hysteresis nonlinearities defined by the expression ?~

- f02~f0~eoq~-"( /f~)~ Vl(~, fl)~Uoq~(t) d~ d]/ ) sin 0 dO dq~. (4.255)

By using Cartesian coordinates, we have e0~o- ex cos ~osin 0 + ~y sin q~sin 0 + ez cos 0,

dfo, dt

= cos ~0sin 0~-~f; + sin q0sin 0~-~f~4- cos 0 dfz dt "

(4.256) (4.257)

By substituting (4.256) and (4.257) into (4.254), after simple transformations we find df - f ( t ) -f0(t),

(4.258)

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

509

where the matrix,4 and its entries are given by

~l-

(

?lxx(~(t)) ?Zxy(~(t)) ?Zxz(~(t))) ?Zyx(~(t)) ~yy(~(t)) ?lyz(~(t)), ?Zzx(~(t)) ?~z~(F~(tl) ?~zz(~(t))

(4.259)

?~xx(F~(t))= foar~fO~ cos 2 qosin 3 0 X (/~

~yy(~(t)) =

Vl(~ ,

~>~

f020 x ( /L

(4.260)

sin 2 qosin 3 0

x (/f~

~lzz(U(t))--

fl)~&~uo~o(t)dc~d~)dOdq0,

Vl(~x,fl)~fluo~o(t)doraft)dO dqo,

(4.261)

COS2 0 sin 0

vl (a, fl)~a~uom(t)do~dfl) dOdq~,

(4.262)

?~xy(~(t))= ?Zyx(~(t))- ~O2'~fO-~cos q~sin q~sin 3 0

axz (u(t)) -- azx (u(t)) --

x (ff~

Vl(~X,fl)~&BUOq~(t)&xdfl)dOdq~, (4.263) ~>B

f02f0

cos q~cos 0 sin 2 0 Vl(C~,fl)~BUOq~(t)&xd~)dOdq~, (4.264)

x (/f~ ~yz (~(t)) - ~zy (~(t)) =

f0210

sin q~cos 0 sin 2 0

x (/f~

Vl(C~,~)}~,Uoq)(t)&xd[3)dOd~0. (4.265) ~>~

Thus the three-dimensional dynamic vector Preisach model of hysteresis (4.254) can be interpreted as a set of three coupled ordinary differential equations (4.258) with hysteretic coefficients (4.260)-(4.265). We next proceed to the discussion of the identification problem for the model (4.258)-(4.265). The solution to this problem is very similar to

510

CHAPTER 4 Vector Preisach Models of Hysteresis

that for the model (4.235)-(4.239) except that the final form of the solution is much simpler in three dimensions than in two dimensions. The experimental data used for the identification of the model (4.258)-(4.265) are measured when the input is restricted to vary along the z-axis, that is, when ~(t) - -~zU(t). As before, two types of relaxation processes are considered. The relaxation processes of the first type occur when the input is made 'infinitely negative' and then monotonically increased to some value a and kept constant thereafter. The relaxation processes of the second type occur when, starting from the state of negative saturation along the z-axis, the input is first monotonically increased to some value a, then monotonically decreased to some value fl and kept constant afterwards. The relaxation times r~ and r~/~ of the above processes can be measured and used for computing the function q(a, fl) defined by (4.247). On the grounds of symmetry, for both types of relaxation processes we have f0 - ~zf0(t),

f(t) - -~zflt).

(4.266)

This results in the reduction of the model (4.258)-(4.265) to the following equation

df

~zz (~(t)) ~ = fit) - fo (t).

(4.267)

Now, by using the same line of reasoning as in the derivation of Eqn. (4.250), we can show that the function P1 (cx, fl), defined by (4.248), is related to the experimental data q(a, fl) by the integral equation f0 ~ Pl(a cos 0, fl cos 0) cos 2 0 sin 0 dO =

q(a, 2~

"

(4.268)

By using the change of variables x = a cos 0,

;~ = -fl,

(4.269)

the integral Eqn. (4.268) is reduced to the form ~0 a x2p1 (x, ,~x) dx - -~-~ ~3 q(a, ,~),

(4.270)

from which we derive the final expression PI(X, ,~x) --

1 d [cx3q(or ,~or 27ca2 da

(4.271)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

511

4.7 GENERALIZED VECTOR PREISACH M O D E L S OF HYSTERESIS. EXPERIMENTAL TESTING There are two ways in which the above vector Preisach models can be further generalized. The first way is to use generalized scalar Preisach models as the main building blocks for the construction of vector models. The second way is to generalize the notion of input projection Uq~(t). We begin with the first approach. To be specific, we shall use the nonlinear (input-dependent) scalar Preisach models. Analysis of generalized vector Preisach models of this type is very similar to that for the 'classical' vector Preisach models. For this reason, our discussion will be concise and will be centered around the description of final results, while filling in the details will be left to the reader. We begin with two-dimensional isotropic models that can be represented in the following mathematical form

f (t) =

-

f (ff-~q~

v(~, fl, u~o(t))~Uq~(t) d~ dfl dq)

)

u,p(t)

4- ~(u(t))fi(t).

(4.272)

The numerical implementation of the above model is substantially facilitated by the use of the following function P(~, fl, u) - f f

v(~', fl', u) d~' dfl',

(4.273)

, I l K aflu

where R~/~u is the rectangle shown in Fig. 4.12. By employing the above function we can find explicit expressions for the double integral with respect to a and fl in (4.272). In addition, the function P(a, fl, u) can be directly used for the identification of the model (4.272). First- and second-order transition curves measured along any fixed direction (for instance, along the direction (0 -- 0) will be utilized for the solution of the identification problem. By using these curves, the following function can be constructed

1

F(a, fl, u) - -~(f~u - f~u),

(4.274)

where f~u and f~u have the meaning of output values along the first- and second-order transition curves, respectively. It can be shown that the function P(a, fl, u) is related to the experimental data (4.274) by the following integral equation 7~

f

~ cos q~P(a cos r 2

cos q~, u cos q~) dq~

F(a, fl, u).

(4.275)

CHAPTER 4 Vector Preisach Models of Hysteresis

512

This equation is very similar to the integral Eqn. (4.132). Thus, the same techniques can be employed for the solution of Eqn. (4.275) as for the solution of Eqn. (4.132). For instance, an approximate polynomial solution of the above equation can be found. The finding of this solution is based on the fact that monomials akfl sum are eigenfunctions of the operator ti: 7~

AiP = f 2 cos q~P(~ cos ~0, fl cos ~o, u cos q~) d~o.

(4.276)

7~

2

In other words, we have A o~kfl s U m -- ,~k+s+mO~k fl s u m,

(4.277)

,~k+s+m -- /_~ (COScp)k+s+m+l dcp.

(4.278)

where: 2

Now, by expanding the right-hand side of the integral Eqn. (4.275) into the series of Chebyshev polynomials

F(a, fl, u) = y ~ a~qmT~(~)Tq(fl)Tm(u),

(4.279)

f.,q,m

we can represent the solution of the above equation in the form P(a, fl, u) = y ~ aeqmt~qm(a, fl, u),

(4.280)

~,q,m

where the polynomials t~qm(a, fl, u) are m a p p e d by operator A into polynomials T~(~)Tq(fl)Tm(u). Polynomials t~qm(a, fl, u) can be determined by using expressions (4.277) and (4.278). As far as expansion coefficients a~qm are concerned, they can be computed by using a formula similar to (4.146):

8 f0/0/0 F (cos 0, cos 0', cos

ae.qm = ~

x cos s cos qO~cos toO" dO dO~dO".

(4.281)

It is also possible to find a closed-form solution to Eqn. (4.275). To this end, the following change of variables is used: x = ~ cos q~,

2 = fl/~,

;~ = u/~

(4.282)

and the above equation is reduced to the Abel-type integral equation fo x v/cz2x_ x 2 P(x, 2x, ;~x) dx = ~F(~, ~ 2a, ;~a).

(4.283)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

513

Using the same reasoning as in Section 4.4, we obtain the following solution of Eqn. (4.283):

1 d fo ~ s2F(s, 2s, Zs) ds. P(~, 2~, Z~) - ~ d~ V~2 _ S2

(4.284)

As is seen from the above discussion, function P(~, fl, u) (and consequently function v(a, fl, u)) can be determined by matching the increments (4.274) between the first-order and second-order transition curves. The function ~(u) in (4.272) can be found by matching an ascending branch of a major loop. This leads to the expression ~(u) - f + + f u . 2u

(4.285)

We next proceed to the discussion of three-dimensional generalized vector Preisach models of hysteresis. These models can be represented as follows

f-"(t) -

/0270rt-dOq)(/[

v(~, fl, UOq)(t))~UOq~(t) d~ aft ) sin 0 dO dq0 uoq~(t)

+ ~(u(t))F~(t).

(4.286)

By introducing function P defined by (4.273) and by using experimental data (4.274) the identification problem for the model (4.286) can be reduced to the following integral equation 2re f0 ~ cos 0 sin OP(oecos O, fl cos O, u cos O) dO - ~(~,/~, u) "

(4.287)

The explicit solution of the above equation can be obtained by using the change of variables

u/~

(4.288)

f0~xP(x, 2x, Zx) dx - ~-~F(a, 0~2 2a, Za).

(4.289)

x = ~ cos O,

2 =///~,

Z=

and by reducing (4.287) to the integral equation

Differentiation of (4.288) yields 1 d [a2F(~ ' 2a, Za)]. P(~, 2~, Z~) - 2 ~ dc~

(4.290)

We next turn to generalized dynamic vector Preisach models of hysteresis. For the sake of notational simplicity, we consider only two-dimensional

CHAPTER 4 VectorPreisach Models of Hysteresis

514

models; extensions to three-dimensional models are straightforward. The two-dimensional generalized dynamic Preisach models can be defined as follows \

f (t) - fo(t) +

Vl (e, fl, Uq)(t))~BUq~(t) de dfl) do,

-b~o --~

u~o(t)

(4.291) where f0 is a 'static' component of hysteresis nonlinearity which coincides with (4.272). By using Cartesian coordinates, the model (4.291) can be reduced to the following ordinary differential equations df -. -. A~-~ =f(t) -fo(t),

(4.292)

where the matrix,4 has the following entries:

xxlO t t/ =

Vl (e, fl, U~o(t))~aBUq)(t)de dfl) dq~, (4.293)

COS 2 (p

u~o(t)

V1(e, fl, Uq~(t))~;~BUq~(t)de dfl) do, (4.294) 2

U~o(t)

?Zxy(~(t)) = ~yx(u(t)) = / _ ~ cos ~osin qo 2

X (ffR

uq~(t)

vl(e, fl, Uq~(t))~;aflUq)(t)dedfl)dq~.

(4.295)

To solve the identification problem for the model (4.292)-(4.295), we restrict input to vary along the x-axis. Then, on the grounds of symmetry, it is easy to conclude that f (t) = -Grit), fo(t) - -exfo(t). (4.296) This results in the reduction of the model (4.292)-(4.295) to the following equation

^ df[ =fit) -fo(t) axx(U(t))--d-

(4.297)

We next introduce the function Pl(e, fl, u): P1 (e, fl, u) - / f R

Vl(e',/J', u) de' dfl', aflu

(4.298)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

515

and experimental data: 1

Q(:r fl, u) = ~(%u - %flu),

(4.299)

where %u and r~flu are relaxation times for the processes, which are similar to those described in detail in the previous section. By using the expressions (4.298)-(4.299) and the same line of reasoning as in the previous section, it can be shown that the function P1 (cx, fl, u) is related to the experimental data (4.299) by the following integral equation

f

2 COS2 (PP1(~zcos (.p, fl cos q~, u cos qo) dq~ - Q(a, fl, u).

(4.300)

/c

By using the change of variables (4.282), the above equation can be reduced to the Abel-type integral equation f0 a

V/~x2X2_ x2P1 (x, ~x, )~x) d x

-

~X2 -~--Q(a, 2a, za),

(4.301)

whose solution is given by 1 d fo a s3Q(s, ,~s, zs) ds. P1 (0r 20r Z~x) - ~0r d~ ?~-2 _ s2

(4.302)

Finally, we shall discuss generalized anisotropic vector Preisach models. Again, for the sake of notational simplicity, we consider only twodimensional models. These models can be defined as /-r ..+

f(t)-L2(ffR

u,p(t)

~(oe, fl, cp, ur

d~ dfl d~p + ~(~(t)). (4.303)

The experimental data defined by the function -*

1 -*

fl, u,

-*

- i(f u ,

(4.304)

will be used for the identification of the model (4.303). As before, we introduce the auxiliary function

P(o~, fl, cp, u) = ffR

:r

~(~'' fl'' cp, u) da' dfl'.

(4.305)

CHAPTER 4 Vector Preisach Models of Hysteresis

516

We employ the following Fourier expansions: OO

P(a, fl, u, q)) -- ~

rJn(a, fl, u)e imp,

(4.306)

F-(a, fl, u, q))= y~ F-n(a, fl, u)e inc.

(4.307)

n~--OO OO

By using (4.304)-(4.307), it can be shown that Pn is related to Fn by the following integral equation cos

n~Pn(a cos ~, fl cos ~, u cos ~) d~ - Fn(a, fl, u).

(4.308)

This equation can be reduced to the Abel-type integral equation /0 ~ Pn(x, -" ,~x, Zx) ~rTn(X) 2 _ x 2 dx = ~1 Fn(a, ,~a, Za)

(4.309)

Various numerical techniques can be used for the solution of this equation. In particular cases when n --- 4-1, the following closed-form solutions can be found: -~ 1 d f ~ s2F+l(S, ,~s, Zs) P+I (a, 2a, Za) = ~a da ]0 2---7~/a - s2 ds. (4.310) As far as the function ~(~) in (4.303) is concerned, the following expression (similar to (4.285)) can be derived: ~(~pU) = f+~p +f~cp 2 '

(4.311)

where the notations in (4.311) are self-explanatory. Next we proceed to the discussion of the second (and probably most fruitful) approach to the generalization of vector Preisach models. This approach is based on the notion of generalized input projection. The corresponding vector model can be written as follows:

f (t)

= ,i-~12

-J4~

~fl

v(a, fl)~l~(t)lg[O(t) - ~b] dad]/ d~b,

(4.312)

where O(t) is the angle between ~(t) and the polar axis. In the case when g(O-q5) = cos(0-q)), the above model is reduced to vector Preisach models extensively studied in this chapter. This justifies the following' cosine-type'

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

517

constraints on the function g(~b): g'(~b)~<0 for 0~r g(0) - 1, g(~) = - 1 ,

g(2)

(4.313) = 0,

(4.314)

g ( ~ - ~b) = - g ( r

(4.315)

and the product Ifi(t)Ig(O- q5) can be construed as a generalized projection of vector input fi(t) on the direction specified by the angle r Function

1

g(r = I cos r (cos ~) is an example of a 'cosine-type' function that satisfies the constraints (4.313)-(4.315). In the above model, functions v(a, fl) and g(r are not specified in advance but rather should be determined by fitting this model to some experimental data. This is an identification problem. To perform the identification of the model, the following experimental data will be used. (a) First-order transition curves that are measured when the input fi(t) is restricted to vary along one, arbitrary fixed direction. By using these curves, we can introduce the function:

1

(4.316)

7(~,/~) = ~(f~ - f ~ ) .

(b) 'Rotational' experimental data measured for the case when the input is a uniformly rotating vector: fi(t) = {urn cos cot, um sin cot}. It can be shown that for isotropic hysteretic media, the output has the form f(t) =f0 +fl(t), where f0 does not change with time, whilef~ (t) is a uniformly rotating vector that lags behind the input by some angle. By using the rotational experimental data, the following function can be introduced:

R(um) =

fi(t) .fl(t)

Um

,

(4.317)

that has the meaning of the projection of fl (t) on the direction of input. Functions v(a, fl) and g(r will be recovered from experimental data (4.316) and (4.317). For the identification as well as computational purposes, it is convenient to introduce the function P(~, fl):

P(~' fl) --/fT (~,~) v(a', fl')d~' dfl',

v(a, fl) = -

O2p(~, fl)

.

(4.318)

518

CHAPTER 4 Vector Preisach Models of Hysteresis

By using the same line of reasoning as before, it can be shown that model (4.312) will match the experimental data (4.316) and (4.317) if functions P(a,//) and g(~b) satisfy the equations: 2

cos ~bP(eg(4~), ~g(~b)) d4~ = ~'(c~, ~),

(4.319)

-2

cos d~P(um, umg(d~)) dd~ = R(um).

(4.320)

Since v(e,/~) is nonzero only within the triangle r = {-e0~
.Jz'(O~,fl) "- y ~ n--0

y~ Uks-(2n+l)o~kfls. k+s=2n+l

(4.321)

Approximation (4.321) contains only odd terms because of the mentioned symmetry properties of ~'(a, ]/). Owing to these symmetry properties, it is also clear that Uks _(2n+1) -- --Usk ..(2n+1)9 An actual polynomial approximation (4.321) can be found by using a Chebyshev polynomial series expansion of ~(~, ]/) as described before. We shall also use the odd extension of R(um) from [0, a0] to [-a0, a0] and the following polynomial approximation: N

a(um)

(4.322)

= y ~ R2n+lU 2n+1 m 9 n--0

We shall next look for P(cr ]/) in the form N

P(a, fl) - y~ n=0

y~.

C:ks"(an+l)o~k~s.

(4.323)

k+s--2n+l

By substituting (4.321), (4.322) and (4.323) into (4.319) and (4.320) and by equating similar terms, we derive ..(2n+1) f ~ ..(2n+1) 2~'ks JO COS ~bg2n+l (~b) d~b = Uks

-2 ~ k+s=2n+l

_(2n+1)~07rCOS~gS (~b) d~b = Cks

(4.324) a2n+l.

(4.325)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

519

From (4.315) we find that the integrals in (4.325) are equal to zero if s is even and

f0

cos qbg s ( q5) d q5 = 2

/0

cos ~gS (~b) d~b

if s is odd.

(4.326)

Next, from (4.324) we obtain (2n+1) ao ' 2n+1 ~0 ~ COS ~g2n+l (~b) d~b. (2n+1) -- ~(2n+1) = 2 Cks ~0,2n+1

(2n+1) aks

(4.327)

From (4.325), (4.326) and (4.327) we derive . (2n+1) ~0 ~ cos ~g2n+l (~) d~b R2n+l -- -~c0,2n+l

-

~0 ~ cos c~gs (~b) d~b, 4 y ~ Ic(2n+1) ks

(4.328)

where Y-~' stands for the s u m m a t i o n over all s that are odd and less than 2n + 1 and such that k + s = 2n + 1. By using (4.327) in (4.328), we find .. ( 2 n + 1 ) _ 2 ~ , . , ( 2 n + l ) ~ a ~ 2 s n + l ) ) ( a ~ ) s ' ~ R2n+l -- -za0,2n+l ~'0,2n+1/ (2n+1) ~(s) ] ' \a0,2n+l COs / which leads to ~(2n+1) c0,2n+l - - _

., (2n+1) R2n+l q- za0,2n+l _(2n+1) (s) " ) ( a6s~ 2 ~ ' ( Uks _(2n+1) (s) ,] UO,2n+l COs-

(4.329)

(4.330)

By assuming that C0,1(1) 1 = 2,

(4.331)

where 2 is some u n k n o w n constant, from (4.330) we derive c(2n+1) _ ~h(2n+l) (4.332) 0,2n+1 . . . . 0,2n+1" ~(2n+1) According to (4.330), constants u0,2n+l can be recursively determined by using the formula ~,(2n+1) u0,2n+l - - _

_(1) and the fact that ,o0,1

_

1.

,. (2n+1) R2n+l -}- za0,2n+l

2E'(a(2n+l))(aoS)s ' k s "~ ..(2n+1) (s) u0,2n+l bos ,]

(4.333)

CHAPTER 4 Vector Preisach Models of Hysteresis

520

After h(2n+l) ~'0,2n+1 are computed, from (4.326), (4.327) and (4.332) we find f0 z~COS ~bg2n+1 (~) d~b - $2n+1 2 '

(4.334)

(2n+1) a0'2n+l $2n+1 -- ~(2n+1) " u0,2n+l

(4.335)

where

According to (4.313)-(4.315), there exists a function z(g) which is inverse to g(~b). By using this function, we shall use the following change of variables in integrals (4.334): ~b = ;~(g),

dq~ = Z ( g ) d g ,

z(g(~b)) = ~b.

(4.336)

By substituting (4.336) into (4.334) and integrating by parts, we transform (4.334) as follows: f_ +1 g2n sin )~(g) dg = 2(2n $2n+1 + 1)" 1

(4.337)

We shall next make another change of variables: g=cos~,

dg=-sin~d~,

(4.338)

which leads to f0 z~COSan ~ sin

6T(6) d~

=

$2n+1 ;~(2n + 1)

(4.339)

where T(6) - sin Z(cos ~).

(4.340)

We shall extend T(~) from [0, ~] as a periodic odd function with half-wave symmetry. As a result, this function can be expressed as N T(~t) - E t2n+l sin(2n + 1)~, n=0 where

2f0 T(6)

t2n+l -- -7~

sin(2n + 1)~ d~.

(4.341)

(4.342)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

521

It is known (see [13]), that

sin(an+1)~_sin~[a2ncos2n~_(n-1)a2n-2 1 COS2n-2 if/ (n--3)22n-4cos2n-4d/

+ 2

--(n--4)aan-6cosan-6~4-...]

3

where (Pk) --

(4.343)

P! . From (4.339), (4.342) and (4.343), we derive k,(p-k)

t2n+l

1[

tn-)aan_2.S2n-1 2

~ 22n

2n 4- 1

1

2n- 1

4-(n-3) $ 2 n22n-4" - 3 2 an-3 --

(n-4) $ 2 n22n-6 - 5 3"2-dn~-511 {- . . . . ~s

(4.344)

From (4.314), (4.336) and (4.340) we find T(2)-sinz(0)-sin(2)=1.

(4.345)

From (4.341), (4.344) and (4.345) we obtain N , ~ - ~--~(--1)ns n--0

(4.346)

Now, the identification procedure can be summarized as follows. First, we find polynomial approximations (4.321) and (4.322). Then, by using coefficients , a2n+l and (4.333), we recursively compute b(2n+1)0,2n+1and $2n+1. Next, by using (4.344) and (4.346), we determine s ,~ and t2n+l. By using (4.327) and (4.332), we compute C ~(2n+1) ks 9 Finally, by using (4.341), (4.340) and (4.336), we compute T(q), ;~(g) and then retrieve the inverse function g(~b). In the particular case when

Uks"(2n+l)

g(~b) = [cos ~p[1/nsign(cos ~)

CHAPTER 4 Vector Preisach Models of Hysteresis

522

the solution of the identification problem can be simplified. Indeed, in this case, the integral Eqn. (4.319) can be written as follows: F(a, fl) = 2

f0~cos (~)P(a COS1/n t~, fl COS1/n r

d~b.

(4.347)

By introducing the following change of variables, ,~ =

X = a COS1/n t~,

fl/a,

(4.348)

and by using the same transformations as before, the following integral equation is obtained from (4.347):

a(a) -

fo ~

(4.349)

N(x) dx, v/a2n _ x 2n

where the following notations are adopted:

N(x) = x2n-lp(x, J,x),

(a2/2n)F(a, ,a.a).

R(a) =

(4.350)

Multiplying both sides of Eqn. (4.349) by a 2n-1/v/S 2n -- a 2n and integrating with respect to a from 0 to s, and then interchanging the order of the double integration for the right-hand side, the following result can be derived: ~0 s

a2n-1

/S2n _ a2n

R(a) da

/ o sN(x) (/x lf0S

an

2n-ld

x/(san _ a2n)( a2n _ xan)

N(x)(~x s ~/( san-xana2 )

)dx

d(a 2n) -

-

(a 2n

(s2n+x2n)2 ) dx. (4.351) 2

Then, by introducing the following variable of integration,

W - a 2n -- (S2n if- x2n ) / 2 ,

(4.352)

we obtain fO s

a 2n-1 R(a) d a /S2n _ a2n

1loS N(x) t f ~n

(s2n-x2n)/2

dw

"J(xRn-s2n)/2 V/( s2n-xan ) _ W2 ) dx

~f0 s N(x) dx. 2n

(4.353)

4.7 GENERALIZEDVECTOR PREISACH MODELS OF HYSTERESIS.

523

By using the notation of (4.350), we obtain

P(o~, 200

1 ['~ nsn-lF(s, 2s) + s n (d/ds)F(s, ,~s) I"C

Jo

ds.

(4.354)

V/~2n _ s2n

From Eqn. (4.354) we conclude that, no matter what value of n is chosen, the model can always match the data obtained from the 'scalar' experiments. In the specific case when n = 1, the final result (4.354) coincides with the final result for the case of the identification of vector models discussed in Section 4.4. In order to complete the identification procedure, the optimum value of the unknown n should be determined by making use of the experimental data obtained from the 'rotational' experiments. One way to do this is by adopting a 'least-squares' approximation. Next, we shall illustrate the above discussion by the following experimental identification and testing of the model (4.312). The identification of the model has been performed for a typical Ampex-641 (7Fe203) magnetic-tape material by using a vibrating sample magnetometer equipped with an orthogonal pair of pickup coils. As mentioned above, the identification problem has been solved by using two sets of experimental data. First, the set of first-order transition curves was measured (see Fig. 4.35). Then, the unknown function P(a, fl) was found as a function of n. Next, the set of 'rotational' experimental data were measured. This was performed by first applying a negative input field, sufficiently large to drive the sample into the state of negative saturation, then increasing it gradually until it reached a certain positive value where it was kept constant while sample (in-plane) rotation was introduced. This was repeated for different positive field values and the o u t p u t / i n p u t phase-lag relationship as a function of the rotating input amplitude was obtained. It was found that the model gives very good matching to the 'rotational' phase-lag relationship when the value of n is chosen to be 3. A comparison between the computed and measured o u t p u t / i n p u t phase-lag values is shown in Fig. 4.36 for two different cases of n in order to demonstrate the impact of the choice of n on the model capability to match 'rotational' experimental results. After the identification process was performed, the experimental testing of the model was carried out in order to estimate its quantitative ability to mimic the property of correlation between mutually orthogonal components of output and input. This property has long been regarded as an important 'testing' property for any vector hysteresis model in the area of magnetics. This experiment was carried out as follows. First, the input field was restricted to the y-axis and was increased from some negative

CHAPTER 4

524

Vector Preisach M o d e l s of Hysteresis

0.06

0.04

3' E .~.

0.02-

r

.9

0.m

(1) E o)

-0.02

--

r

- i

-0.04

-0.06

I , , , I , , , I , , , I , , , I , , , I , -400 -200 0 200 400 600

-600

Magnetic

field

[Oe]

F I G U R E 4.35

60

I

I

t

I

I

I

I

I

I

I

I ,I

I

I

I

I

I

I

I

I

Experiment

(/)

o o

_

,

....

Model

when

n=l

Model

when

n=3

-

I I \X

o

40

__

/

~ \

E~r) _.1

--

6ffl

--

e.12.

--

II

\\\

II

\ \

II / ~

\ x x

"5 20 e~ E Q

//

d--,

0

--

d/

0 I

0

I

I

I

t

t

0.2

t

I

i

i

t

0.4

Rotating

I

0.6

field amplitude

F I G U R E 4.36

i

i

t

I

0.8 [KOe]

t

I

I

1

4.7 G E N E R A L I Z E D VECTOR PREISACH MODELS OF HYSTERESIS.

0.06 ~, E .o

I:' '~'r"'i'r'''

I ''

''

I~

r l ' '-i'~: !

0.04 0.02

0n -0.02

.

.

0

.

.

.

0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

1 1.5 2 Orthogonal Field H x [KOe]

2.5

0.04

O E 5;

"11

0.02 0~

~~'e=~--'~-

'~'~"~'"-"~

- ~

_oo~;,,,, i i~-i i ~i i,,,,,, ii ii, ,l 0

0.5

1

1.5

2

2.5

Orthogonal Field H x [KOe] 0.01 0 E -0.01 :~ -0.02 -0.03

,...,. ,_.I. ,_, _ L , I , _ _ , , -0.04 ----__,., J~_!. ,., j_j_l, 0 0.5 1 1.5 2 Orthogonal Field H x [KOe] 0.02

'' E 0

J'l'

~'~I

,-'2.5

" " '-"'-~[') i--i-'l ~-~ I ~ .-4

~, E 9~. -0.02

~; -0.04 -0.06 0

0.5

1 1.5 2 Orthogonal Field H x [KOe]

[] Experiment

Z~ n = 1

n=3 F I G U R E 4.37

2.5

525

526

CHAPTER 4 VectorPreisach Models of Hysteresis

value (sufficient to drive the sample into the negative saturation state) to some positive value Hy+, and then decreased to zero. This resulted in some remnant magnetization Myr. Then, the magnetic field was restricted to vary along the x-axis. By increasing the field value (now Hx) gradually from zero to very high values, the orthogonal remnant component Myrw a s reduced and the Myr versus Hx relationship was recorded. This experiment was repeated for different values of Hy+ (and consequently different values of Myr). Then by using the model (4.312) the computational results were obtained for the same sequences of field variations as in the experiments. Some sample comparisons between the computed and measured results of this experiment are shown in Fig. 4.37. It is clear from Fig. 4.37 that the model (4.312) gives appreciably better matching with the experimental data than the vector Preisach models discussed in previous sections. This may be due to the presence of an additional unknown function g(~0), which can be determined from the identification process. As a result, the opportunity appears to incorporate more experimental data in the identification process, leading to a more accurate model. The presentation of the material in this chapter is largely based on publications [17-27].

References 1. F. Preisach, Uber die magnetische nachwirkung, Zeitschrift fiir Physik 94, 227-302 (1935). 2. E.C. Stoner and E. P. Wolhfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Philosophical Transactions of the Royal Society of London Series A-Mathematical and Physical Sciences 240 (826), 599-642 (1948). 3. I.A. Beardsley, Modeling the record process, IEEE Trans. Magn. 22 (5), 454-459 (1986). 4. T. R. Koehler, A computationally fast, two-dimensional vector hysteresis model, J. App1. Phys. 61 (4), 1568-1578 (1987). 5. G. Bertotti, Hysteresis in Magnetism, Academic Press, New York (1998). 6. J. C. Slonczewski, IBM Research Memorandum, No. RM 003.111.224 (1956). 7. M. Prutton, Thin Ferromagnetic Films, Butterworth, Washington, DC (1964). 8. G. Friedman, New formulation of the Stoner-Wohlfarth hysteresis model and the identification problem, J. Appl. Phys. 67 (9), 5361-5363 (1990).

REFERENCES

527

9. M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis, Nauka, Moscow (1983). 10. J. A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and Minor loops-Models and Experiments, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 386 (1791), 251-261 (1983). 11. A. Damlamian and A. Visintin, A multidimensional generalization of the Preisach model for hysteresis, Comptes Rendus De L Academie Des Sciences Serie I-Mathematique 297 (7), 437-440 (1983). 12. C. Lubich, Fractional linear multistep methods for Abel-Volterra integral-equations of the 1st kind, IMA J. Numer. Anal. 7 (1), 97-106 (1987). 13. I. S. Gradstein and I. M. Ryzkik, Tables of Integrals, Series and Products, Academic Press, New York (1980). 14. A.W. Joshi, Elements of Group Theoryfor Physicists, Wiley Eastern, New Delhi (1982). 15. N.J. Vilenkin, Special Functions and the Theory of Group Representation, American Mathematical Society, Providence, RI (1968). 16. G.Y. Lyubarski, Applications of Group Theory in Physics, Pergamon Press, Oxford (1960). 17. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, Berlin (1991). 18. I.D. Mayergoyz, Mathematical-Models of hysteresis, IEEE Trans. Mag. 22 (5), 603-608 (1986). 19. I. D. Mayergoyz and G. Friedman, Isotropic vector Preisach model of hysteresis, J. Appl. Phys. 61 (8), 4022-4024 (1987). 20. I. D. Mayergoyz and G. Friedman, On the Integral-Equation of the vector Preisach hysteresis model, IEEE Trans. Mag. 23 (5), 2638-2640 (1987). 21. I. D. Mayergoyz, Vector Preisach hysteresis models, J. Appl. Phys. 63 (8), 2995-3000 (1988). 22. I. D. Mayergoyz, Identification problem for 3-D anisotropic Preisach model of vector hysteresis, IEEE Trans. Magn. 24 (6), 2928-2930 (1988). 23. A.A. Adly and I. D. Mayergoyz, A new vector Preisach-type model of hysteresis, J. App1. Phys. 73 (10), 5824-5826 (1993). 24. I. D. Mayergoyz and A. A. Adly, A new isotropic vector Preisach-type model of hysteresis and its identification, IEEE Trans. Magn. 29 (6), 2377-2379 (1993).

528

CHAPTER 4 VectorPreisach Models of Hysteresis

25. I.D. Mayergoyz, 2D vector Preisach models and rotational hysteretic losses, J. Appl. Phys. 75 (10), 5686-5688 (1993). 26. A.A. Adly and I. D. Mayergoyz, Accurate modeling of vector hysteresis using a superposition of Preisach-type models, IEEE Trans. Magn. 33 (5), 4155-4157 (1997). 27. A. A. Adly, I. D. Mayergoyz, and A. Bergqvist, Utilizing anisotropic Preisach-type models in the accurate simulation of magnetostriction, IEEE Trans. Magn. 33 (5), 3931-3933 (1997).

CHAPTER 5

Stochastic Aspects of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 5.1 5.2 5.3 5.4 5.5 5.6

5.7

Preisach Model with Stochastic Input as a Model of Thermal Relaxations Experimental Testing. Scaling and Data Collapse in Magnetic Viscosity Preisach Model Driven by Continuous-Time Noise. Origin of the Universality of Long-Time Thermal Relaxations Noise in Hysteretic Systems and Stochastic Processes on Graphs Analysis of Spectral Noise Density of Hysteretic Systems Driven by Stochastic Processes Modeling of Temperature-Dependent Hysteresis within the Framework of Randomly Perturbed Fast Dynamical Systems Functional (Path) Integration Models of Hysteresis References

5.1 P R E I S A C H M O D E L W I T H S T O C H A S T I C

529 541 550 562 573

578 587 601

INPUT AS

A M O D E L OF T H E R M A L RELAXATIONS It is well known that the physical origin of hysteresis is due to the multiplicity of metastable states exhibited by hysteretic materials or systems. At temporally constant external conditions, large deviations of random (thermal) perturbations may cause a hysteretic system to move from one metastable state to another. This may result in gradual (slow) changes of the output variable. This temporal loss of memory of a hysteretic system is generally referred to in the literature as 'after effect', 'viscosity' The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

530

CHAPTER 5 Stochastic Aspects of Hysteresis

or 'creep'. This phenomenon can be of practical significance in various engineering applications where hysteresis is utilized. One important example is the magnetic storage technology where the viscosity effect is detrimental as far as the time reliability of recorded information is concerned. Traditionally, the viscosity phenomenon has been studied by using thermal activation-type models (see, for example, [1,2]). It has been realized that these models have some intrinsic limitations. For instance, they are usually regarded as 'noninteracting particle' models and they are valid for hysteretic systems with energy barriers much larger than the thermal activation energy. In this section, we shall explore a new approach [3-6] to the modeling of thermal relaxations based upon the Preisach model of hysteresis. In this approach, random thermal agitations will be modeled by a stochastic input. We shall compare this approach with thermal activation-type models and with some known experimental facts. Consider a deterministic input u(t), which at time t = 0 assumes some value u and remains constant thereafter. In a purely deterministic situation, the output flt) would remain constant for t~0 as well. However, in order to model thermal relaxations, we assume that some noise described by the stochastic process Xt is superimposed on the constant input u. Consequently, the Preisach model is driven by the following random process: Xt = U -Jr-Xt,

Xt = 0,

(5.1)

where Xt stands for the expected value of Xt. The output will also be a random process given by

f t - /f~ >>-~#(~

dad]/.

(5.2)

It is instructive to note that adding noise Xt to the deterministic input u(t) is mathematically equivalent to subtracting the same noise from switching thresholds ~ and//. This is true because (~_Xt,fl_xt)U(t)

--

~ ( u ( t ) + Xt).

(5.3)

Imposing noise on the switching thresholds (barriers) may be more transparent (or justifiable) from the physical point of view, while adding noise to the deterministic input makes the problem more tractable from the mathematical point of view. To simplify the problem, we shall first model the noise by a discrete-time independent identically distributed (i.i.d.) random process Xn. In this case, it can be assumed that process samples remain constant at time intervals At = tn+l - tn and undergo monotonic step changes

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

531

at time tn+l. Accordingly, the Preisach model is driven by the process Xn -- U -}- X n ,

(5.4)

X n - O.

The output process is given by /f~

(5.5)

[.t(~, fl)~flXn da dfl.

We will be interested in the time evolution of the expected value of the output process. Since integration is a linear operation, from (5.5) we derive f n = E{fn} -

#(o~, fl)E{~;~flXn} dadfl. >>,~

(5.6)

Thus, the whole problem is reduced to the evaluation of the expected value, E{~a~Xn}. Since ~ X n may assume only two values, +1 and - 1 , we find -- +1} - P{~aflXn -

E{~aflXn} - P { ~ f l X n

-1}.

(5.7)

It is also clear that P{~aflXn - +1} + P{~aflXn -

- 1 } - 1.

(5.8)

By introducing the notation P{~Xn

- +1} = qa~(n),

(5.9)

- 2qa]~(n) - 1.

(5.10)

from (5.7) and (5.8) we obtain E{~Xn}

We next derive the finite difference equation for q ~ ( n ) . According to the total probability theorem, we have P{~flXn+l - + 1 } - - P { ~ X n + l = q-ll~flXn = q-1}P{~;~Xn = +1} +P{~aflXn+l = +ll~aflXn - -1}P{~flXn - - 1 } . (5.11) It is convenient now to introduce the switching probabilities P++(n)-P{~flXn+l

- +ll~flXn - +1}

(5.12)

P-~; (n) - P{~aflXn+l - q-ll~aflXn - - 1 } .

(5.13)

CHAPTER 5 Stochastic Aspects of Hysteresis

532

Similar meanings hold for the switching probabilities P~fl- (n) and P+fl (n). It is clear that ++ p+P~fl (n)4- ~/~ ( n ) = 1, (5.14) P~-; (n) + P~-~ (n) = 1,

(5.15)

P{~flXn = -1} = 1 - q~fl(n).

(5.16)

By using (5.9) and (5.12)-(5.16), we can transform (5.11) to +-(n))] + P-d;(n) qafl(n + 1) = qa/~(n)[1 - (P2; (n) + P~/~

(5.17)

We next proceed to the evaluation of switching probabilities P 2 ; (n) and P+-(n) e/l 9 It is clear that these probabilities can also be defined as

P-~; (n) = P{Xn+l > ~l~;~flXn = -1}, +-

P~fl (n) = P{Xn+l < fli?aflXn = +1}.

(5.18) (5.19)

In general, these probabilities are difficult to evaluate because multidimensional conditional probability density functions are required. However, the problem is significantly simplified if the noise is modeled by the i.i.d. process. In this case we find

P-~; (n) = P-~+ = P{Xn+l > or -= ~fl (n) = P-~- = P{Xn+l P+-

<

/J} - -

p(x) dx,

(5.20)

/~oo fi(X) dx,

(5.21)

where p(x) is a probability density function. By using (5.20) and (5.21), the finite difference equation (5.17) can be represented as q~fl(n + 1) = r~flq~fl(n) + P;+, (5.22) where

r,fl = 1 - (P;+ + P-~-) =

p(x) dx.

(5.23)

If the probability density function p(x) is strictly positive, then 0 < r~fl < 1.

(5.24)

Equation (5.22) is a constant coefficient first-order finite difference equation whose general solution has the form

q~fl(n) = Arnfl 4- B.

(5.25)

533

5.1 PREISACH MODEL WITH STOCHASTIC INPUT By substituting (5.25) into (5.22), we find p~-+ Pa-+ B - 1 - r~/~ = P~-+ + P J - "

(5.26)

From the initial condition we obtain q~/~(0) - A 4- B,

(5.27)

where qa/~(0) -

1 if (a, fl) ~ S+(0), 0 if (a, fl) e S-(0),

(5.28)

and S + (0) and S-(0) are positive and negative sets on the ~-fl diagram, respectively, at the instant of time t = 0. From (5.25)-(5.27) we derive qa~(n) - [q~/~(O) - q~/~(oo)]rn/~ +

p~-+ P~-+ + P~--'

(5.29)

where q~/~(oo) = lim q ~ ( n ) = n - - + OO

P-~+

(5.30)

P~+ + P~-"

By substituting (5.29) into (5.10), we obtain p~-+ _ p ~ E{~;aflXn} - 2[q~fl(0)- q~/~(oo)]rn/~ +

P~-+ + P~-"

We next introduce the functions p~-+ _ p ~ ~(~, fl)= p~_+ + p + - '

(5.31)

(5.32)

y.

~9(~, fl)= [ 1 ! -1

if (~, fl) E S + (0), if (~, fl) E S-(0).

(5.33)

By using (5.28), (5.32) and (5.33), we can transform (5.31) to E{~a~Xn} - [t9(0r fl) - ~(0r fl)]rn/~ + r

fl).

(5.34)

By substituting (5.34) into (5.6), we finally find

J~n -)?o~ + f L

#(~' fl)[tg(~, fl) - ~ ( ~ , fl) ] rnfl d o~d fl,

(5.35)

CHAPTER 5 Stochastic Aspects of Hysteresis

534

where )~oo = n_,oofJ lim n

#(~, fl)~(a, fl)d~ dfl.

=

(5.36)

It is apparent from (5.36) and (5.32) that the limiting expected value of output foo does not depend on the history of input variations prior to the time t = 0. In this respect, the value, foo, bears some resemblance to an anhysterefic output value. This resemblance is enhanced by the fact that foo = 0 if the expected value u of Xn is equal to zero, and p(x) = p(X) is an even function. This fact can be proved as follows. According to (5.20) and (5.21) we find P~-+ = P+~-. (5.37) By using (5.37), it is easy to check that r

fl) = - r

(5.38)

On the other hand, we recall that (5.39)

~(a, fl) = ~ ( - f l , - ~ ) .

From (5.38) and (5.39), we conclude that #(~, fl)~(~, fl) is an odd function with respect to the line a = -ft. From the last fact and (5.36), we find that )~oo = O.

(5.40)

Next, we shall compare the result (5.35) with thermal activation-type models for viscosity. For this purpose, we replace discrete time n by continuous time t and rewrite (5.35) as

ft

= foo

+ fL

%(or fl)e -r

do~dfl,

(5.41)

where

Z(~, ,8)= #(o~, ,8)[~](o~, ,8) - ~'(~, fl)], 1

~(~,/~)- - ~ In r~.

(5.42) (5.43)

According to thermal activation-type models, metastable equilibrium states of hysteretic systems are separated by energy barriers EB. It is assumed that there is a continuum of these energy barriers and it is postulated (with some physical justification) that the viscosity phenomenon is described by the model (see [1]): fit) = flee) + A

/o

g(EB)e -'~(EB)t dEB,

(5.44)

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

535

where g(EB) is some density of states, 2(EB) --

d.oeEB/kT,

(5.45)

k is Boltzmann's constant, T is the absolute temperature, while A and 20 are some constants. It is clear by inspection that there is some similarity between our result (5.41) and the thermal activation model (5.44). Actually, our model (5.41) can be reduced to (5.44) in the particular case when only symmetrical loops (operators) ~ _ ~ are used in the Preisach model. In this case p(~, fl) = 0(~,/3)6(~ + fl),

(5.46)

where 6(a 4-/3) is the Dirac delta function. By substituting (5.46) into (5.42) and (5.41), after simple transformations we can represent (5.41) as

ft = foo +

2(~z)e-~(~)t da.

(5.47)

Now, by using the change of variables ~ ( ~ ) - 2(EB),

~-

~-1 (,~(EB)),

(5.48)

the expression (5.47) can be reduced to (5.40). This shows that the Preisach model of viscosity (5.41) is reduced to the thermal activation model in a very particular case. This case occurs when only symmetrical rectangular loops are used in the Preisach model. Since it is generally believed that nonsymmetrical loops in the Preisach model account for 'particle interactions', the last reduction is consistent with the generally held opinion that the thermal activation model (5.44) is a 'noninteracting particle' model. We next show that under some assumptions about X(~, fl) the model (5.41) describes in t-type variations for ft. Another (purely stochastic) justification for In t-asymptotics will be given in Section 5.3. The In t-type variations have been observed in many experiments and they are considered to be characteristic of viscosity phenomena. The expression (5.41) can be written as follows E

ft - f~ + f (/Z(~, fl)e-~(~"e)td(z) dfl.

(5.49)

For any fixed fl, we make the change of variables from ~ to ~ in the integral with respect to a:

ft - f~ + f (f ~,(~,fl)e-r

dfl.

(5.50)

CHAPTER 5 Stochastic Aspects of Hysteresis

536

We assume that the function 6(~, fl) can be approximated as

6(r ]/)~

D(fl) for r < r < r r

(5.51)

and zero otherwise. Then, we derive

f

~/(~, fl)e -~t d~ ,~, D(fl)

~ ~2 e-~t - ~ d~ = D(fl)

1

~~2te-V 1t

dr.

(5.52)

Consider such instants of time t that 1

1

(5.53)

~2 << t << ~119 For these instants of time, from (5.52) we derive

f

~/(~, fl)e -r d~ ,~ D(fl)

-

D(fl)

~oo e-V dv

1t

If eV r ltev] -a

V

dv -

e -a

V

dv .

(5.54)

Here a is a fixed small positive n u m b e r and the last two integrals exist in the sense of the Cauchy principal value. Since ~lt << 1, we can use only the first two terms of the Taylor expansion of e -v in the second integral. This yields

f

~(~, fl)e -~t d~ ~ S(]/)lnt 4- C(fl).

(5.55)

By substituting (5.55) into (5.50), we find

ft ~" S ln t 4-C,

(5.56)

which is a famous In t-type variation for ft. It has been observed in experiments (see, for instance, [7]) that the temperature dependence of the decay rate of;~t is satisfactorily approximated by T 1/2 at low temperatures. In the framework of the model (5.41) this fact can be explained as follows. From (5.43) and (5.23), we find

dx]

(5.57)

We subdivide all exponents in (5.41) into two groups: (a) exponents for such ~'s and fl's that u = xt ~ [fl, ~], and (b) exponents for such ~'s and fl's

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

537

that u = xt E [fl, 0r Consider a thermal noise with small variance 0-2. If U = ~:t ~ [fl, ~], then

~

~ p ( x ) d x --> 0

as 0- ---> 0.

(5.58)

This means that according to (5.57) the exponents of the first group decay very fast and they do not contribute appreciably to the 'long-time' decay of the integral in (5.41). If u = ~:t E []~, ~], then

ffl

~ p ( x ) d x --, 1

as 0- ---> 0.

(5.59)

This means that the exponents of the second group decay slowly and they are by and large responsible for the 'long-time' decay of ft. For this reason, only these exponents are discussed below. It is apparent that p(x) d x - 1 -

p(x) d x -

p(x) dx.

(5.60)

For small 0- the integrals in (5.60) are small quantities, which we assume can be approximated as f cx: p(x) d x ~ a(:r

//

p(x) d ~ b(]/)0-.

(5.61)

By substituting (5.61) into (5.60) and then into (5.57), we find ~(or fi) ~ - l n { 1 - [a(~) + b(fl)]0-} ~ [a(a) + b(fl)]0-.

(5.62)

It is reasonable to assume on physical grounds that the variance is proportional to the temperature 0-2 ,~, T. (5.63) From (5.62) and (5.63) we conclude that, for the exponents of the second group, the rate of decay is approximated by ~(~, fl) ~-, T 1/2.

(5.64)

This suggests that the overall rate of 'long time' decay offt may be approximated by T 1/2 at low temperatures. It is also interesting to examine the dependence of the limiting expected value of the output, f ~ , on the variance 0-2 of the noise. Expression (5.36)

CHAPTER 5 Stochastic Aspects of Hysteresis

538

'

1

'

I

'

'

'

"

'

i

"

'

'

m

f~

0.5 m

.

n

O o

,

n

O n

E <

~

-

--0.5

-1

,

,

I

,

i

i

-200

l,

i

I

0

I

I

,,

200

Mean input

FIGURE 5.1

is to be used for this purpose. The noise can be chosen to have normal distribution and ~(a, ]~) in (5.36) can be computed by using formulas (5.32) and (5.20) and (5.21). For illustrative purposes, Preisach function #(a, ]/) can be chosen as

Aexp{

40"2

2+

40"2

2Uc21},

(5.65)

where A is some normalization constant, Uc is the coercive value of a major hysteretic loop, while 0.1 and 0"2 are parameters that are used to control the spread of the function/~(a, fl) along and around the line a = -ft. This Gaussian form of/~(a, ]/) is often used in magnetics [8]. The computations have been performed for different values of 0"1,0"2 and the ratio Z - G. The computations show that for arbitrary fixed values of 0"1 and 0"2 the dependence o f f ~ on u is not sensitive to wide variations of 0.. This is demonstrated in Figs 5.1 and 5.2 where the computed curvesf~ versus u are plotted for the following values of X: 0.5, 0.35, 0.2, 0.15. These curves were computed for the following values of 0"1 and 0"2,respectively: 0"1 "-- 0"2 - - 40, 0"1 - - 80 and 0"2 = 40. The fact that fc~ is insensitive to the variance 0"2 suggests that f ~ can probably be used as another definition of an anhysteretic value. We next consider vector Preisach models driven by vector stochastic inputs as models for viscosity [4,9]. Below we shall only discuss the

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

539 [

1

I

1

L 0.5

o

0

(~

< -0.5

I _J -1

I

5

i

0 Mean input

-200

uc i

I

; l

I-

200

FIGURE 5.2

case of two-dimensional vector Preisach model. Extensions to the threedimensional case will be straightforward and, for this reason, they will be omitted. Consider a deterministic vector input ~(t), which at time t - 0 assumes some value ~ and remains constant thereafter. We assume that some zero mean vector noise described by a discrete-time r a n d o m process is superimposed on ~(t). As a result, the Preisach model is driven by the process

Vn

(5.66)

~dn --- U -}- Wn.

For the output process we have

fn-'- /_~2-~q~(ff~), v(~x,fl, cp)~Vnq~d~xdfl)dq~.

(5.67)

From (5.68), we derive the following expression for the expected value of

fn"

lr

flfn}

--

/__'2-~q~(/~~fl V(~X,fl, q~)E{~,flVnq~}d~xd,)dq~.

(5.68)

By modeling the noise by the i.i.d, vector process with independent Cartesian components and by repeating the same line of reasoning as before, we derive

E{~aflVnq~} -

[O(a, fi, r

- ~(a, fl, q~)]rn/~ + r

fl, q~),

(5.69)

CHAPTER 5 Stochastic Aspects of Hysteresis

540

where ~9(a, fl, q~)=

+1 -1

if (a, fl) e S~(O), if (a, fl) e S~ (0),

(5.70)

P;~ -P~f ~(~, fl, q~)= p~-~- + P ~ '

(5.71)

P~-+ =

pq~(x) dx,

(5.72)

P~-~- =

~ pq~(X)dx,

(5.73)

r~q~ =

pq~(x) dx.

(5.74)

In expressions (5.73)-(5.75), pq~ is a probability density function for Vnq~. This function can be expressed in terms of probability density functions for Cartesian components of vn. Indeed, for Vnq)we have

Vnrp-~-Xncos r -}-yn sin r

(5.75)

where Xn and yn are independent Cartesian components of Vn with probability density functions px(X) and py(y), respectively. According to the well-known result from probability theory, from (5.76) we derive

pq~(V) = cos q~1sin q~ f_~-~ Px( cosx q~) Py( vsin- xq~t dx. o0

(5.76)

By substituting (5.70) into (5.69) and by replacing discrete time n by continuous time t, after simple transformations we obtain

(/L

E{fy} - E l f , } +

~m

)

dq~,

(5.77)

v(a,fl, q~)~(~z,fl, q~)dadfl) dq~,

(5.78)

Z(a, fl, q~)e-~(a'/~'m)t dadfl

where E { f o o } = L ~ "6q~(f L

(5.79) ~(~, #, q~) - - ~

1

In r~#~.

(5.8o)

5.2 EXPERIMENTALTESTING. SCALING AND DATA COLLAPSE

541

By comparing (5.78) with (5.44), it can be concluded that the model (5.78) can be regarded as a vector generalization of the classical thermal activationtype models. For the isotropic model the symmetry property similar to (5.39) holds for v. It can also be checked that for ~ - 0, p~ is an even function if Px and py are even. By using these facts and the same line of reasoning as before, it can be proved that E{f~} = 0 . (5.81) Therefore, for ~ = 0 and regardless of the past history, the limiting expected value of output is equal to zero. In conclusion of this section, it can be remarked that the model (5.35) (or (5.41)) has certain attractive features in comparison with the thermal activation-type model (5.44). First, the model (5.35) explicitly accounts for the specific hysteretic nature of the system as well as for specific input histories. Second, stochastic characteristics of thermal noise explicitly appear in the model, whereas the thermal activation-type model is formulated in purely deterministic terms. Third, the thermal activation-type model (5.44) is intrinsically scalar, whereas the model (5.35) can be generalized to the vector case.

5.2 EXPERIMENTAL TESTING. S C A L I N G A N D DATA COLLAPSE IN M A G N E T I C V ISC OSITY As discussed before, at constant external magnetic fields, large-deviation internal random (thermal) perturbations may cause a hysteretic system to move from one metastable state to another. This may result in slow time variations of magnetization, which are the essence of viscosity. The described temporal variations of magnetization are usually characterized by the following intermediate 'ln t' asymptotics: M ~ Mo - S(H, T ) I n t,

(5.82)

where S(H, T) is called the viscosity coefficient. This coefficient depends on the values of magnetic-field H, temperature T, as well as the past history of magnetic-field variations. The latter means that the viscosity coefficient is also a function of the current state of hysteretic material. The term 'intermediate asymptotics' means that formula (5.82) describes quite well the long-time behavior of magnetization. However, it does not describe properly the ultimate (at t ~ cx~)value of magnetization. In other words, the asymptotic behavior (5.82) breaks down for very long times, at which In t diverges.

542

CHAPTER 5 Stochastic Aspects of Hysteresis

The ability of the model (5.41) to predict the viscosity coefficient (relaxation rate) was first experimentally tested by C. E. Korman and P. Rugkwamsook [10] for ~;-Fe302 magnetic recording materials. By using a vibrating sample magnetometer (VSM), first-order reversal curves were measured. These experimental data were used for the identification of the Preisach model (determination of #(a, ]/)). Then, viscosity measurements were performed under the conditions that the sample was first brought to positive saturation and afterwards the applied field was reversed and held constant at some fixed value, and gradual decay of magnetization was observed. These magnetization decay curves (experimentally measured at constant temperature T and various fixed values of magnetic field H) were approximated by Eqn. (5.82) and the viscosity coefficient as a function S(H) of magnetic field was found. The described viscosity experiments were numerically simulated by using model (5.41) and experimentally measured first-order transition curves. The noise was assumed to be Gaussian and calculations were performed for different values of variances 0-2. The calculations revealed the 'In t' intermediate asymptotics as given by formula (5.82). The slopes of these asymptotics were computed, then they were normalized and it was observed that the normalized c u r v e s S(H)/Smax computed for different variances practically collapsed into one curve. This computed universal curve is almost identical to the normalized curve S(H)/Smax found from the experimental data previously described (see Fig. 5.3). This suggests that if the variance 0-2 of the noise is found by matching Smax, then the model (5.41) will predict the same curve S(H) as observed in the experiments. The 'computational collapse' of c u r v e s S(H)/Smaxprompted the idea to experimentally test the scaling and data collapse of viscosity coefficient S(H, T) for various magnetic materials, and especially for materials used in magnetic recording [11]. Our experiments have been conducted for high-coercivity thin-film CoCrPt recording media by using a vibrating sample magnetometer (VSM) with a temperature-controlled furnace (model Micro Mag 3900 from Princeton Measurements Corporation). First, experiments were performed to demonstrate clearly that the viscosity coefficient strongly depends on the past history of magnetic-field variations. To this end, the viscosity measurements were conducted for two distinct states of magnetic material, which had almost the same values of magnetization achieved at the same value of the external magnetic field but corresponding to two different past histories of magnetic-field variations. These two distinct states and histories of their formations are shown in Fig. 5.4. Figure 5.5 presents the observed time variations of magnetization for the above two states plotted on an In t scale. It is apparent from Fig. 5.5 that the above two

5.2 EXPERIMENTALTESTING. SCALING AND DATACOLLAPSE Simulation with sigma = 10 Oe

543

5e-005

-;

.,I ~ Simulation with sigma = 15 Oe . . . . . . S i m u l a t i o n with sigma = 20 Oe Experiment - - e . - -

0.9

4e-005

,., 0.8 cO

r

m

O

"O

.o_ 0.7

E

"O

O

o 0.6

3e-005

I II

o

"O "O

N ....

0.5

c-

2e-005 ._o

0.4

O o

E o.3

0 Z

9

0.2

.'

o

,,~;

1 e-005 a

0.1 0 - 1 0 ( )0

-ooo

-600

-400

-aoo

o

260

0 400

Applied field, Oe

F I G U R E 5.3 M

P.H

S F I G U R E 5.4

magnetization variations have different (in magnitude and sign) viscosity coefficients. This clearly reveals the strong dependence of the viscosity coefficients on past history.

CHAPTER 5 StochasticAspects of Hysteresis

544

x 10 . 4

-7

Magnetization vs Ln(Time) Measurement of a Hard Disk Sample

!

-7.2 ........................................................ :....................................... -7.4 ............................... ~ E -7.6

................

~"" 1 .......

..................................................................................................

eo ..~

-7.8

N -,~

...............................................................................................

r-

-8 -8.2 -8.4 -8.6

0

9

.

.

i

i

2

4

i 8

6

10

Ln (Time(sec))

F I G U R E 5.5

In subsequent experiments, the viscosity coefficients were measured as a function of H for various temperatures T for the same (and quite simple) histories. Namely, specimens were first driven into the state of positive saturation by applying sufficiently strong external magnetic fields and then the magnetic field was gradually decreased to a desired value at which viscosity measurements were performed. Thus, the viscosity measurements were performed for the states corresponding to the points on the descending branch of the major hysteresis loops. For various fixed values of T, viscosity coefficients S(H, T) are 'bell-shaped' functions of H with the maximum near (but not necessarily equal to) the coercivity. These bell-shaped curves measured for two different samples are shown in Figs 5.6 and 5.7. The samples had coercivities of about 1600 and 2 600 Oe at a temperature of 25 ~ respectively. The material of the second sample (with higher coercivity) had a small value of remanence (Mrt ~ 0.85 x 103 emu/cm2). For this reason, and in order to increase the signal level, the second sample was formed by stacking several layers of the same material. By using the experimental data shown in Figs 5.6 and 5.7, the hypothesis that S(H, T) admits the following scaling: S(H, T) - Smax(T) f ( H , ( T ) )

(5.83)

5.2 EXPERIMENTALTESTING. SCALING AND DATA COLLAPSE x 10 -5

Sample#01

o

M a g n e t i c V i s c o s i t y C o e f f i c i e n t vs M a g n e t i c F i e l d

o

T=25oc

.1~

9

m=5o~

/"

"

T=~O0~

4 /~ Io

~

T=,~ooc

o

T--~oooc

4.5

3.5

~ .......

i..........

i........

!

i

..... !.~...i

I T~

~ ....

!1

.~:..~.i ........ -I

i I li I i I

'I!

I /

"",~ .........!.......... !~~!1~1! ........1 11, i ~! ..... I i 1 ..... ~ _1 ". . . . . . . . . . :. . . . . . . . . . : : " ........

-. . . . . !

545

!

i

A

i

3~ 3 ~ 2.5

----

} ,.s

if

i

:

0.5 0 -1900

i -1800

i,

i

-1700

-1600

i -1500

i

i

-1400

-1300

i -1200

i -1100

I -1000

-900

Magnetic Field H (Oe)

FIGURE

5.6

was tested. In the above formula, Smax(Z)is the maximum viscosity coefficient as a function of T, while H* (T) is the value of the magnetic field at which Smaxis achieved. The last formula suggests that S versus H curves experimentally measured for different temperatures must collapse onto one universal curve when plotted in coordinates: s --

S Smax(t) '

H h -- ~ . H*(T)

(5.84)

This phenomenon of data collapse is the principal significance of scaling and its occurrence was observed for the collected experimental data. Namely, by performing the scaling described above, it was found that the curves shown in Figs 5.6 and 5.7 practically collapsed onto the single curves shown in Figs 5.8 and 5.9, respectively. In addition, the following scaling hypotheses: Smax(Z) = a T ~,

(5.85)

H* (T) - bT ~,

(5.86)

were also experimentally tested and verified with some accuracy. The results of this testing for samples I and 2 are shown in Figs 5.10 and 5.11,

CHAPTER 5 Stochastic Aspects of Hysteresis

546

x 10 -5 4

Sample#02 i

Magnetic i

Viscosity !

o

o

T = 25~

9

3.5

9

9

T = 50~

.

3

~

a

T = 100~

.'

2.5,

v

v

T=150~

0

0

T = 175~

8

Coefficient

vs Magnetic !

Field

x z~

.......................

....................

: .... ilk .... / ~ - I . I . . . . . . . . . . . . . . . . . . . . .

~ 1.5

0.5 ~

i

a

i

=' =,

/

0/ -4000

-3500

-3000

~

~',/ -2500 Magnetic

r

,

-2000

- 1500

- 1000

-500

Field H (Oe)

FIGURE 5.7

respectively. It was found that, for thin-film recording media with coercivity below 2 000 Oe, the exponent ]/was fairly close to -1. For recording media with coercivity above 2 000 Oe, substantial deviations from the above value of/Y were observed. Hypotheses (5.85) and (5.86) along with the formula (5.83) lead to the following self-similar expression for the viscosity coefficient

S(H, T) - aT~f ~

.

(5.87)

The last formula (as well as formula (5.83)) reveals an interesting and peculiar mathematical structure of the viscosity coefficient as the function of two variables: H and T. The essence of this structure is that the normalized viscosity coefficient S/Smax is a function of one variable h - H/H* (T). This structure is quite different from the traditional representation of S(H, T) in terms of irreversible differentiable susceptibility; the representation that is widely analyzed and discussed in the existing literature. This structure is a direct and concise reflection of experimental observations and is not based on any ad hoc assumptions.

5.2 EXPERIMENTALTESTING. SCALING AND DATACOLLAPSE

547

Normalized S vs H Curve of Hard Disk Sample#01 i

i

i

i

i

i

i

i

v_.

T = 25~

~= .... T = 50~ ~-

T = 100~ T = 150~

0.8

--e--

T = 200~

x

E

0.6

0.4

0.2

.5

I

I

I

I

I

-1.4

-1.3

-1.2

-1.1

-1

I

-0.9 H/H*

I,

I

I

I

-0.8

-0.7

-0.6

-0.5

FIGURE 5.8

The experimental evidence of scaling of S(H, T) and data collapse has been presented here only for two samples. However, the described scaling and data collapse were observed for several other samples as well. Those samples had coercivities between 1600 and 2 600 Oe. Scaling and data collapse are typical for critical phenomena where their physical origin is due to the divergence of correlation length [12]. The physical origin of scaling in magnetic viscosity is not clear at this time. However, one may speculate that this scaling is somehow related to the 'granular' structures of hysteretic materials, which ultimately lead to multiplicity of metastable states. In the experimental studies of magnetic viscosity reported above, magnetic samples were subject to external magnetic fields that were constant in time and spatially uniform. In these experiments, the applied fields and sample magnetization were usually aligned, while demagnetizing fields were negligible. This type of experimental study is usually performed by using magnetometers and reveals the intrinsic scalar thermal relaxation properties of the materials. In magnetic recording, thermal relaxations of magnetization patterns recorded on hard disks are of interest. These are data-dependent thermal

CHAPTER 5 StochasticAspects of Hysteresis

548

N o r m a l i z e d S vs H C u r v e of Hard Disk S a m p l e # 0 2 I

I

I

I

I

I

I

I

I

I I I

_

0.8 _

X

E

0.6

----e---

T = 25~

-

T = 50~

-"

T = 75~

=

T = 125~

v--

T=150~

~,

T = 175~

~

_

T=100~

-

~_

_

00 or)

0.4 i

I

0.2

0

i

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

H/H*

FIGURE 5.9

relaxations and they occur under quite different conditions than in the case of intrinsic thermal relaxations. First, the recording media are subject to strong local demagnetizing fields. These fields depend on recorded patterns and they vary with time due to the temporal deterioration of these patterns. Second, the demagnetizing fields are spatially nonuniform. As a result, data-dependent thermal relaxations occur with different local rates. Third, the local demagnetizing fields and magnetization are not aligned. Therefore, data-dependent thermal relaxations are inherently vectorial. Finally, data-dependent thermal relaxations are measured by using a giant magneto-resistive (GMR) head. Signals in these heads are generated by magnetic fields produced by virtual magnetic charges that are proportional to the divergence of magnetization (divM). It is quite possible that, due to the spatially inhomogeneous and vectorial nature of thermal relaxations, very small temporal changes in local magnetization may result in appreciable magnetic charges and GMR head signals. Data-dependent thermal relaxations are usually studied by using spinstands. However, traditional spin-stand studies have only examined the harmonic content (mostly the fundamental harmonic) of recorded patterns.

5.2 EXPERIMENTAL TESTING. SCALING A N D DATA COLLAPSE

549

Ln (Smax) vs Ln (T) of Hard Disk Sample -9.9 Io -10

o'Sample#'01 .

.

.

.

.

.

.

.

.

.

.

.

I .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-10.1

-10.2 o3 v r ._1 -10.3

-10.4

-10.5

5.6

5.7

5.8

5.9

6

6.1

6.3

6.2

Ln (T)

F I G U R E 5.10

Ln (H'ax) vs Ln (T) of Hard Disk Sample 7.9 ......

7.8

o

~ o Samp-'le #01

7.7 7.6 7.5 I,I 7.4 t.._1 7.3

iiiiiiiiiii

i iiiiii

7.2 7.1

6.9 5.6

i

5.7

5.8

5.9

6 Ln (m)

F I G U R E 5.11

6.1

6.2

6.3

550

CHAPTER 5 Stochastic Aspects of Hysteresis

For this reason, the spatially inhomogeneous and vectorial nature of datadependent thermal relaxations has not been captured. It turns out that the spatial and vectorial characterization of data-dependent thermal relaxations can be performed by using the spin-stand imaging technique [13-15]. The detailed exposition of this technique is beyond the scope of this discussion.

5.3 PREISACH M O D E L D R I V E N BY C O N T I N U O U S TIME NOISE. O R I G I N OF THE U N I V E R S A L I T Y OF L O N G - T I M E T H E R M A L RELAXATIONS In Section 5.1, the Preisach model driven by the discrete-time i.i.d, random process is used as a model for thermal relaxations in hysteretic systems. However, actual random thermal agitations are better described as continuous-time noise. From the mathematical point of view, this makes the problem quite complicated. It is shown next that these difficulties can be largely overcome by using the mathematical machinery of the 'exit problem' [16]. The noise Xt in Eqns (5.1) and (5.2) will be modeled by a (continuoustime and continuous-sample) diffusion process, which is a solution to the Ito stochastic differential equation [17]

dXt = b(Xt) dt q- cr(Xt) dWt.

(5.88)

In this equation, Wt is the Wiener process, and its formal derivative is white noise. Formula (5.88) can be construed as a generic equation for dynamical systems driven by white noise, and trajectories of such dynamical systems can be viewed as samples of the stochastic diffusion process. From the purely mathematical point of view, the Ito stochastic differential equation generates complicated diffusion processes by using the Wiener process, which is one of the simplest and most studied diffusion processes. Now we shall return to Eqn. (5.2). Since integration is a linear operation, from (5.2) we derive

ft-- f ~ ~fl #(~, fl)E{4/~xt} do~dfl.

(5.89)

Thus, the problem is reduced to the evaluation of the expected value,

E l ~flxt }. Let

q~,B(t) = Prob{~/~xt- +1}.

(5.90)

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

551

A ~oq~Xt

/k +1 , , JL IF

~

~

Xt

XO

\/

-1 1t`. 1`. i`. i`. i`. i`. i ' / ` . 1 / / ~ / / 1 /

13

11

~-

Xo

~

i

i

I

11

i

i

i

i

i

i

r-

F I G U R E 5.12

Since

~o~,flXtmay assume only two values +1 and -1, we find E { ~ x t } - 2%,fl(t) - 1.

(5.91)

In this way, the problem is reduced to the calculation of qa,~(t). The last quantity can be expressed in terms of switching probabilities P-~(t) and P~-(t), which are defined as follows: P ~ ( t ) - Prob{

k switchings of ~/~ during } time interval (0, t)l~/~x0 = +1 '

P~- (t) - Prob{

k switchings of ~/~ during time interval (0, t)l~/~x0 = - 1 / "

/

(5.92) (5.93)

By using these switching probabilities, we derive

%,~(t) - { Y~.~=0P~ (t) ~ = 0 P2k+l (t)

if ~a/~x0 = +1, if ~a/~x0 - - 1.

(5.94)

The last expression is valid because occurrences of different numbers of switchings are nonintersecting (disjoint) events. Next, we shall discuss the mechanism of switching. It is clear from Fig. 5.12 that the first switching occurs at the moment when the stochastic process Xt, starting from the point x0, exits the semi-infinite interval (fl, ~ ) . Then, the second switching occurs at the moment when the process xt, starting from the point x = fl, exits the semi-infinite interval (-oo, a). The third switching takes place at the moment when the process xt, starting from the point x = a, exits the semi-infinite interval (//, cx~). It is apparent

552

CHAPTER 5 Stochastic Aspects of Hysteresis

that the mechanism of all subsequent even switchings is identical to the mechanism of the second switching, while all subsequent odd switchings occur in the same manner as the third switching. Thus, switchings of rectangular loops ~a]~ are closely related to the exit problem for stochastic processes. This problem is one of the most studied problems in the theory of diffusion processes and the mathematical machinery developed for the solution of this problem will be utilized in the calculation of probabilities 4Pk (t). The exit problems just described can be characterized by exit times rxi , which are random variables. In the above notation for the exit times, subscript 'x' means that process xt starts from point x, while superscripts '4-' mean that upward and downward switchings, respectively, occur at these exit times. Next, we introduce the functions v+(t, x) = Prob{~xi~t},

(5.95)

V + (t, x) - e(t) - v + (t, x),

(5.96)

where e(t) is a unit step-function. It is clear that V+(t, x) = Prob{z x4-Kt},

(5.97)

which means that V+(t, x) has the meaning of a cumulative distribution function for the random variable ~x~. This, in turn, implies that /9+ (t, x) = 3V+ (t, x) 0t

(5.98)

is the probability density function for the random variable Zx ~. It is apparent from (5.96)-(5.98) that p+(t, x) can be easily computed if v + (t, x) are somehow found. It turns out (and this is a well-known result from the theory of stochastic processes) that v4-(t, x) is the solution to the following initial boundary value problem for the backward Kolmogorov equation: ~V40"2(X) ~2V+ ~V4= }- b ( x ) - (5.99) c~t 2 cgx2 3x ' v(O, x) - 1,

v(t, c4-) - O,

(5.100)

where c+ are the exit points for the process Xt, which are equal to a - u0 and f l - u0, respectively. Next, we shall show that switching probabilities P~ (t) can be expressed in terms of v + (t) and p+ (t). Note that, according to (5.96)-(5.98), p4- (t) are related to v+(t) as follows: p+(t) = ~[e(t) - v+(t)].

(5.101)

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

553

A ~or ~Xt /\

+1

I X

I X+dX

~

t

-1 - -

F I G U R E 5.13

It is clear from the very definition of v • (t, x) that P ~ (t) - v m (t, 0).

(5.102)

It is apparent from Fig. 5.13 that the occurrence of exactly one downward switching is the union of the following disjoint elementary events: downward switching occurs in the time interval (2, 2 + d2) and then no upward switching occurs up to the time t. Owing to the strong Markov property of Xt, the probability of this elementary event is given by p-(2, 0)v + ( t - 2 , / / - u0)d2.

(5.103)

Now the probability P+ (t) of exactly one downward switching can be found by integrating (5.103) from 0 to t: P+ (t) = fo t p-(2, O)v+ ( t - 2, f l - u0)d2.

(5.104)

In other words, P~-(t) is the convolution of p-(t, O) and v + (t, fl - u0) P~- (t) - p- (t, 0) ,v + ( t , / / - u0).

(5.105)

By using similar reasoning, we can derive P1 (t) - p+ (t, 0),v-(t, a - u0).

(5.106)

Next, consider the probability P~-(t) of the occurrence of exactly two switchings starting from the initial state ~flx0 - 1. According to Fig. 5.14 this occurrence can be considered as the union of the following disjoint elementary events: downward switching occurs in the time interval (2, 2+d2) and then exactly one upward switching occurs up to the time t. The probability of these elementary events is given by p- (2, 0)P l ( t - 2) d2.

(5.107)

CHAPTER 5 Stochastic Aspects of Hysteresis

554 /k

~{o~13Xt J\ +1 = t

i X+dX -1

t

--

FIGURE

5.14

Now, by integrating (5.107), we find (5.108)

P~- (0 = fO t p - (,~, 0)P i- (t - ,b d,~. From (5.106) and (5.108) we obtain

(5.109)

P~ (t) = p - (t, 0) ,p+ (t, ] / - u0) , v - (t, a - u0). By using the same line of reasoning, we derive

(5.110)

P2 (t) = p+ (t, O ) , p - ( t , ~ - u o ) , v + (t, f l - uo).

For the sake of conciseness, we introduce the notations: p+ (t, 0) = p~ (t),

p+ ( t , / / - uo) = p+ (t),

p- (t, a - u0) = p - (t), (5.111)

v+ (t, O) = v:~ (t),

v + (t, fl - uo) = v + (t),

v-(t, ~ - u0) = v-(t). (5.112)

Now, by using the same line of reasoning as before and the induction argument, we can easily derive the following expressions for the switching probabilities: 2k-2 terms ,,

P ~ ( t , uo) - P o ( t ) , p + ( t ) ,

p-(t),p+(t),.

.. , p - ( t ) , p + ( t )

, v - (t), (5.113)

2k terms P2k+l(t, uo) = p-~(t), p - ( t ) ] , p - ( t ) ,

.. . , p - ( t ) , p + ( t )

,v-(t).

(5.114)

By substituting (5.113) and (5.114) into (5.94), we obtain the expression for q ~ ( t ) in terms of infinite series of iterated convolutions. These series can

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

555

be reduced to geometric ones by employing Laplace transforms: ~(s)~(s)-

It is clear that

p(t)e -st dt

Jo

(Re s > 0),

(5.115) (5.116)

v(t)e -st dt.

]~(s) I < 1.

(5.117)

By using these Laplace transforms, from (5.102), (5.113), and (5.114) we obtain / ~ (s) - ~ ( s ) ,

(5.118)

~P~k(S) - ~9o(S)~9+(s)~;-(s)[~-(s)~+(s)] k-l,

(5.119)

P2k+l (s) -- ~ (s)~-(s)[~-(s)~ + (s)] k.

(5.120)

From (5.120) and (5.94), we derive

~/a/~(s)

-

~-~(s)~- (s) 1 - ~- (s)~ + (s)

if ?~flx0 = - 1 .

(5.121)

A similar expression can be derived for the case 9~/3x0 - +1. According to (5.101) ~+ (s) - 1 - s~ + (s).

(5.122)

Thus, the problem of computing F/a/~is reduced to the problem of determining ~+(s). This can be accomplished by using the initial boundary value problem (5.99)-(5.100). The complexity of this task will depend on the nature of the stochastic process Xt, which models the noise in hysteretic systems. It is natural that the stochastic process that models the noise must be a stationary Gaussian Markov process. According to the Doob theorem [17], the only process that satisfies these requirements is the Ornstein-Uhlenbeck process. This process is the solution to the following Ito stochastic differential equation: dXt = - b X t dt + a d Wt,

(5.123)

where 1/b has the meaning of the correlation time. (This means that Xt and Xt, are only significantly correlated if it - t'l<,l/b.) The backward Kolmogorov equation for the Ornstein-Uhlenbeck process has the form (~V+

0-2 0-2V+

c~t = - 2

c~x2 - b x

(~V+

Ox

(5.124)

CHAPTER 5 StochasticAspects of Hysteresis

556

This equation should be considered jointly with initial and boundary conditions (5.100). By applying the Laplace transform to (5.124) and (5.100), we arrive at the following boundary value problem for ~+(s):

2 d2v+(s, x) _ bxd~;+(s, x_______~) 0" _ s~;+(s ' x) - - 1 2

dx 2

~;+(s, c +) - O,

dx

~;+(s, (x~) - 1/s.

'

(5.125) (5.126)

The solution to the boundary value problem (5.125)-(5.126) can be written in the form

~)&(S,X)_ !(1_e[X2-(c+)2]/4,~2 E)-s/b(x/~) ) ,3 ~D_s/b(C&/,~,) '

(5.127)

where 1)_s/b(X/2) are parabolic cylinder functions, while ,,1.= o-/J~.

Expressions (5.122), (5.121), and (5.127) jointly with (5.89) and (5.91) outline the main steps of computing)~t. It is apparent that the case of continuous-time noise is computationally expensive. Some sample examples of computations for this case can be found in [18,19]. In spite of its complexity, the case of continuous-time noise is convenient for the discussion of the origin of the universality of intermediate asymptotics M ",, M o - S(H, T ) I n t. (5.128) The very fact that intermediate asymptotics (5.128) has been observed for various magnetic materials as well as for superconductors and other hysteretic systems reveals its universality. This raises the question of the physical origins of this universality. The purpose of the subsequent discussion is to demonstrate that this universality can be traced back to the asymptotic temporal behavior of internal thermal noise [20]. This demonstration also suggests that ~F~ t-type relaxations may actually occur in the case of stationary Gaussian thermal noise. Our analysis also yields a simple expression for the viscosity coefficient in terms of differential susceptibility and temperature T. Experimental data supporting the theoretical analysis are also presented. Consider a hysteretic material that is first driven into the state of positive saturation. Then, the external magnetic field is reduced to some value -H0 close to coercivity - H c and maintained constant (in time) thereafter. Owing to the inherently present thermal noise, gradual relaxations

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

557

FIGURE 5.15

of magnetization take place. To mathematically describe these relaxations, we shall use the Preisach model: f/"

M t - JJ #(~z, fl)~a~Ht d~z dfl,

(5.129)

Ht -- -Ho if- ht,

(5.130)

and ht is a stochastic process that models the thermal noise. It is instructive to note again that adding noise ht to -H0 is mathematically equivalent to adding noise -ht to switching thresholds a and ]~. It seems natural to assume that ht is a stationary Gaussian process with decreasing in time correlations. In this case, the following temporal asymptotics is valid for ht with probability one [21] h t - max Ihrl ~ v/2R(0)lnt, 0Kr<~t

(5.131)

where R(0) is the value of the autocovariance function of ht at zero. It is this asymptotic behavior of ht that is ultimately responsible for long-time relaxations of magnetization. Our reasoning is based on the diagram technique for the Preisach model. According to this technique, the state of the Preisach model at the moment when the external magnetic field is reduced to - H 0 is represented by the diagram shown in Fig. 5.15(a). Let M0 be the magnetization in this state. At subsequent instants of time, this diagram will be modified due to the time variations of noise process ht. This will lead to fluctuations

558 AMt

CHAPTER 5 Stochastic Aspects of Hysteresis of magnetization

Mt - Mo - ~ t .

(5.132)

Consider the nature of these fluctuations during the time interval 0~
~

(5.133)

2 f/f~ #(o~, fl) do~dfl.

The trapezoid f] can be represented as a difference of two right triangles: F ( - H 0 - h~n) and F(-H0). This means that

AMt "~2(//F(_Ho_h,m ) #(a,fl)dad~-ffr(-H0)

#(a, fl) da dfl).

(5.134)

It is known from the theory of the Preisach model that p/.

2 ]/ #(c~,fl) da dfl - M~- - M + (H), JJF (S)

(5.135)

where M~ is the positive saturation value of the magnetization, while M + (H) is the magnetization value on the descending branch of the major hysteresis loop. By using Eqns (5.134) and (5.135), we derive AMt ~ M + (-S0)

- M + (-S0

-

hm)

dM+ (-Ho)h' m. dH

(5.136)

By invoking formula (5.131) and the fact that ~aM + (-H0) is the differential susceptibility )~(-H0, T) evaluated at -H0 and temperature T, we transform formula (5.136) as follows: ~/It

"~" v/2R(0)

Xa(-Ho, T)v/-~ t.

(5.137)

Furthermore, R(0) can be interpreted as being proportional to the total energy of the thermal noise. Indeed, R(0) -

F

oo

S(co) d o - 7T,

(5.138)

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE x 10 - 4

559

CoCrPt Material Hard Disk Sample at T=25~

3

2 .............................:

................................................. ~.....................................,..........................~

.............

"~" 0 E

(I) "~ cO ~N C

-I

-2

--3

-4 ............................................................................. ~

..........................................................................................

-5 .......................................... ~

.......................................~ i

-7 2

2.2

2.4

2.6

i 2.8

............. :

i 3

3.2

Sqrt (Ln(Time(sec)))

FIGURE 5.16

where S(~) is the spectral density of noise process ht. By using Eqn. (5.138) in Eqn. (5.137) and taking into account Eqn. (5.132), we arrive at Me ~ MO - ,~~ / ~ Za ( - Ho , T) ~ n

t,

(5.139)

where 2 is some constant. This is the final expression for the intermediate asymptotics of thermal relaxations in hysteretic systems. The immediate question is how this expression stands the test of experiment. To test the ~ dependence of long-time thermal relaxations, the measurements of these relaxations for thin-film CoCrPt magnetic recording materials and YBaCuO superconducting materials have been performed by using a vibrating sample magnetometer (VSM) (Micro Mag 3900 from Princeton Measurements Corporation). The measured values of M t were plotted on ~ t scale and the linear dependence of M t versus ~ t has been consistently observed. Some sample results of these measurements are presented in Figs 5.16 and 5.17. To test the temperature dependence of intermediate asymptotics (5.139), we point out that for very low temperatures Za(-H0, T) does not depend on T for magnetic materials. This means that asymptotics (5.139) predicts dependence of the ~ t decay of magnetization. This prediction was made earlier (see [1]) on the basis of the thermal activation model and is

CHAPTER 5 Stochastic Aspects of Hysteresis

560

High Tc Superconductor 0.15

E

9

I

Sample

at T=30

0.14

................................... ! .................................... i ..........

0.13

.......................................................................-~....................................~.....................................i..................................

0.12

~

0.11

................................... . ....................................--....................................,.....................................i..................................

................................................................................................................~..................................

O

,= ~

"

N

" O'}

K

I

~

0.1

................................................

~

0.09

................................... ! .................................... .~.........................

.

0.08

................................... .................................... ~.................................... ~

}

2

9 99

.

.

9

.

.

i

i

2.1

2.2 Sqrt

..............................................................

.................... :.................

"-'--'--

J

2.3

2.4

2.5

(Ln(Time(sec)))

F I G U R E 5.17

consistent with the reported experimental observations (see [7]). The temperature dependence given by Eqn. (5.139) has also been tested for high temperatures in the following manner. Equation (5.139) can be written in the form Mt ~" Mo - S(-Ho, T)~/--~, (5.140) where the slope S(H, T) can be retrieved from time observations of thermal relaxations for various values of H and T. Equations (5.139) and (5.140) suggest that S(H, T) - 2 ~ z a ( H , T). By integrating the last equality with respect to H over a sufficiently large range H_ KH~
fHH+ S(H, T) dH 2-

~/T [M+(H+, T) - M + ( H _ , T)]

.

(5.141)

If Eqn. (5.139) predicts the correct temperature dependence, then 2 in Eqn. (5.141) must remain constant. This has been verified over the temperature range of about 150 ~ for samples of high- and low-density CoCrPt recording materials. The results of this verification are presented in Fig. 5.18. It is apparent from Fig. 5.18 that 2 remains fairly constant and exhibits slight decay with increasing T. This decay can be attributed to the fact that S(H, T) becomes very sharply peaked around coercivity with the increase

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE 10

!

!

i

!

561

! I

High Density Hard Disk Samp,el

9

[]

I Low Density Hard Disk Sample I i

8 ............................................................................................................................................................................ i.........

7 ............................................................................................................................................................................ i ........ . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . .

. . . . . . . .

"~

5

. . . . . . . . . . . . . . . . . . . i ..................... i ..................... i ..................... ~..................... i ..................... i ..................... i ..................... i . . . . . . . .

~

4

.....................................................................................................................................................................................

1

o

280

300

320

340

360

380

400

420

440

Temperature (K)

F I G U R E 5.18

in T. This makes the measurement of max S(H, T) less accurate and eventually leads to the underestimated values of the integral in Eqn. (5.141). The derivation of Eqn. (5.139) is based on asymptotics (5.131). The proof of Eqn. (5.131) rests on the Gaussian nature of 'low probability tail' of the probability density function for ht. The Gaussian nature of stationary noise is customarily justified on the grounds of the central limit theorem. This theorem holds for normal and moderately large deviations. However, very low probability tails may appreciably deviate from Gaussian tails. If the very low probability tails have asymptotics p(x) = e -anxlv and the process ht has sufficiently strong mixing properties, then it can be shown that Eqn. (5.131) can be modified as follows: max Ih~l ~ b(ln t) 1Iv.

(5.142)

0 ~ t

By using Eqn. (5.142) and literally repeating the previous line of reasoning, the following intermediate asymptotics can be derived Mt ~ Mo - S(ln t) 1/v.

(5.143)

Particularly, in the case of exponential (v - 1) tails, Eqn. (5.143) leads to In t intermediate asymptotics. Experimentally, it may be quite difficult to distinguish between (in t) 1Iv and v / ~ t (or In t) thermal relaxations. This

CHAPTER 5 Stochastic Aspects of Hysteresis

562

is because In t is a slowly varying function within the time range of the validity of intermediate asymptotics.

5.4 N O I S E IN HYSTERETIC SYSTEMS A N D S T O C H A S T I C PROCESSES O N G R A P H S In the previous section, the Preisach model driven by a (continuous-time and continuous-sample) diffusion process is discussed. The mathematical treatment of this model is reduced to the analysis of random switching of ~flxt which, in turn, is reduced to the 'exit problem' for diffusion processes. Although the concept of random switching as an exit problem is quite transparent from the physical point view, its mathematical implementation leads to an infinite series of iterated convolutions. It turns out that the mathematical treatment can be appreciably simplified by using an entirely different approach based on the theory of stochastic processes on graphs. This theory has only recently been developed [22] and applied to the study of random perturbations of Hamiltonian dynamical systems [23]. The main purpose of this section is to demonstrate that the mathematical machinery of this theory is naturally suitable for the analysis of random output processes of hysteretic systems. As a by-product of this demonstration, we derive analytical expressions for stationary characteristics of these processes. These expressions are of interest in their own right. The discussion presented below closely follows the paper [24]. This discussion is based on the following simple fact. The output it ~Xt is a random binary process. This process is not Markovian. However, the two-component process yt (itt) is Markovian. This is because the rectangular-loop operators describe hysteresis with local memory. This means that joint specifications of current values of input and output uniquely define the states of this hysteresis. The two-component process yt is defined in the four-edge graph shown in Fig. 5.19. The binary process it assumes constant values on each edge Ik of the above graph. This justifies the following concise notation for the transition probability density: -

-

-

-

p(t, YlY0)I~*xk -- p(k)(t, x[~0). It is obvious that p(1) _ p

for

x~fl,

p(4) _ p

(5.144)

for x~>a,

p(2) q_ p(3) = p for x ~ [fl, a], where p is the transition probability density of the process assumed to be known.

xt,

(5.145) which is

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES 13

i=+1

11

/ i=-1

563

12

i=-1

14

i=+1

FIGURE 5.19

According to the theory of Markovian processes, the following equality is valid for

p(k).

4

4

E fkfCgP(k)dx--E fk(Lf)p(k)dx.

k=l

0t

(5.146)

k=l

Here,

1

~2

L = -~cr2 (x) ~

if- b(x)

0---~

is the generator of the semigroup of the process Xt, while f is a function that is continuous on the entire graph and sufficiently smooth inside the edges and satisfies certain 'gluing' (interface) conditions at a and ft. These interface conditions follow from the Markovian nature of the process yt on the entire graph [22]. In our case, the process yt 'spends zero time' at the graph vertices. In this situation, the interface conditions can be written as follows [22]:

E '~kJ-~x k

-- O,

,~,kj~O.

(5.147)

oj

Here, fIk = flIk, summation is performed over all edges connected to a graph vertex while the derivatives are taken along the edges in outward directions with respect to It is known [22] that constants are (roughly speaking) proportional to the probability that the process will 'move' from vertex along the edges It is clear that in our case there is zero probability that the process yt will move from the vertex fl along the edge I3, while the random motions along the edges I1 and 12 are equally probable. A similar assertion is valid for the vertex a. As a result, we arrive at the following interface conditions:

Oj,

Oj.

Oj

|

dx ~

2kj

Ik.

_

|

~x

- df141 dx

(5.148)

CHAPTER 5 StochasticAspects of Hysteresis

564

while the values of the derivatives dflB/dXlB and dfi2/dxl~ are entirely arbitrary. It is understood that differentiation in (5.148) is performed in the direction of increasing values of x. By integrating by parts in the equality (5.146) and by taking into account the interface conditions (5.148) and the choice inherent in f, dfI3/dxlB and dfI2/dxl~, one finds that the transition probability density p(3) satisfies the forward Kolmogorov equation (~ (b(x)p ~xx

~p(3) 1 c32 r = 2 ~?x2 ( ~

(3))

(5.149)

and the following boundary conditions: P(3)ie = PI~-

p(3)l] / = 0,

(5.150)

It is also tacitly understood that the standard r initial condition is imposed at Y0. A similar initial boundary value problem can be stated for p(2). However, p(2) can also be found by using formula (5.145). The solution to the initial boundary value problem (5.149)-(5.150) can be found in terms of parabolic cylinder functions and their Laplace transforms in the case when xt is the Ornstein-Uhlenbeck process (dxt -- b ( x t - xo) dt + e dWt) with expected value x0. Simpler analytical results can be obtained for stationary densities Pst _(3) and Pst _(2)" In this case, we have to deal with the following boundary value problem for the ordinary differential equation 1 d2 d (5.151) 2 dx 2 (Cr2(X)Pst)(3) (3) "- 0, ~xx(b (X)Pst) p(3)

_(3)

(5.152) Although the analytical solution to the above boundary value problem can be written out for any stationary diffusion process xt, below we present this solution only for the case of the Ornstein-Uhlenbeck process: st (fl)--" 0,

p(3) st (X) -- Pst(X)q0(X, 0r fl)'

Pst ( ~ ) - - P s t (00"

_(2) (X) -- Pst (x)[1 -- q0(X, 0r ]~)], Pst

(5.153)

where /)st(X)-qP(X,

/ ~ff2b e_b(x_xo)2/~2

0~,fl) -- /; eb(y-x~162 dy f; e b(y-xO)2/rr2 dy

(5.154) (5.155)

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES

565

If we consider the probability current

0-2 dp (k)(x) _ b(x - xo)p (k) (x) 2 dx

Jk(X) --

then it is easy to conclude from formulas (5.153), (5.154), and (5.155) that Jl(X) -- J4(x) -- 0, while J2(x) = -J3(x) ~- 0. Thus, there exists a nonzero probability current circulating in the loop formed by edges 13 and 12. The existence of the circulating-loop current can be considered as the manifestation of the lack of detailed balance in the two component process yt. The existence of this circulating current can also be traced to energy losses associated with the random switchings of rectangular loop 9~. This dissipated energy is extracted from noise, which is the only source of energy present in the discussed problem. The situation here is somewhat analogous to the one observed for stochastic resonance, where a feeble deterministic signal alone cannot affect switchings. These switchings are assisted by the internal noise and they are accompanied by the extraction of energy from the noise [25]. It is instructive to compute the expected value it of the binary output process and its variance 0-2. It is clear that -- P~tfl(it - 1)

it -- Est(9~flxt)

-

P~flt (it -

-1),

(5.156)

which is equivalent to it --

2P~flt (it - 1) - 1.

(5.157)

It is apparent that

P~tfl( i t

-- 1)

-

_(3) Pst (x) dx +

fist (X) dx,

(5.158)

which leads to

it -- 2

Pst-(3)(X) dx -}-

Pst(X) dx - 1

(5.159)

When it is computed, 0-2 can be calculated as follows: 0-2 _ 1 - 72.

(5.160)

Calculations are substantially facilitated by the observation that

foX ey2 dy - --~erfi(x),

(5.161)

f e-Xaerfi(x) dx - ~x2 2F2 ( 1, 1; ~, 3 2; --X 2) ,

(5.162)

CHAPTER 5 StochasticAspects of Hysteresis

566

/ i/,;':. .i 0.8 r ........... ::............ I ........... i ............ i~;[i.-"- .... i

o.o ~ ........... ~............ I ........... ~............ ~i':....:...d.... !

!

~:..

i ....... ~

=~ ..... -I

.~........', ........... 4

!

!

o.,

........... i .......................

i ............ !J: ........ ! ....................

i...........

0.2

...........

::. . . . . . . . . . . . . . . . . . . . . . .

: ............

:. . . . . . . .

9 ....................

-o-

o.s

!

:

:"

:

-e-

1.0

:

:

i:

: ::............

_o..~ :

:

: ::-

: -it i r.--.'-..:i;r.:

_o,

.......

-0.8

. . . . . . . . . . . . . . . . . . . . ........ .. . .-. .

-0.6

............

!:." .:"i] .

-1

,.. v -2

v

v _ _

' iF

~-- ~----

! ......................

.,.~j:!................................................. . . . . . . . :,

...."

-

-1.5

-1

-().5

0

0.5

1

1.5

2

V

FIGURE

5.20

where erfi(x) and 2F2 are 'imaginary error function' and 'generalized hypergeometric function', respectively. By using the formulas presented above, the expected value it and variance cr2 have been computed as functions of the expected ('bias') value x0 of the input process xt for the case of hysteretic nonlinearities represented by symmetric rectangular loops ~,_~. The results of the calculations are shown in Figs 5.20 and 5.21, respectively. These results are plotted for normalized values of input bias v = x0/a and normalized values of switching thresholds ~ = a/2, where ~2 _ ry2/b is the variance of the stationary distribution of the input Ornstein-Uhlenbeck process xt. The dependence of it on x0 can be interpreted in magnetics as an 'anhysteretic' magnetization curve. This anhysteretic curve depends on the noise variance. This dependence is especially appreciable when the noise standard deviation 2 is comparable with the switching threshold value a. Next, we shall apply the obtained results to the case of a hysteresis loop with 'curved' ascending and descending branches (Fig. 5.22). This type of hysteresis loop is exhibited, for instance, by Stoner-Wohlfarth particles when the applied magnetic field is restricted to vary along one direction. Suppose that the loop shown in Fig. 5.22 is driven by the Ornstein-

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES 1

567

m ~ m

0.9 0.8 0.7 0.6

%

0.5

0.4 0.3 0.2 0.1 0 -2

~~'~" -1.5

-1

-0.5

0 v

0.5

1

1.5

- v

" ~ ,~ 2

FIGURE 5.21

-Uhlenbeck process Xt and we are interested in the stationary distribution of the output process tt. T h e process Yt is not a binary one. However, it admits the following representation in terms of the binary process

it -- ~o~flXt: yt =

d + (Xt) --d- (Xt) 2

it +

d + (Xt) if-f- (Xt) 2

'

(5.163)

where the meaning o f f + (xt) and f - ( x t ) is clear from Fig. 5.22. By using formula (5.163) and the appropriate change of variables, we obtain the following expression for the stationary distribution density ~s+ (y) of the process yt along the descending branch: ~s+t(y)

_(3)

--Pst(g+(Y))

d g + (Y)

dy

(5.164)

Here, g+ (y) is the inverse of the function y - f+(x). The last formula is valid for fl < x < ~. For x > ~, density/9 st (3) should be replaced by Pst. In a similar way, the density ~t (Y) along the ascending branch can be computed.

CHAPTER 5 Stochastic Aspects of Hysteresis

568

f+(zt)

S

~

J

A

< FIGURE

J

f -(xt)

5.22

The results obtained for ~a/; operators can also be used to compute the stationary characteristics of the output processft of the Preisach model driven by the stochastic process xt:

~t _ [ [ ~(0~,~)Est(7~xt) dd~

d~ dfl,

(5.165)

where Est(7a~xt) can be evaluated by using expressions (5.153)-(5.162). If it is desired to evaluate the stationary value of the second moment Est (ft2) of the output processft and its variance, the following integral must be evaluated:

Est(f2) -- /~x>.~l /~a>/~aEst(7~176

xt)

xfl(~162]~1)fl(~162f12)d0r dill d0r dfl2,

(5.166)

where Est(.) stands for the stationary expected value. To compute Est (70r1/~1x t ~ 0~2fl2xt )" we consider the three component Markovian process

(:i) xt where i~ - ~ l & X t and i 2 - ~2&xt. Depending on the relation between c~1, //1, e2 and//2, this process is defined on graphs shown in Fig. 5.23. By using the same line of reasoning as before, one can easily arrive at the following _(k) expressions for the stationary densities Pst (zt)IIk = Pst (x).

5.4 NOISE IN HYSTERETIC SYSTEMS A N D STOCHASTIC PROCESSES 13

16 ~~ ( X

(X1/

llxx ~ 1 ~

569

2 /17

14 12

(a)

15

15 ~2N~/~lX2

13 O~1

14

,/ ~//NN

(b)

16

~2 11

\

12

14

O~1

Ill

O~2

~2

12

/

17

(X1

(c) FIGURE 5.23

In the first case (Fig. 5.23(a)) w h e n two rectangular loops do not overlap (ill < ~1 < f12 < ~2), w e have p(3) p(stk) -- Pst f o r k - 1, 4, 7, st (X) -- Pst(X)q~(X, 0r i l l ) , (5.167) p(6) st ( X ) -

Pst(X)qO(X, 0r

f12),

/9(2) ^(3) st - - P s t - - P s t ,

p(5) (6) st -- /gst -- /)st 9

(5.168) In the second case (Fig. 5.23(b)) w h e n two rectangular loops completely overlap (ill < f12 < 0r < C~1),we have p(k) _(3) (x) -/9st (x)q)(x, ~1, ill) (5.169) st --/)st for k - 1, 8, Pst _(2) _(3) Pst -- [ s t -- Pst '

p(st4) - - P s_(3) t

^(5) --Pst '

-(5) (X) _ Pst ~(3)(x)q)(x, 0r Pst

f12)

(2) (X)q)(X, C~2, f12) ' /)st(7) (X) -- l) st

(5.170)

CHAPTER 5

570

S t o c h a s t i c A s p e c t s of H y s t e r e s i s

(7) p(st6) m P^(2) st --Pst "

(5.171)

Finally, in the third case (Fig. 5.23(c)) when two rectangular loops partially overlap (ill < / J 2 < ~ < ~162 w e have

p(stk) -- Pst

for k = 7,

p(3) ^(2) ' st -- Pst -- Pst

^(2) (X) -- Pst (X) [1 -Pst

q~(x,~l , ill)], (5.172)

(4) (X) -- P s t ( X ) q ) ( x , Pst

0r

p(5) ^(2) (4) st - - P s t - - P s t --Pst '

^(6) (4) Pst m P s t _ P s t 9

]~2),

(5.173) (5.174)

It is worthwhile noting that in the last case there is no graph edge corresponding to i~ -- - 1 and it2 = 4-1 because these simultaneous values of i~ and it2 are not consistent with the definition of rectangular-loop operators ?~1#1 and ~2/J2. By using the above expressions for Pst^(k),Est(~alfllXl~o~2flaxt) can be computed. As an example, consider the particular case of the Preisach model when all rectangular ~-loops are symmetric: ~_~xt = ~=xt. Mathematically, this case is obtained when the 'weight' function #(a, t) has the form

/~(a, t) = ~(a)6(a 4- fl),

(5.175)

and the Preisach model is reduced to

ft =

f0 ~(e)~xt

de.

(5.176)

For many magnetic materials, the weight function #(e, fl) is usually narrowly peaked around the line e = -ft. For these materials, formulas (5.175) and (5.176) can be regarded as fairly good approximations. At the same time, the calculations are considerably simplified because, in the case of symmetric loops, any two loops completely overlap. As a result, one has to deal with the three-component Markovian process ~.t defined only on the graphs shown in Fig. 5.23(b). For the case of model (5.176), formulas (5.165) and (5.166) can be written as follows: --

f0

~(~x)Est(~xt)

d~,

(5.177)

Est(f2)-a~o~~ Est(~;axt), formulas (5.153)-(5.162) Est(~;axt~;a,xt) in (5.178), we first remark that

To compute

Est(~axt~a, xt) - 2P(~;axt~a,x t -

(5.178) can be used. To evaluate 1) - 1.

(5.179)

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES ! 9

! .

.

.

571

I . .

. . . . . . . . _

.: .

.

.

.

.

.

.

.... 9

.,_~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~,'.

, *9

/ i

9

/

1

...........

:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

..~-

9

; ......

.....,

,i, . .

.

9

.,-,

9

~,,

.

.

.

.

.

,,

_,...,

:

.

':~:7~ .

:

,

.

_

.

--

,--

--

w

.

J

l

.

'

?~" . . . . . ,; . . . . ; . . . . . . . :l

~. . : . : . . :

.

.........

!. . . . . . . . . . . .

.

.

"

.

w 0 ~ ~ ' / l : . ~

I

'

"'i

j

9.

..

.j.

-1

9

9

..

I

9

!

9

.

9

-2

~-

""

,d.

_

/L - - - - : -

.,.,

.,-

-.-

9

"

#

.:

,,.

--tl-0.5 ~ 1.0

I

9

"-".-:

'...J'9"

-:3

t

-2

9 9

9 ,....

_..

_-

,,...,,.,

-

-'~"

"9

--

9

i

i

i

-1.5

-1

-0.5

--

1.5 2 . 0 2 . 5

3.0

I

0

0.5

1.5

1

V

FIGURE 5.24 To find P(~axt~;ovxt), one has to integrate the appropriate Pst-(k)(X) over those edges of the graph shown in Fig. 5.23(b) on which ~xt and ~&,xt have the same signs. This leads to the formula

Est(~xt~a, xt) = 2

[f_~' Pst [a Pst(3)(x) dx 4- [ c o ~(5) (x) dx +

Pst (x) dx

!

0C!

A(6)(x)d x

4-

Pst 0~ I

A(2)

Pst(X)d x

Pst (x) dx +

+ 0~

- 1.

OO

(5.180)

The integrals in the last expression can be evaluated by using formulas (5.169)-(5.171) and taking advantage of relations (5.161) and (5.162). Some sample results of calculations are shown in Figs 5.24 and 5.25 for ?tst and Est(f2), respectively. In these calculations, it is assumed that ~(~) - I and ?tst and Est(f 2) are computed as functions of normalized values of input bias v - xo/~o for various normalized values of ~0 - ao/2, where ~2 is the variance of the stationary distribution of xt. The values ofy~tst and Est(f2)

CHAPTER 5

572

9

.....

Stochastic Aspects of Hysteresis

~.

8

.........................................

~. . . . . . .

....... i .... i ................. i

I

I

I

I

....... ~ ! i ! ,

~ ...........

,.. : - \....,.,.,.,.,.../. ,:::,

i

!

.... :,,,,,,^,~, 2 ' --.--.-.'---.--.~-;

-2

I

! ~ : :

i

!

!

-1.5

-1

.

.

.

.

.

.

.

.

.

.

.

.

.

\

I

....... ~ - :

~!i!,

~!i!~ ...... \..~.l.i.R

~

........................................

i,'

I

i

.

I

~.... i.. . . i . . . t ',!

.......................

.

!

x

3-

.

'7..~!,,' ..... ~......... !. ...... ,~!-...7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 ......................................... 4

--2.0 - -2.5 .-. 3.0

i

.

I

~1.5

"

I

'

I ~ 1". 0 0.5

7 ..................................... , ...... ,I........... t: ...... .,

III

.

/ -.

/ ~,

6 ................................

'

..4. . . . . .

x ,~

\

-0.5

........................

/ /

\ qii~/

~ ...........

::

i

i

i

i ....

i .....

!

i

1

1.5

~

0

.

0.5

2

V

FIGURE

5.25

12k+1

13

t1"

/

I I I I I I I

I1

FIGURE 5.26

h a v e also b e e n n o r m a l i z e d b y 2 a n d ,~2, respectively. It is w o r t h w h i l e to n o t e that, as e v i d e n t f r o m Fig. 5.25, the a s y m p t o t i c v a l u e s of E s t ( f 2) are e q u a l to ~ a n d coincide w i t h a s y m p t o t i c v a l u e s of (ftst) 2. This g u a r a n t e e s zero a s y m p t o t i c v a l u e s for v a r i a n c e 0-2

ft"

5.5 ANALYSISOF SPECTRALNOISE DENSITYOF HYSTERETIC SYSTEMS 573 The described formalism of stochastic processes on graphs can be further extended to compute higher-order moments of the output process ft. This extension is more or less straightforward in the case of the model (5.176). In this case, the relevant multicomponent Markovian processes are defined on the graph shown in Fig. 5.26 and, by using the same line of reasoning as before, we can derive the following explicit expression for p st (2k+1)(x) for edges I2k+1 k P ((2k+1) x)st

-- Pst (X) I-I ~(X, ~j, --~j) . j=l

(5.181)

Similar expressions can be derived for other graph edges.

5.5 ANALYSIS OF SPECTRAL N O I S E D E N S I T Y OF HYSTERETIC SYSTEMS D R I V E N BY S T O C H A S T I C PROCESSES The behavior of hysteretic systems (materials) is usually affected by the internal noise that causes thermal relaxations in these systems. This implies that outputs of hysteretic systems are stochastic processes. In this section, a method for the calculation of the spectral density of such output processes is presented. The discussion is centered on the case of a rectangular hysteretic loop driven by a diffusion stochastic process. This case is important because complex hysteretic systems can be modeled through the Preisach formalism as weighted superpositions of rectangular loops. The problem of calculation of the spectral density of the output process is of considerable mathematical complexity because this process is not Markovian in nature. This difficulty is overcome by using the mathematical machinery of stochastic processes on graphs. In addition, the so-called 'effective' distribution function is introduced that appreciably simplifies the calculation of spectral density. As a result, remarkably simple explicit expressions for the spectral density in terms of parabolic cylinder functions are derived for the case when the input stochastic process is the Ornstein-Uhlenbeck process. The discussion in this section follows the paper [26]. Consider a rectangular loop hysteretic nonlinearity driven by a diffusion stochastic process xt: it = ~flxt. (5.182) The input stochastic process (noise) Xt leads to random switching of rectangular loop ~ . Thus, the output it is a random binary process. The main goal of this section is to compute the spectral density of it. The process it is not Markovian. The efficient way to study the random process it is to

574

CHAPTER 5 Stochastic Aspects of Hysteresis

consider as before the two-component Markovian process yt:

yt -- xtit) '

(5.183)

which is defined on the four-edge graph (see Fig. 5.19). Process Yt can be analyzed by using the mathematical theory of Markovian processes on graphs. This approach will be pursued in our calculations of the spectral density of it. To start these calculations, we shall first define the autocovariance matrix for yt. Consider random vectors ( ix)

Y~-Y=

and

y 0 - y ' = ( i ' ) x' '

(5.184)

generated by the process yt at times t -- r and t = 0, respectively. Then, the autocovariance matrix C(z) can be defined as follows: C(r) = ((y - (y}) (y' - (y'))T),

(5.185)

where the symbol '{ }' is used for the notation of average value, while superscript 'T' denotes a transposed vector. By using the joint probability density function p(i, x, r; i~, x ~, 0) and stationary distribution function ps(i, x) of the process yt, the autocovariance matrix can be computed as follows: C(~)

- f f ~ ~ y(y')Wp(i, X, ~; i', X', O) dx dx' J J

i

i~

-(/~yps(i,x)dx)(f i

E(y')Tps(i',x')dx'). i'

(5.186)

Since the process yt is Markovian, its joint probability function can be represented as the product of the transition probability function p(i, x, r.]i~, x ~, O) and the stationary probability function. This leads to the following simplification of formula (5.186):

"C(r) = ff ~ ~y(y')T[p(i,x, r]i',x', O)- Ps(i,x)]Ps(i',x') dx dx'. i i'

(5.187) To further simplify the calculations, the following two-component 'effective' distribution function is introduced:

g(i'x"c)-(gl(i'x'~) )ga(i,x,"c)

- f Ey'[p(i,

x, r[i', x', 0) -

ps(i, x)]Ps(i', x') dx'.

(5.188)

5.5 ANALYSISOF SPECTRAL NOISE DENSITY OF HYSTERETIC SYSTEMS 575 If this 'effective' distribution function is somehow found, then the autocovariance matrix can be computed as follows: C(r) - 1 ~

ygT (i, x, r) dx.

(5.189)

i

Next, we shall derive the initial boundary value problem for g(i, x, ~) on the graph shown in Fig. 5.19. On each edge of this graph, the transition probability density function satisfies the forward Kolmogorov equation

8p(i, x, z[i', x', O) 8~

+'Lxp(i, x, rli', x', O) - 0,

(5.190)

A

where Lx is the second-order differential operator whose structure is determined by the nature of the process xt. The transition probability density function satisfies the obvious initial condition

p(i, x, 0[i', x', 0) = ~ii,~(x- X')

(5.191)

and certain boundary conditions at graph vertices x = ~ and x = ft. These 'vertex' boundary conditions are discussed in the previous section and they express the continuity of the transition probability function when the transition from one graph edge to another through a vertex occurs without switching of the rectangular loop ~/~, and a zero boundary condition is imposed on the third graph edge connected to this vertex. In addition, the probability current is conserved at each vertex. The stationary probability function satisfies the equation A

LxPs(i,x) - 0

(5.192)

on each edge of the graph and the 'vertex' boundary conditions. By using the formulas (5.190)-(5.192) and the definition (5.188) of g(i, x, ~), the following initial boundary value problem on the graph can be derived for the 'effective' distribution function: g(i, x, ~) satisfies the equation ?g(i, x, "c)

A

+ Lxg(i, x, "c) = 0

(5.193)

on each graph edge, the following initial condition g(i, x, ~:)l~=o = (Y - (Y))Ps(i, x) and the 'vertex' boundary conditions.

(5.194)

CHAPTER 5 StochasticAspects of Hysteresis

576

It is well known that the matrix of spectral density is related to the autocovariance matrix through the Fourier transform:

A S(co) = F

C(z)e -j~~ dr

(5.195)

(30

A

The element $11 (09) of the matrix S(co) can be construed as the spectral density of the output process it. By introducing the Fourier transform of the' effective' distribution function g(i, x, ~:)e-jc~ dr, G(i'x'c~176 t G 2 ( ix,, o9) -- fo ~176

(5.196)

and using formulas (5.189) and (5.195), we find: $ 1 1 ( c o ) - 2 R e ( f y ~ i G l ( i , x , co)dx). i

(5.197)

Thus, if Gl(i, x, co) is found, then the spectral density of the process it can be computed. To find G1 (i, x, co), we perform Fourier transformation of the initial boundary value problem (5.193)-(5.194). As a result, we arrive at the following boundary value problem for G1 (i, x, co): it satisfies the following equation

jCOGl(i, x, 09) + L"xGl (i, x, 09) - (i - (i))ps(i, x)

(5.198)

on each graph edge and the 'vertex' boundary conditions. Thus, by solving the boundary value problem (5.198) and by using formula (5.197), the spectral density of the output process it can be computed for any input process xt. Analytical results can be obtained when xt is the OrnsteinUhlenbeck process. This process is very appealing as a noise model because of its stationary and Gaussian nature. In the case of the OrnsteinUhlenbeck process the operator Lx has the form: A

LxGl(i, x, co) -

0-2

2

d2G1 (i, x, o9) b d dx 2 q- dx [(x - x0)GI(/, x, o9)],

(5.199)

where x0 is the expected value of Xt. Formulas (5.198)-(5.199) lead to a linear second-order, inhomogeneous differential equation whose solution has two distinct components: particular solution of the inhomogeneous

5.5 ANALYSISOF SPECTRALNOISE DENSITYOF HYSTERETICSYSTEMS 577 equation and general solution of the homogeneous equation: j09G 0

0"2 d 2 G ~

if-

2 dx 2

b d [(X -- x0)G ~

-

(5.200)

0

dx

It is apparent from formula (5.192) that a particular solution of the inhomogeneous equation has the form - J ( i - (i))ps(i, x). Because the particular solution is purely imaginary, it does not contribute to the spectral density computed in accordance with formula (5.197). Thus, G1 in formula (5.197) can be replaced by GO that satisfies Eqn. (5.200) and can be expressed as follows:

GO-Jl

-~

a2 daG~ 2 dx 2

b d~

u~ ((x

- x~176

"

(5.201)

By substituting expression (5.201) into formula (5.197), performing integration and using the 'vertex' boundary condition, the following formula can be derived for the spectral density: $11 (09) - -

2a 2 dG 0 dG~ ] - - -09 Im[F~dG0 (1, a+, 09) - - ~ x (1, a-, 09) + d x - (1, fl+, 09) " (5.202)

The solution of Eqn. (5.200) can be expressed in terms of parabolic cylinder functions. This fact, along with formula (5.202), leads to the following analytical expression for the spectral density

$11 (09) - -

20"2 09

Im

I~ (

dzk (~X,09) + Ck(09)-d-xx dzk (fl, 09)t l , bk(09)-d-xx

(5.203)

k=l

where functions Zk(X, 09) are related to the parabolic cylinder functions yk(x, 09) by the formula:

Zx(X, 09) - Yk

(x -- Xo), 09 e -~2

(5.204)

and coefficients bk and Xk are found from the 'vertex' boundary conditions. The functions yl and Y2 are the parabolic cylinder functions that vanish at +cx~ and -cx~, respectively. The sample results of the computation of the spectral density are shown in Fig. 5.27.

CHAPTER 5 StochasticAspects of Hysteresis

578

3.5 \

.

.

.

.

.

3 "'",.~ 2s

o~=b=c~=l

',.\

"- , :",.,~-"',, \:\ "',,, ",..\ ", '..X 1.5 -",, '",XX,

Xo=0.2 .... Xo=O.4 ---e- Xo=0.6 ---- Xo=0.8 4 - Xo=l.0

2

~,,. "".~,.

.....

0.5 0.5

1

1.5

2

2.5

03

FIGURE 5.27

5.6 M O D E L I N G O F T E M P E R A T U R E - D E P E N D E N T HYSTERESIS WITHIN THE FRAMEWORK OF RANDOMLY PERTURBED FAST DYNAMICAL SYSTEMS In the previous sections, the theory of stochastic processes is used to study noise-induced thermal relaxations in hysteretic systems within the framework of the Preisach model of hysteresis. In this and the next sections, the theory of stochastic processes will be used to construct novel models of hysteresis. We begin in this section with the discussion of models for temperature (noise)-sensitive rate-independent hysteresis. It has been experimentally observed that hysteretic properties of materials are sensitive to temperature, i.e. to the level of internal thermal noise. At a given temperature, various materials exhibit rate-independent deterministic (that is, fully reproducible) hysteresis. However, many hysteretic properties, such as coercivity, saturation magnetization, loop shape and others, change appreciably with temperature. Usually, temperature dependence of hysteresis is taken into account within the framework of the Preisach model in ad hoc ways without any effort to understand how sensitivity to noise and deterministic properties of rate-independent hysteresis can be compatible and coexist. In the following discussion, the modeling of temperature (noise)-dependent deterministic hysteresis is

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

579

developed within the framework of randomly perturbed fast dynamical systems. By using the mathematical machinery of the large deviation theory, it is demonstrated that, for specific interplay of the fast system dynamics and the noise strength, randomly perturbed fast dynamical systems may exhibit rate-independent deterministic hysteresis sensitive to noise. The theory of large deviations of randomly perturbed dynamical systems is extensively developed in [26]. The subsequent discussion closely follows the papers [27,28]. To emphasize the sensitivity of deterministic hysteresis to noise, the discussion is deliberately centered around the case of dynamical systems that under purely deterministic conditions do not exhibit hysteresis. It is demonstrated that noise-induced deterministic hysteresis may occur in such systems. This phenomenon is of interest in its own right, and it clearly reveals sensitivity of deterministic hysteresis to noise. First, the case of dynamical systems with two attractors is studied. Then, the case of dynamical systems with many attractors is discussed and the mathematical machinery for identifying the attractors through which such systems evolve in time is presented. In the case of temperature (noise)-sensitive rate-independent deterministic hysteresis one deals with four distinct time scales. The first is the time scale of fast internal dynamics of the hysteretic system. The second is the time scale on which observations (measurements) are performed. This time scale is much larger than the time scale of systems dynamics so that every observation can be identified with a specific stable state of the system. The third is the time scale of input variations. This time scale is much larger than the observation time scale so that every measurement can be associated with a specific value of input. In this way, every measurement establishes the connection between a specific input value and the respective state of hysteretic system. Finally, the largest time scale is the time scale on which thermal relaxations occur. It is demonstrated that for a given noise level all attractors can be subdivided into two distinct groups. The first group consists of attractors that participate in fast switchings that occur on the first time scale. These switchings lead to deterministic hysteresis. The second group consists of attractors that participate in slow switchings that occur on the fourth time scale. These switchings are known as thermal relaxations. As temperature (noise level) is changed, the 'boundary' between these two groups is shifted and this results in temperature-dependent deterministic hysteresis. To start the technical discussion, let us consider one of the possible mechanisms that leads to the occurrence of rate-independent hysteresis. Suppose that the state of a system is defined by a set of interior parameters X = ( X 1 . . . . . X n) E [~n, and the evolution of the system is governed by the

CHAPTER 5 Stochastic Aspects of Hysteresis

580

g

T A

(a)

gl

c

-..~_ ,. )X

l

(b)

FIGURE 5.28

equations:

~ X t - b(Xt, gt),

Xo E ~n

(5.205)

where gt is an exterior parameter-input, while b(x, g) is a sufficiently smooth vector field in R n+l. It is further assumed that 6 << 1, which implies the fast dynamics of interior parameters in time. The case when this dynamic is appreciably faster than the temporal variations of input gt is discussed below. Let us also assume that the system (5.205) with a 'frozen' input gt -- g has a finite number s of asymptotically stable equilibriums Kl(g) . . . . . K~(g)(g) and that any trajectory of this system is attracted to one of Ki(g) (with the possible exception of trajectories belonging to separatrix surfaces). An observation (measurement) Yt performed on the system can be mathematically interpreted as a certain function of interior parameters Xt. Since there are no instantaneous observations, the above function will include some integration (averaging) over some time h. If h is small on the time scale of input variations and large on the time scale of Xt-variations, then the relation between Yt and gt will be essentially rate-independent. When input gt is changed in time, the vector field b(x, g) may have bifurcations that lead to hysteresis. A typical example of this situation for the one-dimensional case is shown in Fig. 5.28(a), where curve g is defined by equation b(x, g) = 0, and it is assumed that b(x, g) < 0 and b(x, g) > 0 above and below this curve, respectively. It is apparent that the vector field b(x, g) has one stable equilibrium for gt ~ (gl, g2) and two stable equilibria for gt ~ (gl, g2). If the initial value of input go > g2, then the evolution of the system for decreasing gt essentially follows the left branch MBA of the curve E. The state of the system is switched to point D on the right branch

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

581

of g when gt reaches (from above) the value of gl, and the system follows the right branch (DN) towards N as the input is further decreased. When the variation of input is reversed, the state of the system moves upward along the right branch of g until the switch to the left branch occurs at point C. It is apparent that the evolution of this system results in a rateindependent hysteretic relation between observation Yt and input gt. Next, consider the case when the field b(x, g) has no bifurcations. The one-dimensional example of this situation is shown in Fig. 5.28(b), where the set of g of solutions of equation b(x, g) = 0 consists of three disjoint curves x = ?0(g), x = ~2l ( g ) , x = ?2(g) with ]2l(g) ( ? o ( g ) ( 7 2 ( g ) . It is assumed that points of x -- 71 (g) and x = 72(g) correspond to stable equilibria, while points of ?o(g) are unstable equilibria. It is apparent that, for sufficiently slow input variations, the system will essentially move only along one curve (?l(g) or 72(g)) and no hysteretic relation between Yt and gt can be observed. It will be demonstrated below that small random perturbations may result in observable deterministic hysteresis between Yt and gt, although the original deterministic system has no bifurcations. Hysteresis appears due to bifurcations in the randomly perturbed system. First, the case of hysteresis induced by noise in systems with two asymptotically stable equilibria will be discussed. Then, the case of many equilibria will be treated. Consider small white-noise perturbations of dynamical system (5.205)

(~J(y - b(X~, gt) + v/-d o'(X~)IfVt.

(5.206)

Here, Wt is the vector Wiener process in R n, while r~(x) is a nondegenerate square matrix of order n with bounded Lipshitz-continuous entires, while e is a small parameter. We next assume that the following relations for small parameters and e hold e, ~ $ 0, l i m ~ - 1 ] In ~] -- c > 0. (5.207) By introducing the time r - ~, eI - ~ and by fixing gt - g, the stochastic differential equation (5.206) can be written as follows

'g - b(X "g, g) +

(5.208)

Next, we shall introduce the (normalized) action functional defined as follows [26]: Sg0,T((p ) _ 21 ~0 T [o._l(q)s) (~bs _ b((fis,

g))]2 as,

(5.209)

if ~ ~ Co, T is absolutely continuous, and Sg0,T is infinite for all other functions from CO,T. The action functionals is the main mathematical tool of

CHAPTER 5 Stochastic Aspects of Hysteresis

582

the theory of large deviations. It is, roughly speaking, a measure of the 'difficulty' for a sample of stochastic process X~'g to pass through a small neighborhood of function qh. This function determines the'most probable' transition route between equilibrium points as well as the asymptotics of switching (transition) times between these equilibria. By using this functional, functionals V12 (g) and V21 (g) can be introduced: V12(X) -- inf{g0,T(~0)" q~0 -- },l(g), (PT -- },2(g), T > 0},

(5.210)

V21(g) = inf{g0,T(q~)" q~0 -- },2(g), OPT = },l(g), T > 0}.

(5.211)

When the vector field b(x, g) is potential: b(x, g) = -VxU(x, g) (which is always true for the one-dimensional case) and a is the unit matrix, then the above functionals can be expressed as the following potential differences: V12(X) -- 2[U(},0(g), g) - U(}, l(g), g)] and V21(g) - 2[U(},0(g), g) -

U(},2 ( g), g) ]. Finally, let us introduce the sets: G12 - {g: V12(g) < c, V12(g) < V21(g)},

(5.212)

G21 -- {g: V21 (g) < c, V21(X) < V12(g)},

(5.213)

where c is defined in (5.207). It is known from the theory of large deviations [26] that for the process X g'e' the transition times between equilibrium points },i(g) and ?7(g) (i, 7 e {1, 2}, and i # 7) have (in the sense of logarithmic equivalence) the exponential asymptotics exp{ F1 FiT(g)} for ~' $ 0. This means that the transition times TiT for the process Xt 'g have the asymptotics b exp{ 7'~Vd( g) }. By taking into account relation (5.207), we find In ~ } _ (~1 vii(g) c .

T d ,-, ~exp ViT(g)---d-

(5.214)

It follows from the last expression that TiT tends to infinity if ViT(g) > c, and it tends to zero if ViT(g ) < c. Thus, it can be concluded that if the input gt is monotonically increased and the state of the system moves upwards along the curve },2(g), it is switched to the curve },1(g) as soon as the boundary of the set G21 is crossed. Similarly, if the variation of the input is reversed and the system tracks the curve }'1 (g), it is switched to the curve },2(g) as soon as the boundary of the set G12 is crossed. It is apparent that the above switchings result in a rate-independent hysteretic relation between observation Yt and input gt. It is also clear that the input switching ('coercive') values as well as the shape (width) of the corresponding hysteresis loop are determined by constant c in (5.207) and, consequently, by

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

583

the interplay of fast system dynamics and the noise strength. It is worthwhile to remark that trajectory X~ of randomly perturbed system (5.206) is close to 71 (gt) or 72(gt) in some integral norms that filter out 'short-lived' excursions of stochastic process X[. Such integral norms naturally appear in functionals that relate observation Yt to the state X[ of the system due to some finite time of any observation. Now, consider the case when for any fixed gt = g the dynamical system (5.205) has an arbitrary finite number s of asymptotically stable equilibria Kl(g), K2(g) . . . . . K~(g)(g). Let us suppose that as the input is changed with time, these equilibria trace disjoint curves 71 (g), 72(g) . . . . . 7e(g) (g). It is apparent that, for sufficiently slow input variations, the deterministic system essentially moves along one of those curves and no hysteresis is observed. The situation may change when our dynamical system is randomly perturbed and its motion is governed by Eqn. (5.206). In this case, the presence of noise may result in switching from one 7-curve to another and, in this way, produce hysteretic behavior of the system. The immediate task is to describe the mathematical machinery that allows one to determine the switching values of input and to identify the 7-curves involved in switchings. This mathematical machinery is based on the notion of the action functional (5.209) that is used to define the set of functions

Vij ( g )

-

inf { S g0,T(~)" ~0 -- 7i(g), ~(T) -- 7j(g)}.

(5.215)

It turns out that the time sequence (order) of noise-induced switchings in the randomly perturbed dynamical system (5.206) is in a way not random and completely governed by deterministic functions Vij(g). To fully describe the time sequence of noise-induced switchings, it is necessary to introduce the hierarchy of cycles [26] defined by functions Vij(g) on the set of L of equilibrium points and the notion of/-graphs defined on L. Each point i from L is considered as rank 0 cycle and the exit rate ei from this cycle is given by the formula

ei - min Vij. j: j#i

(5.216)

The 'next' state to i is defined as the state k = Jl(i), where the above m i n i m u m is reached. The transition i --~ k is the most probable (with probability close to one). It is assumed below that our dynamical system (5.205) is generic in the sense that the above m i n i m u m as well as all similar minima considered in the subsequent discussion are achieved only at one equilibrium state. A cycle rank I is defined as an ordered subset of v equilibrium points from L with the property that for any point i of this subset we have J~(i) - i. It is clear that the ordering on rank-1 cycles is introduced by

CHAPTER 5 Stochastic Aspects of Hysteresis

584

the transition operation Jl(i). To introduce such characteristics as the main state of a cycle, the stationary distribution rate on a cycle and the exit rate from a cycle, the concept of/-graphs [26] is very instrumental. Consider an arbitrary subset A of L and some equilibrium point i ~ A. A set of directed edges (arrows) connecting some points j ~ A is called an /-graph qi if one edge starts from any point of this graph j ~ A (j # i), and from any point of this graph there is a directed path to i along the edges of the graph. The set of all/-graphs is denoted as Qi(A). Now, the main state of rank-1 cycle C is defined as the state j" = M(C) where the following minimum is reached

2 - min min

j~C g~Qj(C)

y~

Vmn.

(5.217)

(m--+n)~q

The stationary distribution rate mc(i) of the rank-1 cycle C is defined by the formula mc(i) = min ~ Vmn -- ,~. (5.218) q~Qi(C) (m--,n)~q

Finally, the exit rate ec from the cycle C is given by the expression

ec(i) =

min [mc(i) + Vij].

i~C, j~C

(5.219)

For any rank-1 cycle C (p) there is such i* ~ C (p) and such j* belonging to the rank-1 cycle C (s) = J2(C (p)) that the above minimum is reached. In this sense, the cycle C (s) is the 'next' to the cycle C (p), and the transition C (p) --+ C (s) is the most probable (with probability close to one) among all possible transitions between C (p) and other rank-1 cycles. A cycle rank 2 is defined as an ordered subset of ~ rank-1 cycles with the property that for any rank1 cycle C (p) of this subset J~2(C(p)) = C (p). It is clear that the ordering on rank-2 cycles is introduced by the transition operator J2(C(P)). For rank-2 cycles, we can introduce main states, stationary distribution rates and exit rates by using formulas (5.217), (5.218), and (5.219), respectively. The only qualification is that C in the above-mentioned formulas is a rank-2 cycle. By literally repeating the same line of reasoning, we can introduce cycles of higher ranks until we reach some rank at which there exists only one cycle consisting of all equilibrium points. For this cycle the exit rate is defined as infinite. The described hierarchy of cycles is illustrated by Fig. 5.29 where four rank-1 cycles, two rank-2 cycles and one rank-3 cycle are depicted. Now, the important notion of observable equilibrium states of dynamical systems (5.206) can be introduced. Consider an arbitrary value g of input and some equilibrium state i that exists for that input value. For this

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

585

FIGURE 5.29

state, we can introduce the ordered sequence of cycles Ck of all possible ranks that describes the sequence of the most probable (with probability close to one) transitions initiated from i. The above sequence of cycles generates the sequence of exit rates ek given by formula (5.219). Let ej be the smallest exit rate from the above sequence such that

c < ej,

(5.220)

where c is the constant from (5.207). Let j* be the main state of the cycle Cj. Then, by using asymptotics similar to (5.214), it can be shown that a very fast (theoretically instantaneous) transition from the equilibrium state i to the equilibrium state j* is induced by the noise. This transition occurs via very fast intermediate transitions between equilibrium states that form cycles Ck with k < j. In this sense, the equilibrium state i, as well as the intermediate states participating in the transition, are not observable, while the state j* is observable. It is clear that the observability of the state is determined by the value of constant c in (5.207) and, consequently, by the interplay of fast system dynamics and the noise strength. It is also clear that for any instant of time there is a finite number of observable equilibrium states Jn (n = 1, 2 . . . . . e * ( g t ) ) . A particular observable state, occupied by the system at a given instant of time, is determined by the past history of system dynamics. Suppose that the system (5.206) is in some observable state Jn and suppose (for the sake of being specific) that the input is monotonically decreased. If this input decrease does not violate inequality (5.220) and the condition that Jn is the main state of the cycle Cj, then the state of the system will move along the ?-curve to which the observable state Jn belongs. However, as soon as the input gt reaches the

586

CHAPTER 5 StochasticAspects of Hysteresis

value at which the state Jn is no longer observable, the system is switched to another observable state Jm belonging to another ~-curve. The new observable state Jm is determined by using the same algorithm as above with only the difference that the ordered sequence of cycles Ck describes now the sequence of the most probable transitions initiated from Jn" It is this change in the initial state of transitions that leads to branching and hysteresis phenomena. Indeed, if after switching to the state Jm input is somewhat further decreased and then reversed, then for the monotonically increasing input the system (5.207) may not switch back to the observable state Jn at the same input value as the switching from Jn to Jm occurred. This is because the equilibrium state Jm may loose its 'observable' status at a different input value than happened for the state Jn" It is apparent from the above discussion that inequality (5.220) subdivides all equilibrium states (attractors) into two distinct groups: equilibrium states that participate in fast switchings that lead to deterministic hysteresis and equilibrium states that form cycles Ck with k > j. These equilibrium states participate in slow switchings that are usually known as thermal relaxations. In the previous discussion, emphasis has been given to the noiseinduced hysteresis in dynamical systems that under purely deterministic conditions do not exhibit hysteresis. However, it is clear from the presented reasoning that in dynamical systems, which under purely deterministic conditions do exhibit hysteresis, the presence of noise may substantially alter the structure of hysteresis. For instance, in one-dimensional dynamical systems (5.205), which exhibit hysteresis due to bifurcations between two existing asymptotically stable equilibrium states, the presence of noise may appreciably alter switching input values and the shape of the hysteresis loop. In dynamical systems (5.205), which exhibit hysteresis due to bifurcations between many existing asymptotically stable equilibrium states, the presence of noise may make only some of those states actually observable and, in this way, appreciably change the structure of hysteresis. It is apparent from the presented discussion that the effect of small noise on hysteresis in dynamical systems is very pronounced for fast systems. Since fast systems are typical for rate-independent hysteresis and since some small noise is always inherently present in physical systems, the study of rate-independent hysteresis in fast dynamical systems without noise may be inadequate. The presented discussion has been based on relation (5.207) between small parameters ~ and e that characterize the fast dynamics and the noise strength, respectively. The question is to what extent this relation is natural. By using results from [29], it can be demonstrated that for other relations between ~ and e deterministic hysteresis either does not exist (strong noise) or it is insensitive to noise (very weak noise). Under the condition (5.207),

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

587

hysteresis still exists in a deterministic sense and it is quite sensitive to noise. This is consistent with many experimental facts which demonstrate that deterministic hysteresis is temperature sensitive. In this sense, the condition (5.207) can be regarded as natural for fast dynamical systems employed to describe noise-sensitive deterministic hysteresis. Finally, it is important to note that usually observation Yt is a scalar or a vector whose dimension is much smaller than the dimension of the state vector Xt. For this reason, the observation of the (Yt) versus input (gt) relation gives a 'reduced' description of the dynamical system. This reduced description may often lead to the situation where many different state vectors Xt result in the same observation Yt. This, in turn, may be the origin of the nonlocal memory of the hysteretic relation between Yt and gt.

5.7 F U N C T I O N A L

(PATH) INTEGRATION

MODELS

OF

HYSTERESIS The Preisach model is designed as a continuous superposition of the simplest rectangular loop operators ~ . These operators can be construed as elementary building blocks of the Preisach model. A natural way to generalize the Preisach model is to consider more sophisticated elementary hysteresis operators and to design hysteresis models as continuous superpositions of such elementary operators. In this section we pursue this approach and consider functional (path) integration models of hysteresis that are designed as superpositions of elementary hysteresis operators generated by continuous functions. A physical interpretation of the path integration models as well as their various connections with the classical Preisach model are presented. The discussion in this section follows (to a certain extent) the paper [30]. Consider a continuous function g(x) on some closed interval [x_, x+] that satisfies the condition

u_ ~g(x_) ~g(x) Kg(x+) = u+.

(5.221)

Such a function will be called a generating function, while x_ and x+ can be termed as lower and upper saturation values, respectively. An elementary hysteresis operator ~gu(t) can be associated with each generating function by traversing its upper or lower envelopes (see Fig. 5.30). This can be done as follows. Suppose that at time to the input u(t) assumes some extremum value u0 and

uo = g(xo).

(5.222)

CHAPTER 5 StochasticAspects of Hysteresis

588

Uo

| | |

X_

X0

X+

FIGURE 5.30

If u0 is some minimum value, then for the subsequent monotonic increase of input the upper envelope g+0 (x) = max g(x) [x0,x]

(5.223)

is traversed. On the other hand, if u0 is some maximum value, then for the subsequent monotonic decrease of the input the lower envelope guo (x) = m i n g(x) [x,x0]

(5.224)

is traversed. This means that for monotonic input variations the elementary hysteresis operator ~gU(t) is defined as follows:

~gU(t) =

x (t) x-(t)

if u(t) is monotonically increased, if u(t) is monotonically decreased.

(5.225)

Here x + (t) and x-(t) are the solutions of the following equations, respectively: g+ (x + (t)) - u(t),

(5.226)

guo (x- (t)) - u(t).

(5.227)

Since upper g+ (x) and lower gu0 (x) envelopes usually have 'horizontal' parts parallel to the x-axis, solutions of Eqns (5.226) and (5.227) may not be unique for some values of u(t). This difficulty can be removed by using minimal and maximal solutions of Eqns (5.226) and (5.227), respectively: x+(t) = min{x: g+(x) - u(t)},

(5.228)

x - ( t ) - max{x: guo(X) - u(t)}.

(5.229)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

589

Ut

U2 x

U_ X_

X2

Xt

X1

FIGURE 5.31

The elementary operator ~gu(t) has been so far defined for monotonic input variations. This definition can be extended to the case of piecewise monotonic inputs by consecutively applying the definition (5.225) for each time interval of monotonic variation of u(t). The ambiguity of choosing x0 in Eqn. (5.222) can be removed if it is agreed that the evolution is started from the state x_ (or x+) of negative (or positive) saturation. The given definition of elementary hysteresis operator ~gu(t) is illustrated by Fig. 5.31. It is clear that elementary operator ~gu(t) is rate-independent. This is because the output value x(t) depends only on the current value of input u(t) and the past history of input variations but does not depend on the rate of input variations. It is also clear that the operator ~;~u(t)has local memory. This is because the simultaneous specification o t output and input uniquely defines the state of elementary hysteretic nonlinearity ~gu(t). Finally, it is clear that the operator ~gu(t) exhibits the 'wiping-out' property (see [3]). In a way, the wiping-out property can be regarded as a consequence of local memory. It is important to note that not all parts of g(x) are accessible. For instance, point A shown in Fig. 5.31 is not accessible. This part of g(x) will not be traversed for any input variations. In this sense, the same elementary hysteresis operator ~gu(t) is defined on the equivalence class of functions g(x) with the same accessible parts. Now, consider some set G of generating functions g(x) and some measure/~(g) defined on this set. Then, the functional (path) integration model of hysteresis can be formally defined as follows:

fit) -/c ~gu(t) d#(g).

(5.230)

CHAPTER 5 Stochastic Aspects of Hysteresis

590

The above model is quite general. Its structure depends on the measure #(g) introduced on the set G. Next, we demonstrate that the classical Preisach model is a particular case of the path integration model (5.230). To this end, consider the subset Gp of G that consists of functions with two vertical parts (x_ = - 1 , u ~ ) and (x+ = 1, u~fl) separated by inaccessible parts 2(x) with ] / < 2(x) < a (see Fig. 5.32). It is clear that for such functions

~;gu(t) - ~;~u(t)

for

g(x) ~ A~,

(5.231)

where A~/~is the equivalence class of functions from Gp that have the same values of a and ft. Now, consider the measure #(g) that is concentrated on the subset Gp. Since Gp is the union of nonintersecting equivalence classes A~/~, the model (5.230) can be written as follows: (5.232)

f(t)= f~>~(fA~ ~gu(t)d#~(g))d~

where #~/~(g) are the measures on the equivalence classes A~/~ induced by the measure #(g) on Gp. By using (5.231) in (5.232), we obtain

fA

~gu(t)d#~(g)=(fAd#~(g))~u(t)=#(a,fl)~u(t),

(5.233)

where the following notation is introduced (5.234)

#(~'//) = fA d/~/~(g).

By substituting (5.233) into (5.234), we end up with the classical Preisach model

- [ [ #(~, ~)~&~u(t)d~ dfl. ,Ida

(5.235)

In the case when the vertical parts in Fig. 5.32 are replaced by curved paths (see Fig. 5.33), the corresponding elementary hysteresis operator ~/gu(t)can be represented as follows:

x(t) -

1;+ (Ut) -- 1;- (Ut) ,, ~~

2

+

V+ (Ut) -}- V - (Ut)

2

'

(5.236)

where v+ (u) and v- (u) are the inverses of g+ (x) and g - (x), respectively.

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

591

u (x

-1

m,,,..

X

FIGURE 5.32

u

g+(x)

~X

/~ g-(x) FIGURE 5.33

By choosing measure ~(g) concentrated on the set of functions shown in Fig. 5.33 and literally repeating the same line of reasoning as before, it can be demonstrated that the path integration model (5.230) is reduced to the Preisach model with the input-dependent measure (see [3]). It has been demonstrated above that the path integration model (5.230) is reduced to the classical Preisach model if the measure #(g) is concentrated on functions g(x) such that ~gu(t) - ~a~u(t). Below, it will be shown that this reduction is also possible in cases when the measure #(g) is concentrated on functions g(x) such that ~gu(t) ~ ~a~u(t). This will further emphasize the generality of the classical Preisach model. In general, it is not immediately obvious how to generate the measure on the functional set G and how to carry out functional integration in (5.230), in other words, how to compute output fit). It turns out that the above difficulties can be appreciably circumvented if the set G is interpreted as a set of samples of a stochastic diffusion process, that is, the Markovian process with continuous samples generated by the Ito

CHAPTER 5 StochasticAspects of Hysteresis

592

Ul

X

U_ X_

U_

Xt

i

T X_

x

I

xt

Xl

FIGURE 5.34

stochastic differential equation

dgx = b(gx, x) dx + a(gx, x) d Wx.

(5.237)

Here x must be construed as fictitious 'stochastic' time that must not be confused with real physical time t. In the above case, the measure #(g) is the stochastic measure that, in principle, can be generated by using the transition probability density function for the process defined by Eqn. (5.237). This is the consequence of the Markovian nature of the process gx. However, there is no need of doing this because, as will be demonstrated below, the output f(t) can be interpreted as an average level-crossing (stochastic) time. As a result, the mathematical machinery developed for the solution of level-crossing (exit) problems can be extensively used for the output calculations of functional integration-type models (5.230). First, we shall discuss the meaning of ~;gu(t) when the generating function g(x) is a sample of the diffusion stochastic process. This meaning is different for monotonically increasing and monotonically decreasing inputs. To illustrate this, let us consider Fig. 5.34 where a particular sample g(x) of the diffusion stochastic process that starts from the negative saturation value u_ is shown. It is clear from this figure and the definition of ~gu(t) that for monotonically increasing input u(t) the elementary operator ~gu(t) has the meaning of the first level-crossing time, where the level is equal to the current value of input u(t). This time is a random variable

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

593

and, consequently, the output of the functional integration model (5.230) is equal to the average value of the first level-crossing time. This is true for any value of input until u(t) reaches some maximum value Ul. It is clear from Fig. 5.34 and the definition of ~gu(t) that for the subsequent monotonic decrease of input u(t) the elementary operator ~gu(t) has the meaning of the last level-crossing time, where the level is equal to the current input value u(t). More precisely, this is the last time of crossing the level u(t) before the first time of crossing the level Ul. This last level-crossing time is also a random variable. Thus, for monotonically decreasing input, the output of the functional integration model (5.230) is equal to the average value of the last level-crossing time. It is this difference in the meaning of output values of the path integration model (5.230) for monotonically increasing and decreasing inputs u(t) that results in hysteresis. Next, we shall present a mathematical formalism that supports the statements outlined above. Let T(~, x~]fl, xfl) be the notation for the probability density of the first crossing time x~ of the level a under the condition that the sample of the stochastic process crossed the level fl < a at the time xfl. It is clear that

Jxx~ T(~x, x~]fl, xfl) dx~ 1. (5.238) fl It is apparent that the probability densities p~(x~) of xa and pfl(xfl) of xfl are related by the expression

p~(x~) -

f0x~T(~x, xalfl, xfl)pfl(xfl) dxfl.

(5.239)

Consider the set G of all samples of the diffusion process gx that satisfy the condition (5.221). Let us analyze mathematically how the output of the path integration model (5.230) changes when the input ut is increased from u_ to some maximum value Ul and then is decreased to some minimum value u2. During the monotonic input increase, the elementary operator xt - ~gut has the meaning of the first time of crossing the level ut. By using the notations pfl(xfl) = pu_ (x_) and p~(x~) = Put (Xt), according to (5.239) we find

Put (Xt)

--

~

Xt

T(ut,

XtlU-,

x-)pu_ (x_) dx_.

(5.240)

The probability density function Pu_ (x_) must be chosen as a part of the characterization of the initial state of lower saturation. For instance, it can be chosen as ~(x - x_). After that, Eqn. (5.240) permits one to compute the unknown probability density Put (Xt) provided that the first-time

CHAPTER 5 StochasticAspects of Hysteresis

594

level-crossing problem has been preliminarily solved and the function T(ut, xtlu-, x_) has been found. By using the probability density Put (xt), the output value of the path integration model (5.230) can be computed as follows:

f(ut) =

f0

xtput (xt) dxt.

(5.24:1)

Formulas (5.240) and (5.241) can be used to compute Put (xt) and f(ut) for all values of ut between u_ and Ul. In this way, the ascending branch of the major loop can be computed. For ut = ul, we have

pu~(Xl) =

r(ul, x l l u - , x - ) p u _ ( x - ) d x - .

f0

(5.242)

Next, consider the monotonic decrease of input from Ul to u2. For this input variation, the elementary operator xt = ~xut has the meaning of the last time of crossing the level ut before the level Ul is reached for the first time. This means that probability density Put (xt) of the last level-crossing time xt satisfies the integral equation

Pul (Xl) --

0 xl

T(Ul, Xl Jut, xt)Put (xt) dxt.

(5.243)

This integral equation can be (in principle) solved for any value of Ut between ul and u2. In this way, Put(Xt) c a n be found and used in formula (5.241) for the calculation of the output value f(ut) along the descending branch attached to the previous ascending branch at the point ut = Ul. After Pu2(X2) is found by solving integral equation

Pul (Xl) -- ~0xl T(Ul, X1 lU2, x2)Pu2 (X2) dx2,

(5.244)

it can be used in the formulas

Put (xt) -

f0xtr(ut, xtlu2, x2)Pu2(x2) dx2,

(5.245)

PuB(x3) -

fox3r(u3, x31u2, x2)Pu2(x2) dx2

(5.246)

for the computations of Put (xt) and PuB(x3) for the third hysteresis branch when input u(t) is monotonically increased from u2 to u3. Similarly, PuB(x3) can be used in the integral equations

PuB(x3) -

f0x3r(u3, x31ut, xt)Put (xt) dxt,

(5.247)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

Pu3 (X3) --" ~0X3Z(u3, x31u4, x4)Pu4 (x4) dx4

595

(5.248)

for the computations of Put (Xt) and Pu4(X4) for the fourth branch of the hysteresis loop when input u(t) is monotonically decreased from u3 to u4. It is clear that the computations described above can be recursively used to find any branch of hysteresis described by the path integration model (5.230) with stochastic measure. It is important to note that in the case when u3 = Ul the probability densities Pu3(x3) and Pul (Xl) coincide. This directly follows from the coincidence of Eqns (5.244) and (5.246) for the above case. The coincidence of Pu3(X3) with Pu~ (Xl) for u3 - Ul implies the validity of the 'wiping-out' property for the path integration model. This fact can also be deduced from the validity of the 'wiping-out' property for each elementary hysteresis operator ~gu(t). Now, consider the particular case when the stochastic process gx is homogeneous (translationally invariant) with respect to 'stochastic time' x. Such a process is described by the Ito stochastic differential equation dgx = b(gx) dx 4- cr(gx) dWx.

(5.249)

Owing to the translational invariance, the conditional first-time levelcrossing probability density T(~, x~]fl, xfl) has the property T(~, x~lfl, xfl) = T(~, x~ - xfllfl, 0).

(5.250)

Next, we shall use this property to compute the expressions for ascending f + (ut) and descending f - ( u t ) branches of hysteresis loops formed when the input u(t) is monotonically increased from some minimum value U2k to some maximum value U2k+l and is then monotonically decreased back to U2k. For the calculation off + (ut), formulas similar to (5.241) and (5.245) are appropriate. This leads to the expression

f+ (lit) --

XtPu+t(Xt) dxt

- fo ~176 xt (fo xt T(ut, xt - X2k]U2k, O)Pu2k(X2k) dX2k ) dxt. (5.251) By using Fig. 5.35, the above double integral can be transformed as follows: f + (ut) --

xtT(ut, xt - X2klU2k, O) dxt Pu~ (X2k) dx2k

2k --

2k

(Xt -- X2k)T(ut, Xt -- X2klU2k, O) dxt Pu2k(X2k) dx2k

CHAPTER 5 StochasticAspects of Hysteresis

596

Xt

Xt = X2k

" X2k

FIGURE 5.35

-t-

T(ut, xt - X2klU2k, O) dxt X2kPu2k(X2k) dx2k. 2k

(5.252) By using the change of variables (5.253)

Z -~ Xt -- X2k,

from (5.252) we find T(ut, ZIU2k, O) dz X2kPu2k(X2k) dX2k

f + (Ut) --

q-

zT(ut, Z[U2k, O) dz Puak(X2k)dx2k.

(5.254)

By using formulas (5.238) and (5.241) in the first integral and the normalization condition

/o ~ Puak(X2k) dX2k

--

1

(5.255)

in the second integral, from (5.254) we find J:+ (Ut) -- f(U2k) q- F(ut, U2k),

(5.256)

where ~(Ut, U2k) --

zZ(ut, ZlU2k, O) dz.

~0~176

(5.257)

It is clear from (5.256) that the current value of output on the ascending branch is determined only by the current value of input ut and the last minimum value U2k. We shall next establish a similar result for the descending branchf-(ut) of the hysteresis loop. According to (5.241) and (5.247)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

597

Xt X2k+l ~ / t / /

// /

/

/

= X2k+1 FIGURE 5.36

we have f(U2k+l)

/0

X2k+lPu2k+l(X2k+l) dX2k+l \

oo (foX2k+l Z(U2k+l, X2k+l [Ut, xt)Put (Xt) dxt) dX2k+l O X2k+l fO

Z(U2k+l, X2k+l -- Xt[Ut, O)put (Xt) dxt) dX2k+l.

X2k+l

(5.258) By using Fig. 5.36, the last double integral can be transformed as follows:

f(U2k+l) --

X2k+lZ(U2k+l, X2k+l -t

Xt[Ut, O) dX2k+l Put (xt) dxt. (5.259)

By using the change of variables Z ~- X2k+l --

Xt,

(5.260)

from formula (5.259) we derive

(/0

f(U2k+l) --

)

zr(u2k+l, zlut, O) dz p~ (xt) dxt

Jr-

T(u2k+l, Z]Ut, O) dz XtPut(Xt) dxt.

(5.261)

Now, by using the same reasoning that was used to simplify formula (5.254), we obtain d(U2k+l) --

.~'(U2k+l, ut) + f-(ut),

(5.262)

CHAPTER 5 StochasticAspects of Hysteresis

598

which leads to

f - (Ut) - -

f(U2k+l)

-- .T(Uak+l,

Ut).

(5.263)

Formulas (5.256) and (5.263) show that the shapes of generic ascending and descending branches of a minor hysteresis loop are the same regardless of the past input history. The past history is reflected in the values of f(u2k) and f(U2k+l ). In other words, the minor hysteresis loops corresponding to different past histories are congruent. Since the wiping-out property and the congruency of minor loops (formed for the same back-and-forth input variations) represent the necessary and sufficient conditions for the description of hysteresis by the Preisach model, we conclude that the path integration model (5.230) with the stochastic measure corresponding to the homogeneous diffusion process homogeneous diffusion process is equivalent to the Preisach model. It is clear from the previous discussion that the output calculations for the path integration model can be performed if the function T(~, x~l//, x/~) is known. This function can be computed by solving the exit problem for the stochastic process defined by Eqn. (5.237). This, in turn, requires the solution of initial boundary value problems for the backward Kolmogorov equation similar to those discussed in Section 5.3. In particular, by using the mathematical machinery of the exit problem, the closed-form expressions can be derived for the weight function #(~, fl) of the Preisach model, which is equivalent to the path integration model (5.230) with the stochastic measure generated by the process (5.249). Below, we present the final results; the mathematical details of the derivation can be found in the paper [30]. Consider the function

(/0=

0(g) = exp - 2

o.2(g,)

d=)

(5.264)

then the function Y'(a, ]/) from (5.257) and #(~, ]/) can be computed as follows: Y'(~, fl) - ~

0(g') dg'

)(/=" 0(g') dg' ) r~2(g)0(g d= ) ,

(5.265) #(~x, fl)= ~(~x)~,(fl) Y'(a,/~),

(5.266)

where K~/~ =

~(g) dg.

(5.267)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

599

8

.0

/

-4

-8 -4

I 2

0

-2

FIGURE

L

5.37

As examples, consider the following cases: (a) g x is the Wiener process (b - 0, r~ - 1). Then 6(g) - 1, Karl - o ~ - fl and 1 1 #(~, fl) = 3r(~, fl) = ~ ( ~ _ fl)2. (5.268) o

Thus, the Preisach weight function is simply a constant and all hysteresis branches are parabolic (see Fig. 5.37). (b) g x is the diffusion process with constant drift [b(g) - 2~ (Z > 0), r~ = 1]. In this case we have 6(g) - e-g/x,

(5.269)

Karl - e - f l / z - e - a / z ,

(5.270)

a-flcoth(~xfl ) - 1 fl(0r

]~) - -

2Z

(5.271)

,

sinh2 (~zfl) 3r(~,fl)-4[(

~fl)coth(

a -2Z fl)-11

.

(5.272)

Typical hysteresis branches computed for this case are shown in Fig. 5.38. (c) g x is the Ornstein-Uhlenbeck process (b(g) - - ~, cr - 1). In this case, ~(g) - e g 2 / Z , (5.273)

CHAPTER 5 Stochastic Aspects of Hysteresis

600

_

0

-4-

-8

I

-4

i

-2

'

0

I

2

U

FIGURE

5.38

and #(a, fl) and Z(a, fl) are obtained by inserting the last expression in the formulas (5.265)-(5.267). The described functional (path) integration model (5.230) admits the following physical interpretation. It is known that hysteresis is due to the existence of multiple metastable states in the system free energy F(X) (the temperature dependence is tacitly understood), which means that the system may be trapped in individual metastable states for long times. Consider a simple case where the state variable X is a scalar quantity and the relevant free energy in the presence of the external magnetic field H is ~(X; H) = F(X) - HX. The metastable states available to the system are represented by ~-minima with respect to X for which O~/OX = 0, 0~/r 2 > 0. When H is changed with time, the number and the properties of these minima are modified by the variation of the term - H X . The consequence is that previously stable states are made unstable by the field action and the system moves to other metastable states through a sequence of (Barkhausen) jumps. Because the condition 3~/OX = 0 is equivalent to H = OF/OX, one can analyze the problem by using the field representation shown in Fig. 5.39. The response of the system, expressed in terms of H(X), is obtained by traversing the upper and lower envelopes of OF/OX for increasing and decreasing H, respectively. From the physical viewpoint, this construction amounts to assuming that the system, once made unstable by the action of the external field, jumps to the nearest available energy minimum, which means that one excludes dynamic effects that could aid the system to reach more distant minima. It is clear from the above description

REFERENCES

601

FIGURE 5.39

that OF/OX and H are similar to the generating functional g(x) and input u(t), respectively, within the framework of the function (path) integration model (5.230). The functional integration model itself can be interpreted as the average hysteresis response of a statistical ensemble of independent (elementary) systems evolving in a random free-energy landscape. This interpretation can be of importance in applications where randomness due to structural disorders plays a key role in the appearance of hysteretic effects. A particularly important example is the motion of magnetic domain walls in ferromagnets, where various forms of structural disorder (point defects, dislocations, gain boundaries, etc.) are responsible for the random character of OF/OX. There are classical papers in the literature [31,32] where the domain wall picture has been applied to the prediction of coercivity and magnetization curve shapes, starting from some assumption about the properties of F(X). Equations (5.264)-(5.267) provide a general solution for the case where the process OF/OXis Markovian, continuous, and homogeneous. In particular, the proven equivalence of Markovian disorder to the Preisach model gives a sound statistical interpretation of the latter.

References 1. S. H. Charap, Magnetic viscosity in recording media, J. App1. Phys. 63(6), 2054-2057 (1988). 2. R. Street and J. C. Wooley, A study of magnetic viscosity, Proc. Phys. Soc. London A 62(357), 562-572 (1949). 3. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, Berlin (1991).

602

CHAPTER 5 StochasticAspects of Hysteresis

4. I. D. Mayergoyz and C. E. Korman, Preisach model with stochastic input as a model for viscosity, J. Appl. Phys. 69(4), 2128-2134 (1991). 5. I.D. Mayergoyz and C. E. Korman, On a new approach to the modeling of viscosity in hysteretic systems, IEEE Trans. Mag. 27(6), 4766-4768 Part 2 (1991). 6. I. D. Mayergoyz and C. E. Korman, The input dependent preisach model with stochastic input as a model for aftereffect, IEEE Trans. Mag. 30, 4368-4370 Part I (1994). 7. V. M. Tobin, S. Shultz, C. H. Chan, and S. B. Oseroff, Magnetization time decay in CrO2 tape, IEEE Trans. Mag. 24(6), 2880-2882 (1984). 8. J.A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and minor loops-models and experiments, Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences 386(1791), 251-261 (1983). 9. G. Friedman and I. D. Mayergoyz, Stoner-wholfarth hysteresis model with stochastic input as a model of viscosity in magnetic-materials, IEEE Trans. Mag. 28(5), 2262-2264 Part 2 (1992). 10. C. E. Korman and P. Rugkwamsook, Identification of magnetic aftereffect model parameters: Comparison of experiment and simulations, IEEE Trans. Mag. 33(5), 4176-4178 Part 2 (1997). 11. I.D. Mayergoyz, A. A. Adly, C. Korman, M. W. Huang, and C. Krafft, Scaling and data collapse in magnetic viscosity, J. App1. Phys. 85(8), 4358-4360 Part 2A (1999). 12. N. Goldenfeld, Lectures on Phase Transmitions and the Renormalization Group, Addison-Wesley, Reading, MA (1992). 13. I. D. Mayergoyz, C. Serpico, C. Krafft, and C. Tse, Magnetic imaging on a spin-stand, J. Appl. Phys. 87(9), 6824-6826 Part 3 (2000). 14. I. D. Mayergoyz, C. Tse, C. Krafft, and R. D. Gomez, Numerical technique for integration of the Landau-Lifshitz equation, J. Appl. Phys. 89(11), 6991-6993 Part 2 (2001). 15. C. Tse, D. I. Mircea, I. D. Mayergoyz, P. Andrei, and C. Krafft, Spatial and vectorial characterization of thermal relaxation using the spinstand imaging technique, J. Appl. Phys. 91(10), 8846-8848 Part 3 (2002). 16. I.D. Mayergoyz and C. E. Korman, The Preisach model with stochastic input as a model for aftereffect, J. Appl. Phys. 75(10), 5478-5480 Part 2A (1994). 17. C.W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, Berlin (1983).

REFERENCES

603

18. C.E. Korman and I. D. Mayergoyz, Preisach model driven by stochastic inputs as a model for aftereffect, IEEE Trans. Mag. 32(5), 4204-4209 Part I (1996). 19. C. E. Korman and I. D. Mayergoyz, Review of Preisach type models driven by stochastic inputs as a model for after-effect, Physica B 233(4), 381-389 (1997). 20. M. I. Freidlin and I. D. Mayergoyz, Rotationally symmetric solutions of the Landau-Lifshitz and diffusion equations, J. App1. Phys. 87(9), 5511-5513 Part 2 (2000). 21. H. Cramer and M. Leadbetter, Stationary and Related Stochastic Processes, Wiley, New York (1967). 22. M. I. Freidlin and A. D. Wentzell, Diffusion-processes on graphs and the averaging principle, Ann. Prob. 21(4), 2215-2245 (1993). 23. M. I. Freidlin, Markov Processes and Differential Equations: Asymptotic Problems, Birkh~iuser-Berlin, Berlin (1996). 24. M. I. Freidlin, I. D. Mayergoyz, and R. Pfeiffer, Noise in hysteretic systems and stochastic processes on graphs, Physical Review E 62(2), 1850-1855 Part A (2000). 25. L. Gammaitoni, P. H~inggi, P. Jung, and F. Marchesone, Stochastic resonance, Review of Modern Physics 70(1), 223-287 (1998). 26. I.D. Mayergoyz and M. Dimian, Analysis of spectral noise density of hysteretic systems driven by stochastic processes, J. Appl. Phys. 93(10), 6826-6828 (2003). 27. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin (1998). 28. M.I. Freidlin and I. D. Mayergoyz, Hysteresis in randomly perturbed fast dynamical systems, Physica B 306(1-4), 15-20 (2001). 29. M. I. Freidlin and I. D. Mayergoyz, Modeling of temperaturedependent hysteresis within the framework of dynamical systems, J. App1. Phys. 91(10), 7640-7642 Part 2 (2002). 30. M.I. Freidlin, On stochastic perturbations of dynamical systems with fast and slow components, Stochastic and Dynamics 1(2), 261-281 (2001). 31. G. Bertotti, I. D. Mayergoyz, V. Basso, and A. Magni, Functional integration approach to hysteresis, Physical Review E 60(2), 1428-1440 Part A (1999). 32. L. Ne61, Theories des lois d'aimantation de Lord Rayleigh, Cah. Phys. 12, 1-20 (1942). 33. H. Kronmuller, in Magnetic Hysteresis in Novel Magnetic Materials, Kluwer, Dordrecht (1997).

CHAPTER 6

Control of Systems with Hysteresis Stavros A. Belbas, Mathematics Department, University of Alabama, Tuscaloosa, AL 35487-0350, USA

Contents 6.1 6.2

Introduction Optimal Control of Systems with Simple Hysteretic Elements 6.3 Optimal Control of Systems with Hysteron-Type Operators 6.4 Optimal Control of Systems with Preisach Hysteresis 6.5 Hysteresis from Optimal Control, and Hysteresis for Optimal Control References

605 607 614 622 632 634

6.1 I N T R O D U C T I O N Classical control theory deals with systems described by ordinary differential equations that involve a control function. The problems of classical control theory are concerned with obtaining specified performance of the solution by choosing an appropriate control function, or by choosing a control strategy in the form of a feedback operator, or some sort of decision rule based upon total or partial observations of the current state of the system, as well as on other types of information. It has been understood that realistic systems cannot always be described by smooth and memoryless models, like systems of ordinary differential equations, but they can, and in many cases must, include operators with discontinuities, discrete control actions coupled with continuously evolving state and continuous controls, and memory terms like Volterra integral terms or general causal functional terms. It is an ongoing challenge to extend the results and methods of smooth and memoryless control systems to systems with discontinuities and memory. The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

606

CHAPTER 6 Control of Systems with Hysteresis

Among such systems, hysteresis plays a prominent role, because of its ubiquitous appearance in many applications, including modern actuator and sensor technology, hysteretic modeling of certain mechanical systems like systems with friction or systems with rheological properties that exhibit hysteresis, in environmental problems like groundwater motion and contaminant transport with hysteresis in the relationships between quantities of hydraulic significance during wetting and de-wetting of a porous medium, hysteresis in biological systems, in economic systems, etc. For applications to controlled ODEs, with hysteresis modeling of mechanical systems, see [1]. For rheological models with hysteresis, see, e.g., [2]. For applications to groundwater models, see, e.g., [3-5]. For applications to the modeling of actuators in elastic systems, see [6]. For applications to 'smart materials', see [7,8]. For identification problems (which are closely related to control problems), see [9,10]. Hysteresis exhibits these characteristics in the context of optimal control theory: it involves discontinuities and non-smooth nonlinearities; it has memory effects (in the case of Preisach hysteresis and its extensions and generalizations); and it involves set-valued controlled systems (for Preisach and related operators), since the evolution of a hysteretic nonlinearity depends on the evolution of the active set (relays that are in the 'on' position) on the Preisach plane, and consequently the study of optimal control problems for such systems involves set-valued analysis. Just like most branches of mathematics and the exact sciences, the study of control systems with hysteresis nonlinearities has a prehistory and a history. The prehistory is not easy to trace, but we can mention, for example, the classical older works on nonlinear controlled systems, that typically include some techniques and results for the analysis of systems with hysteretic relays. A typical representative of such works is the classical book of Popov [11] on nonlinear control systems. The history of control of systems with hysteresis is, in our opinion, easy to trace. We take as a starting point the work of Krasnosel'skii and Pokrovskii, which is described in their monograph [12], but which was also available, in journal articles, before the publication of [12]. We consider as 'history' of the control of hysteresis the works that appeared after the work of Krasnosel'skii and Pokrovskii, which make use of results, concepts and methods that are either available in [12] and related works or were developed in works following the impetus of [12], and also make use of the essential ideas of optimal control theory, namely, the extremal principle (usually called 'maximum principle') of Pontryagin, Boltyanskii, Mishchenko, and Gamkrelidze and Bellman's method of dynamic programming for systems of ordinary differential equations with hysteresis nonlinearities. With these stipulations, it is easy to pin-

607

6.2 OPTIMALCONTROL OF SYSTEMS

point the beginning of the history of the control of hysteresis, it is the work of Brokate [13] on control of ODE systems with hysteresis nonlinearities. Naturally, there is a 'gray area' in the transition from prehistory to history, but any further historical inquiry would be beyond the scope of this introduction. Subsequent contributions to the optimal control of systems of ordinary differential equations with hysteresis, and their discrete-time analogs, include the works [3,14-22]. In particular, [14] addresses the question of existence of optimal controls for systems with hysteresis. A number of other authors have worked on particular cases of systems with hysteresis. The mathematics of hysteresis is covered mostly in [12], [23] and its precursor [24], and [25-27]. Additional, state-of-theart information can be found in [28,29]. Early works include [30-34]. The references given in this work are not presented as an exhaustive or nearly exhaustive bibliography on control of hysteresis; rather, they represent the works (among those works that are known and available to us) that seem to us relevant to the specific issues that we present in this report. The type of optimal control problems that arise in systems with hysteresis nonlinearities belong to the category of systems with both continuous state and discrete changes in system behavior, i.e. to the general category of hybrid and discontinuous systems [35-37]. However, the specific properties of hysteresis nonlinearities are such that previously existing methods are not applicable, and a direct study of such systems is necessary.

6.2 O P T I M A L C O N T R O L O F S Y S T E M S HYSTERETIC ELEMENTS

WITH

SIMPLE

As is well known, the standard Preisach model of hysteresis involves a weighted superposition of a family of rectangular relays. Every one of these rectangular relays is a 'primitive' element in that it cannot be further decomposed into simpler elements. We use the term 'simple hysteretic elements' to refer to operators that cannot be reduced to simpler terms and, at the same, have been traditionally labeled as 'hysteresis' in the optimal control literature. 6.2.1 CONTROL OF A SINGLE RELAY

First, we consider a simple rectangular element R ~/~, an operator that depends on two real-valued parameters ~ and fl that satisfy a > ft. The effect of R a~ on a continuous input signal z(.) is defined by the conditions:

9 (R~flz)(t) is continuous at points t where ~ ~- z(t) ~ fl;

CHAPTER 6 Control of Systems with Hysteresis

608

9 (Ra~z)(t) takes the values 0 or 1, and is a right-continuous function 9

9

that has a limit from the left at every point t; if, at some time-instant t, (R~z)(t -) - 0andz(t) = ~, then (R~z)(t) (R~z)(t +) - 1; if, at some time-instant t, (Ra~z)(t -) - I andz(t) - t, then (Ra~z)(t) -

(Ra~z)(t +) - O. Naturally, the definition of (R~z)(t) requires an initial value, 0 or 1, at some initial time to in the case z(to) ~ (fl, ~). Therefore, (R~z)(t) is really (R~z)(t; to, v0), defined for t>~to, with (R~z)(to; to, vo) = vo. Unless we specifically w a n t to emphasize this dependence of (Ra~z)(t) on some initial value, we shall suppress the variables to and v0, and we shall write simply

(Ra~z)(t). This operator R ~/~ is usually called a non-ideal relay. A different but equivalent definition is given in [12], where also several properties of nonideal relays are established. We consider a controlled ordinary differential equation that includes a simple hysteresis operator acting on the control: dx(t)

dt

=fit, x(t), u(t), (R~u)(t)); x(to) = xo.

(6.1)

The control u is one-dimensional, and it is postulated to be a continuous function. We consider the problem of minimizing a cost functional of the form: J := fo(x(T)) 4-

f;

F(t, x(t), u(t), (R~u)(t)) dt.

(6.2)

We seek the appropriate set-up for obtaining dynamic p r o g r a m m i n g equations for this model. Clearly, the quantity (R~u)(t) takes values in the set {0, 1}, and the admissible values of u(t) that are consistent with the values of (R~u)(t) are: if (R~u)(t) = 0, then u(t) ~ ( - ~ , a); if (Ra~u)(t) = 1, then u(t) ~ (fl, +oo).

(6.3)

We set U0 := ( - ~ , ~), U1 " - (fl, 4-(X)). We parameterize the system (6.3) by initial time, initial state, and initial value of R~u; thus, we consider the parameterized system:

3x(t, s, ~, v) =fit, x(t, s, ~, v), u(t), (Ra~u)(t)), for s < tKT; Ot x(s +, s, ~, v) - ~; (R~u)(s) - v ~ {0, 1}; u(s) ~ Uv.

(6.4)

6.2 OPTIMAL CONTROL OF SYSTEMS

609

We define the parameterized cost functional

Is T F(t, x(t, s, ~, v), u(t), ( R ~ u ) ( t ) ) dt.

J(s, ~, v" u(.)) " - Fo(x(T, s, ~, v)) +

(6.5) subject to the same initial conditions at time s as in (6.4). The HamiltonJacobi functions (value functions) are denoted by Wv, v E {0, 1} and they are defined by Wv(s, ~) = inf l(s, ~, v; u(.)). (6.6) u(.):u(s)~uv Then a formal application of the principles of dynamic programming yields the following equations for Wv: min

inf ~Uv

C~Wv(S,~) + ~s

f(s, ~, a, v) + f(s, ~, a, v) ,

~

W v ( S - ~ ~) -- W(vq_l)mod2(Sq-~

r

-- 0;

W v ( Z , ~) -- Fo(r

v E

{0~ 1}. (6.7)

The symbol (v + 1)mod 2 refers to arithmetic modulo 2, i.e. (2k)mod 2 = 0, (2k + 1)mod 2 = 1. Another approach is possible. If the controls that are used for the derivation of the dynamic programming equations are taken as absolutely continuous functions, then we can use the current value of the control as part of the state, and the derivative of the control as a new control variable; thus the state dynamics equation (6.4) is augmented with u(t) -- u(t; s, a, b); cgu(t; s, a, b)

0t

= p(t); u(s; s, a, b) - a; p(s) = b.

(6.8)

The corresponding Hamilton-Jacobi functions will now depend on three variables (in addition to the index v): Wv = Wv(s, ~, a). Again, the principle of dynamic programming gives the following equations: inf / c~Wv(s, ~, a) cgWv(s, ~, a) ~gWv(s, ~, a) b b . -~s + c9~ f(s, ~, a, v) + cga +F(s, ~, a, v) } - 0; a ~ Uv, v - O, 1; W0(s-, ~, a-) = Wl(S +, ~, a); Wl(S-, ~,/1+) = W0(s +, ~, fl).

(6.9)

610

CHAPTER 6 Control of Systems with Hysteresis

6.2.2 A VARIANT OF A SINGLE RELAY

Thus far, we have used a relay that operates on the principle that a switching may occur when the input reaches the boundary of a certain set, namely, the set (-cx~, ~), or the set (fl, +cx~). This can be formalized by defining the access time, subsequent to time t, of the signal's u (.) reaching the boundary of an open set I, say r(t, u,/), by r(t, u,/) "= sup{s 9s > t, and u(s1) E IVs1 E (t, s)}.

(6.10)

Clearly, the definition (6.10) is non-trivial only when u(t) ~ I. According to the definition of the operator R ~/~, if, at some time t, u(t) < a and (R~u)(t) = 0, then a switching in the value of Ra~u will occur, for the first time after time t, at the time r(t, u, ( - ~ , a)), and, similarly, if u(t) > fl and (R~u)(t) = 1, then the first time, after t, that Ra~u switches values is the instant ~(t, u, (fl, + ~ ) ) . There is, however, another related but different operator, which we shall denote by Qa/~. This operator is based on the exit times r~(t, u, I), defined by rift, u, I) := inf{s : s > t and u(s) r I}. (6.11) In a way analogous to the case of access times, the definition of the exit times is also non-trivial only when u(t) ~ I. The operator Q~/~ is defined in a way analogous to R aB, by substituting exit times in lieu of access times; thus, if, at some time t, u(t) < a and (Q~u)(t) = 0, then a switching in the value of Qa~u will occur, for the first time after time t, at the time rift, u, (-cx~, a]), and also, if u(t) > ]/and (Q~u)(t) = 1, then the first time, after t, that Qa~u switches values is at the exit time r~(t, u, [fl, +cx~)). We consider the same system dynamics as (6.1) and the same cost functional as in (6.2), but with Qa]~ instead of R a]~. Then the dynamic programming equations become quite different from those that correspond to the relay R ~/~. In this case, it seems appropriate to increase the regularity of the control functions that are used in the derivation of the dynamic programming equations, and to require those control functions to be twice continuously differentiable. Thus, the controlled system becomes dx(t)

dt

= fit, x(t), u(t), (Q~u)(t))~

du(t) = p(t); dt dp(t) = q(t). dt

(6.12)

6.2 OPTIMAL CONTROL OF SYSTEMS

611

The Hamilton-Jacobi functions Wv are now parameterized by s, ~, a, b, which have the same meaning as before, except that now b, representing the value of the state component p(t) at t = s, is part of the variables inside Wv. The possible values of q(s) will be denoted by c. For definiteness we shall assume that p(.) takes values in an open interval Ip, and q(.) takes values in a closed interval Iq. According to the definition of Q~fl, a switching in the values of Q~flu takes place if (i) u(t) - ~ and p(t) > 0, or (ii) u(t) = fl and p(t) < 0. The dynamic programming principle now gives inf

+

c~Wv(s, ~, a, b) 4- cgWv(s, ~, a, b).f(s, ~, a, v) cgWv(s, ~, a, b) I c~b q + F(s, ~, a, v) J - 0, for a ~ Uv, b ~ Ip"

Wv(T, ~, a, b) = F0(~); W0(s-, ~, ~-, b) = Wl(S +, ~, a, b) for b > 0; Wl(S-, ~, fl+, b) = W0(s +, ~, fl, b) for b < 0.

(6.13)

Two remarks are in order about the system (6.13). First, at points where ~Wv(S,~,a,b) # 0, the dynamic programming equation gives optimal controls cgb of 'bang-bang' type, i.e. a minimizer of the Hamiltonian (the expression in curly brackets in the first equation in (6.13)) takes only the extreme possible values of q, the maximum value (say, qmax) if ~Wv(S,~,a,b) < 0, and cgb the minimum value (say, qmin) if OWv(s,~,a,b) ~b > 0. Second, a slight change in the definition of the relay, from switching at access times to switching at exit times, increases the dimensionality of the dynamic programming equations by inserting the additional variable b in the Hamilton-Jacobi function. 6.2.3 MULTIDIMENSIONAL SYSTEMS

The above treatment can be extended to the case of systems with multidimensional state and multidimensional control, with simple hysteretic elements acting on 'observation states' (to be defined below), which are functions of both the original state and the control. The resulting dynamic programming equations are based on an extended state that includes the original state, the original control, and certain derivatives of the 'observation state'. We consider the following system of controlled ordinary differential equations: dx(t) dt =fit, x(t), u(t), (Q~y)(t), (R~z)(t));

CHAPTER 6 Control of Systems with Hysteresis

612

y(t) = g(t, x(t), u(t)); z(t) = h(t, x(t), u(t)); x(to) = x0; x(t) = (x i(t) 9 l ~ i ~ n ) , f ( t ,

x0 = (x~ 9 l ~ i ~ n ) ; u 1

x, u, Q, R)

(u i .

= ~ i (t, x , u ,

Q, R) 9 l K i K n ) ;

l~i~m);

y(t) = (yJ(t) " I K j K M ) ; g(t, x, u) = (gJ(t, x, u) " I<~jKM): = (aJ'IKj~M):

1KjKM); aJ > ]~u

fl-(/~"

Q~fl y _ (Qd ~ yj . 1 Kj ~ M); z(t) - (zJ(t) 9 1KjKN): h(t, x, u) = (hJ(t, x, u) 9 I ~ j ~ N ) ;

= (?J" 1KjKN); ~ - ( ~ J " I~jKN); ~ > 6JYj; (6.14)

R ~ 6 z - (R~JYz 9 I K j K N ) . We set

p(t) "=

cgu(t) 0t

We call the functions y(.) and z(.) the observation states, by analogy with a similar situation in the theory of partially observable systems, even though, in the present context, they do not have the meaning of actual observations. We shall need the time-derivatives of y and z; these are easily calculated

as c~yJ( t ) cgt

cgt

cggJ + ~cggJ - c f k (t x, u Q , R ) + ~T pk(t) UU ~ dX,~-

cgzJ(t) c~hJ cghJ k (~hJ ~pk cgt = cgt + -~xkf ( t ' x ' u ' Q ' R ) + -~uk

~ ( t , x, u, p Q, R) ;

(t) =_ pJ (t, x, u, p, Q, R).

(6.15) In (6.15) above, we use strictly the tensor convention of summation over repeated upper and lower indices, and we have suppressed some of the variables to avoid messy-looking formulas. The functional that is to be minimized is given by l " - Fo(x(T)) + ~0 T F(t, x(t), u(t), (Q~fly)(t), (R?6z)(t)) dt.

(6.16)

We shall use two multi-indices v = ( v J ' l ~ j K M ) ~ {0, 1}M, w - ( w J : l ~ j ~ N ) ~ {0, 1}N; the symbols v and w represent the possible values of (QC~fly)(t),

6.2 OPTIMAL CONTROL OF SYSTEMS

613

and (R~&z)(t). The symbol (1)j will denote the M- or N-dimensional array (M or N, according to the context) with the jth entry equal to I and all other entries equal to 0. The Hamilton-Jacobi function contains the variables s, ~, r/, f, a, b, which represent the values of t, x(t), y(t), z(t), u, p, respectively, and is also indexed by two subscripts v and w, which represent the values of (Q~fly)(s), (R~flz) (s), respectively. We set

M Nx (Uvi

Uvw "= x

j--lj'=l

x Uj).

(6.17)

/,/int,~int

The symbols signify limits of a quantity, depending on r/', f', as 01', f') --* (r/, f) while (r/', f') ~ Uvw. For 01, f) ~ c~Uvw, we shall d e n o t e b y BvwO1, f) the set of those values of b = p(s) that guarantee exit of y; from Uvj, provided yJ (s) ~ CgUvj; those values of b are characterized by the condition pJ > 0 if vJ = 0 and vJ < 0 if vJ = 1; it is clear that Bvw(rl, f) also depends on all the other variables that appear inside Wvw, but we do not explicitly show that dependence. Finally, we shall denote addition modulo 2 by the symbol @2. The dynamic p r o g r a m m i n g equations are:

~,~f J cgWvw(S, ~, r/, f, a, b) 4c' / c~s

+

4-

~Wvw(S, ~

""

.)

(~Wvw(S, .~.... ).fi (s, ~, a, v, w)

q)J(s,~,a,b,v,w)+

~Wvw(S, ~

""

.)

pJ(s,~,a,b,v,w)

OWvw(S, ~.... ) bk -Jr-OWvw(S, ~.... ) ck 4- F(s, ~ a v, w) } - 0 Oak

cgbk

' ,

for (r/, f) ~ Uvw;

Wvw(S-, ~, //int ~int, a, b) - Wv, w,(S +, ~, 11, f, a, b) if (r/, f) ~ cgUvw and b ~ Bvw(r !, f),

where v " - v •a (j ~ 2 (1)j) , w' - w Oa (j (~E)2 (1)j) :rlJE((gUvw)J :zJE(~Uvw)J Wvw(T, ~, ri, f, a, b) - Fo(~); ri - g(s, ~, a), f - h(s, ~, a).

(6.18)

614

CHAPTER 6 Control of Systems with Hysteresis

We note that certain simplifications are possible if the operator Q ~ is omitted from the state dynamics, or if the hysteresis operators depend only on the state x but not on the controls u. The problems we have examined up to this point are related to the' exit time problems' of optimal control; however, the problems above are different from exit time problems in a substantial way. In exit time problems, the exit event depends only on the behavior of the state and its time-derivative; in the problems we have examined up to this point in this section, the exit events depend on the choice of the controls, i.e. it is a part of the control policy to decide whether a switching (exit from a set Uvw) will take place. The papers [15-17] deal with optimal control problems that are closer to the standard exit time problems of classical optimal control. In those problems, the relays depend only on the state, but not explicitly on the control.

6.3 O P T I M A L C O N T R O L O F S Y S T E M S W I T H HYSTERON-TYPE OPERATORS For the purposes of this exposition, we use the non-standard terminology 'hysteron-type operators' to denote a class of hysteresis operators that includes the so-called 'stop' and 'play' operators, and are akin to the 'hysteron' nonlinearity of [12]. This class of simple hysteresis operators can be defined in a general way as described below, with possible additional conditions for specific hysteresis models. 6.3.1 A CASE OF MARKOVIAN HYSTERESIS

We consider one-dimensional input signals, say z(t), OKtKT, and onedimensional output, say w(t), O~t~T. On the zw-plane, we consider an open domain f2 whose boundary, 0f2, consists of two connected nonintersecting arcs, which we conventionally denote by (c~), (Co). The open set f2 may be bounded or unbounded. There are two families of curves, parameterized by a one-dimensional parameter p, say (c~), (c~), 0 < p < 1, with the property that, for every point P in f2, there exists a unique + p+(P) ~ (0, 1) and a unique p-(P) ~ (0, 1), such that both c u r v e s (Cp+(p)) and (cp_(p)) contain the point P. Each of the curves (c~), (Co), (c~), (Cp), can be described as w = g(z) for some continuous function g; g depends on p and on the superscript + or - , thus we have the functions g~, go, g+, gp" The effect of the hysteresis operator is that, as long as the input signal z(t) takes values in the projection of h onto the z-axis, the response w(t)

6.3 OPTIMAL CONTROL OF SYSTEMS

615

is such that the point (z(t), w(t)) moves on a curve with a superscript ' + ' dz(t) < 0. However, if _~(t d t_) > 0, and on a curve with a superscript ' - ' if --a/this description is not limited to differentiable inputs; if z(.) is piecewise monotone, then (z(t), w(t)) moves on a curve with superscript ' + ' when z is non-decreasing, and on a curve with superscript ' - ' when z(.) is nonincreasing. The definition of the associated hysteresis operator clearly also depends on some initial conditions, as was the case for the relays of Section 6.2. Sometimes, a hysteresis operator of the 'hysteron- type' class is extended, by a substitution operator, to the exterior of the domain f~, i.e. w(t) = h(t, z(t)), for some function h, when z(t) falls outside the projection of h onto the z-axis; such extensions are described in [38], and also have been used in [19 ]. A hysteresis operator of 'hysteron type', as described above, is Markovian in the sense that knowledge of the current value of input and output, and knowledge of the time-derivative of the input, implies knowledge of the forward time-derivative of the output. Therefore, the associated optimal control problems fall within the scope of dynamic programming methods. Here, we shall outline the dynamic programming equations for a controlled system with a hysteron-type operator acting on the control. We shall follow the ideas of [19] but we shall present the optimality conditions for a continuous-time problem, whereas the problem in [19] was a discrete-time problem. We consider a controlled ordinary differential system dx(t) = f i t , x(t), u(t), (Hu)(t)); x(O) = xo dt

(6.19)

where the state x(.) is n-dimensional and the control u(.) is one-dimensional (i.e. a real-valued function). The objective is to minimize a cost functional J of the form

J "- Fo(x(T)) 4- ~0 T F(t, x(t), u(t), (Hu)(t)) dt.

(6.20)

The Hamilton-Jacobi function depends on four variables: s, the current time; ~ = x(s), the current state; a = u(s), the current value of the control; and b = (Hu)(s), the current value of the hysteretic function of the control. The formulation of the dynamic programming equations is facilitated by introducing some assumptions and notation. We assume that each arc (c~) (i.e. the closure of (c~))intersects both curves (c~-) and (Co) transversely. We denote by (Z#io, W#io) the points of intersection of (c~) with

CHAPTER 6 Control of Systems with Hysteresis

616

(c~). We denote by (cpi0) the following sets: (c~0) "= {(z, w) ~ h ' ( z , w) ~ (c~)or [z>~Z-~oand (z, w) ~ (c~)]}; (c~-0) "= {(z, w) ~ h ' ( z , w) ~ (c;) or [z<.Z-~o and (z, w) ~ (Co)]}.(6.21) For consistency of notation, we set

Co0)-= Co)

(6.22)

We extend the functions g~,_ 0 < p < I to functions gp~0_defined on (c~0)ras follows: +

for z < Zoo +r for z)Zpo '

g,+o(z)= { _

gp0 (z) =

{

g

go (z)

I,

for z > z~0 for zKZ-~o "

(6.23)

For consistent notation, we also set g~0 - g ~

(6.24)

With this notation, the dynamic programming equations are:

~W(s, 3s

a,

b)

inf [V~W(s , ~, a, b)f(s , ~, a', g++(a,b),O(a')) + min /|a':a'>~a

-t-F(s, ~, a', gp+(a,b),O(a'))], inf [V~W(s, ~, a, b)f(s, ~, a", gp_(a,b),O(a"))

a" :a" ~
+F(s, ~, a', gp_(a,b),O(a'))l

W(O, ~, a, b) = Fo(s, ~).

}= 0 (6.25)

The first equation in (6.25) holds for (a, b) ~ ~. The two infima in (6.25) are taken over a', a" in the projection of f~ on the z-axis and also satisfying the indicated inequalities a'>~a, a"~
6.3 OPTIMAL CONTROL OF SYSTEMS

617

6.3.2 BROKATE'S OPTIMAL CONTROL PROBLEM AND ITS CONNECTION TO THE METHOD OF DYNAMIC PROGRAMMING

Brokate [13,39] initiated the study of optimal control problems of ODE systems with hysteresis nonlinearities by considering a system with the hysteron-type operator acting only on the state. Brokate's model fits into the framework of the problems introduced above in Section 6.3.1. The hysteresis operator is defined within a set h on a zw-plane, if z denotes the input signal and w the output of the hysteresis operator. Instead of parameterizing the interior curves (c~) by a real parameter p, as we have done in Section 6.3.1, Brokate describes those curves as families of solutions of two ordinary differential equations. Now, this is just a matter of the manner of description of those curves. In principle, given a family of curves (Cp) described by w - ~p(z, p) (6.26) we can calculate the derivative of w with respect to z along the curves (Cp) as

dw d z (Cp)

=

0~p(z, p) Oz

(6.27)

and then eliminate p between (6.26) and (6.27) to come up with the differential equation dw - ~(z, w). (6.28) dz Conversely, the general solution of a differential equation like (6.28), under standard conditions on the function 6, is a one-parameter family of curves. We shall describe a model that is, in some respects, more general than Brokate's model, but in some other respects less general, and we shall do so in a way consistent with our notation. An open set f2 on the zw-plane is bounded by two curves (c~) given by equations w - ~p~(z) (6.29) and such that r > q0o (z)Vz. Through every point (~, o9) in f2, two curves (c+(~, o9)), (c-(~, o9)) are defined as solutions of the initial value problems: dw = ~+ (z, w); WIz=~ - o9. (6.30) dz It is assumed that each problem in (6.30) has a unique solution, for every initial point (~, o9) in f2, and that a solution exists forwards and backwards in z, and it gives an arc in h extended from some point on (Co) to some point on (c~-).

CHAPTER 6 Control of Systems with Hysteresis

618

For a piecewise monotone continuous input signal, the hysteretic output w(t) = (Hz)(t) is defined by the requirement that the point (z(t), w(t)) fl moves on one of the curves with superscript '+' when z(.) is increasing, and one of the curves with superscript ' - ' when z(.) is decreasing. Now we consider an ODE system dx(t) dt =fit, x(t), y(t), u(t)); x(0) = x0;

y(t) := (H(r~))(t), (~(t) : - g(t, x(t)). (6.31) Here, x(.) takes values in IR n, u(.) takes values in a compact subset U of IR m, and cr and y are real-valued functions, and H is the hysteron operator defined above. The objective of the optimal control problem is to minimize a functional J given by J "= f0(x(T), y(T)) +

J0' f(t, x(t), y(t), u(0) dt.

(6.32)

This problem can be treated by the method of dynamic programming. We assume that the function g is continuously differentiable with respect to all its variables. Then, for x in f~, the evolution of y(.) can be described by dy(t) = ~,+(g(t x(t)) y(t))

dt

'

'

[ 3g(t,3tx(t)) + Vxg(t, x(t))f(t, x(t), y(t), u(t))] + -~,- (g(t, x(t)), y(t))

l

Og(t, x(t)) + Vxg(t x(t))f(t, x(t), y(t), u(t)) 3t

l-

(6.33)

where, for any real number r, r + := max(r, 0), r- := m a x ( - r , 0). We parameterize the problem by initial time s and initial values ~ : x(s) and r/:= y(s). The corresponding Hamilton-Jacobi function is W(s, ~, r/). At each point (s, ~, r/), we partition the set U of admissible control values into three mutually disjoint subsets Uo(s, ~, ~1., U1 (s, ~, r]), U-1 (s, ~, r]), defined by: ! Og(s, + V~g(s, ~)f(s, ~, ~l, a) = 0 } ; UO(S, ~, ~I) := {a U" / Os ! + V~g(s, ~)f(s, ~, r/, a) > 0 } ; UI(S, ~ t]) := {a U" '

l

OS

U-l (S, ~, r]) "-- {a ~ U" Og(S,os~) + V~g(s, ~)f(s, ~, r/, a) < 0 }. (6.34)

6.3 OPTIMAL CONTROL OF SYSTEMS

619

For brevity, we shall denote by # the expression ag(s,~) + V~g(s ~)f(s, ~, ~I, a) r and by 9~ - ~(s, ~, r/, (#)+, (~)-, a) the right-hand side of (6.33). Then the dynamic programming equations take the form: min

inf

i~{0,1,-1} a~Ui(s,~,rl)

{ c~W(s, ~, rl) + V~W(s, ~, rl)f(s, ~, rl, a) dS

+VnW(s, ~, ~)V/(s,~, ~, (#)+, (#)-, a) + F(s, ~, ~, a)}

- O;

(6.35)

W(T, ~, ~) = Fo (~, ~).

In addition, we need boundary conditions for W(s, ~, ri) on the boundary 0 f 2 - (c~)U(Co). For (g(s, ~), r/) ~ (c~-), we have r/ = q~(g(s, ~)),andfora U1 (s, ~, r/), where ~/- q~- (g(s, ~)), the total derivative (D/Ds) W(s, x(s), y(s)), evaluated at x(s) - ~, y(s) - q;-~(g(s, x(s)), is:

,~W(s,

~,~(g(s,~)))

8s + v,~ W(s, ~,~ (g(s,

(ag(s,as ~) +

+ V~W(s, ~, q~-~(g(s, ~)))f(s, ~, q~ (g(s, ~)), a)

,;)))(~-)' (g(s, ~))

V~g(s, ~)f(s, ~, qY~(g(s, ~)), a) ) .

(6.36)

On the other hand, for a ~ Uo(s, ~, rl) U U-1 (s, ~, r/), with r / - q~-(g(s, ~)), the total derivative (D/Ds)W(s, x(s), y(s)) is:

aW(s, ~, ~) + V~W(s, ~, rl)f(s, ~, ri, a) + V~W(s, ~, rl)~P(s, ~, rI, (#)+, ~s (6.37) (6)-, a) [ rl=r ~(s, ;)) 9 Consequently, the boundary conditions for W on (c~-) are: min{

inf

{ 0W(s, ~, q~- (g(s, ~)))

aEU1 (s, ~, q~-~(g(s, ~))

~S

+V~W(s, ~, q~-~(g(s, ~)))f(s, ~, q~-~(g(s, ~)), a)

+v~w(s, ~, G-(g(s, ,;)))(G-)'(~(s, ,;)) (ag(s, ~) 8s

+ V~g(s, ~)f(s, ~, q~ (g(s, ~)), a)

+F(s, ~, q~-~(g(s, ~)), a},

)

CHAPTER 6 Control of Systems with Hysteresis

620

a~Uo(s,r

inf

(S,r

{ OW(s, ~, ~1) + YeW(s, ~, ~l)f(s, ~, ~1,a)

r

+V,W(s, r rl)W(s, r rl, (~)+, (~)-, a) + F(s, r rl, a) } ],=r162

O.

}

(6.38) The boundary conditions on (co ) are obtained from (6.38) above by interchanging the roles of Ul(S, ~, r/) and U-l(S, ~, ~/) and replacing c#~ by cp0 9 The dynamic programming equations obtained above allow a formal derivation of Hamiltonian equations and an extremum principle akin to Pontryagin's m a x i m u m principle. We define three Hamiltonians:

hl (s, r r/, a, p, q) := pf(s, r rl, a) +q6+ (g(s, r rl) [ 3g(s'

+ Vcg(s' r

r rl' a)] + F(s' r rl' a)'

for a ~ UI(S, r r/);

h-l (S, r rl, a, p, q) "= pf(s, r rl, a) +q6-(g(s, r r/)[ 3g(s' r

+ Vcg(s, r

r rl, a)] + F(s, r rl, a),

for a ~ U-1 (s, r r/);

ho(s, r rl, a) "= pf(s, ~, 11, a) + F(s, ~, 11, a), for a ~ Uo(s, r ~1).

(6.39)

The variable p is an n-dimensional row vector; p plays the role of Vr W(s, ~, 11) and q plays the role of 3W(s, ~, ~1)/&l. Further, we define the minimized Hamiltonians,

hm,i(s, ~, rl, p, q) := hm(s, ~, rl, p, q) :--

inf aEUi(s,r

hi(s, ~, ~/, p, q, a), i = 0, 1, -1;

min hm,i(s, ~, rl, p, q). iE{0,1,-1}

(6.40)

Now, we assume that there is an optimal control function u*(t), OKtKT, with corresponding trajectory (x*(t), y* (t)), O~tKT. We set p*(t) := VxW(t, x*(t), y*(t));

q*(t) := 3W(t,x*(t), y*(t)) ~y

(6.41)

assuming existence and continuity of the indicated derivatives. Then we have, from (6.38), dp*(t) = -Vxhm(t, x* (t), y* (t), p* (t), q* (t)); dt

6.3 OPTIMAL CONTROL OF SYSTEMS

621

cghm(t, x*(t), y*(t), p*(t), q*(t)) cgy

dq* (t)

dt

hm(t, x* (t), y* (t), p* (t), q* (t)) =

min

a~Ui(t,x* ( t),y* ( t) )

hi(t, x* (t), y* (t), p* (t), q* (t))

= hi(t, x* (t), y* (t), u* (t), p* (t), q* (t)), for u* (t) ~ Ui(t, x* (t), y* (t)) (6.42) for OKt < T, (x*(t), y*(t)) ~ f2. On the boundary curve (c~), the Hamiltonian hi is replaced by h~where h + (s, ~, r/, a, p, q) := pf(s, ~, ~1,a)

1

+q(~)' C4(s,O) I ~g(s,_ds~) + V~g(s, ~)f(s, ~, ~1,a)~ + F(s, ~, rI, a) (6.43) and we define

h +, (s, ~, rI p, q) := 1

'

inf

a~Ul(S,~,rl)

h~- (s, ~, r/, a, p, q);

h + : - min{h +, h-i, h0}. (6.44) Then the Hamiltonian equations and the extremum principle (6.42) hold with hi, hm replaced by h~-, h +, for O~t < T, (x*(t), y*(t)) ~ (c-~). Analogous statements hold for OKt < T, (x*(t), y*(t)) ~ (Co). At the final time T, we have

p*(T) - VxFo(x*(T), y*(T)), q*(T) - cgFo(x*(T), y*(T)). cgy

(6.45)

In the particular case in which the function g(t, x) is independent of t, we can obtain a somewhat different form of the Hamiltonian equations. We define the Hamiltonians:

H(s, ~, ri, a, r) := rf(s, ~, rl, a) + F(s, ~, rl, a) for a ~ U; Hm,i(s, ~, rl, r) :=

inf

H(s, ~, ri, a, r);

aEUi(s,~,rl)

Hm(s, ~, ~1, r) := inf H(s, ~, rI, a, r). aEU

(6.46)

Here, the variable r plays the role of the 'total derivative' of W with respect to ~,i.e. theroleofD~W(s, ~, r/) - - V~W(s, ~, ~/)+ 0W(s,~,,) 0, 6+ (g(~), r/) V~g(~). If we set r*(t) := DxW(t, x*(t), y*(t)) (6.47)

622

CHAPTER 6 Control of Systems with Hysteresis

then we find, from (6.38), dr*(t)

dt

= -DxH(t, x* (t), y* (t), r* (t))

(6.48)

at points of differentiability of r* (t), and the extremal condition

H(t, x* (t), y* (t), u* (t), r* (t)) = Hm(t, x* (t), y* (t), r* (t))

(6.49)

dy at points of continuity of ~ with (x, y) in f2. The definition of r* implies the followingjump conditions at times tij at which u* (t~ ) ~ Ui(tij, x* (tij), y* (tij) ) and u*(t~) ~ Uj(tij, x*(tij), y*(tij)), i ~=j, with i and j taking the values 0, 1, and -1:

r* (t~) - r* (tT~) = 2ijVxg(x* (tij)); ,~ij .__ cgW(tij, x* (tij), y* (tij)) cgy [qJ(g(x* (tij)), y* (tij)) - 6~ (g(x* (tij)), y* (tij))]; 61 ~ ~y+~6 -1 ~ ~-~ 60 ~- 0.

(6.50)

The problem formulated and studied in [39] has g(t, x(t)) = bx(t) for a constant row vector b, so that Vxg(t, x) = b and jump conditions of the form (6.50) hold with the constant vector b in place of Vxg(x* (tij)). The problem in [39] involves additional ingredients such as a final time constraint (which modifies the final conditions on the co-state) and a change of the time scale. (The relationship between dynamic programming and the effect of final time constraints on the extremum principle has been analyzed in [37].) The problems in [13] contain more general hysterons, such as intersecting or coalescing boundary curves (c~).

6.4 O P T I M A L C O N T R O L OF S Y S T E M S W I T H P R E I S A C H HYSTERESIS Systems with Preisach hysteresis operators are the most crucial category of hysteretic control systems. The Preisach model of hysteresis, and certain variants of the standard Preisach model, have been used to model many kinds of physical systems with hysteresis. From the point of view of control theory, an important category is the control of elastic systems via actuators that exhibit hysteresis.

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

623

In this section, we shall make reference to, without repeating all the details of, the definition and basic properties of the Preisach hysteresis operator. We shall take for granted the information contained in the essential reference [23], and we shall also make reference to other works that contain results about the Preisach operator and systems that include Preisach hysteresis nonlinearities. The Preisach hysteresis operator has global memory, it stores information about the entire history, from some initial time to the current time, of the dominant local extrema of the input signal. For this reason, it is clear that the use of dynamic programming arguments necessitates the extension of the original state space to a substantially larger space, so as to include, in the extended state, relevant information about the past history of the input signal. It turns out that a suitable way to produce a Markovian model (in a substantially broader space than the original state space) is to formulate the Preisach operator in terms of the active set on the Preisach plane, and to formulae the controlled dynamical system as a system with a set-valued state. This creates some conceptual questions, because (i) we need to define the concept of a derivative of a set-valued function of time, and (ii) the Hamilton-Jacobi function, as a function of the extended state, is a real-valued set function, not necessarily additive, and we need a suitable concept of derivatives for non-additive set-functions. The concept of derivatives for set-valued functions that we introduce here, and the almost identical concept introduced in [21], are new concepts, different from other definitions of derivatives of a set-valued functions. The one paper that is conceptually somewhat close to our treatment is [40]. It is expected that the concept of a set-valued state may be useful for more general systems with a set-valued state, and not only for Preisach hysteresis. The papers [15-18], and particularly [17], deal with controlled hysteretic systems with hysteresis acting on the state, and the analysis is carried out from the point of view of 'viscosity solutions', a form of generalized solutions for dynamic programming equations. We remark that, some cases of Preisach operators with a singular measure, instead of a continuous function, supported on a monotoneconnected curve on the Preisach plane, such as the version of StonerWohlfarth models introduced by G. Friedman [41], result in a dynamical system description of the corresponding optimal control problems, with one-dimensional (rather than set-valued) state, which can be treated by easier methods than the set-valued state evolution that we consider here. The double and multiple extensions of the Preisach model, introduced in [42,43], can be treated by the method described in [20], which we shall not repeat here.

CHAPTER 6 Control of Systems with Hysteresis

624

6.4.1 SET-VALUED DERIVATIVES

In this section, we shall give an account of an approach that is conceptually based on [21]. Our exposition below is technically different from [21], however. Let H denote a Preisach hysteresis operator with weight function w(a, fl), flo~fl < ~ 0 . This means that, for a continuous input z(.), the hysteresis output is

(Hz)(t)=/fp(~o,~o) w(a, fl)(R~z)(t) da dfl

(6.51)

where P(a0, fl0) := {(a,

fl)

~ Ia2 " ~0 ~0r > fl~/J0}

(6.52)

and each R ~/~is an operator of the type defined in Section 6.2. It is assumed that initial conditions for R~z are chosen so as to make (R~z)(s), at some initial time s, a measurable function. Now, if the input signal z(.) is piecewise monotonic, we assume that the specific construction given in [23] is used to define the hysteresis output Hz. This means that, if at some initial time to we have z(to) < rio, then all relays (R~z)(to), (~, fl) e P(~o, fl0), are in the output state 0; analogously, if z(to) > s0, then all relays (R~z)(to) are in the output state 1; for other values of z(t0), all relays (R~z)(to) with aKz(to) are in the output state 1, and all other relays, at time to, are in the output state 0. As z(t), t > to, increases or decreases, new relays are being set to the 1 or the 0 output state, producing, at each time instant t, the staircase curve that separates the values of (~, fl) ~ P(a0, rio) for which (Ra~z)(t) = I from those values for which (R~z)(t) = 0. The rule is this: if z(t) increases from z(tl) to z(t2), t2 > tl, i.e. if z(.) remains increasing for t ~ (tl, t2), then all relays with z(h) < a < z(t2) will have been switched from output state 0 (at time tl) to output state I (at time t2); similarly, if z(t) is decreasing for t2 < t < t3, then the relays that correspond to (a, fl) with z(tg)~fl < z(t2) will have been switched from output state I (at time t2) to output state 0 (at time t3). If we denote by S(t) the active set at time t, i.e.

S(t) "- {(~, fl) ~ P(~o, rio) " (R~z)(t) - 1}

(6.53)

then we plainly have

(Hz)(t) = / fs

(t)

w(~, fl)d~dfl.

(6.54)

Thus the hysteretic output Hz is completely characterized by the active set S(t). Consequently, a dynamical system that includes Preisach hysteresis

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

625

nonlinearities can be interpreted as a system in which part of the state is set valued, i.e. the active set S(t) becomes part of the state of a dynamical system. We are led, therefore, to consider dynamical systems with a set-valued state. Consequently, we need a suitable concept of derivatives of set-valued functions of one real variable. The research literature contains m a n y concepts of derivatives of set-valued functions. However, we need a concept of derivatives that is suitable for the situation that arises in Preisach hysteresis operators and we develop such a concept from first principles. Also, in anticipation of having to apply dynamic p r o g r a m m i n g arguments, we need a concept of derivatives of real-valued set functions, which m u s t have the property of compatibility with the derivatives of set-valued functions, in the sense that a version of a chain rule should hold. It will be seen that the appropriate concepts of derivatives can be extended to more general cases than just the Preisach hysteresis operator. Given a set-valued function, say Q(t), whose values are open sets in IR n, we represent the two set-theoretic differences Q(t+~t) \Q(t), Q(t) \ Q ( t + &t) as unions of simpler sets. The m a g n i t u d e of a set is generally measured via w h a t we shall call a gauge function g, i.e. a function defined on b o u n d e d open sets in Ia n and taking values in [0, +cx~), that has the following properties:

A c B =~ g(A)~g(B); g(Q) = g({x}) - O, Vx ~

I a n"

An ,[, A =~ g(An) --> g(A).

(6.55)

We note that, if we omit the condition g({x}) - 0, the remaining properties of g are the defining properties of Choquet capacities. The gauge function is used to measure the 'size' of a set. We shall say that a set Z(t, ~t) is og(l~tl ~) if g(Z(t,~t)) i~tl~ --+ 0 as ~t ~ 0. We have: DEFINITION6.4.1. We say that the set-valued function Z(t) is differentiable at t - to if, for all ~St sufficiently small, we have

Z(to + ~t)\Z(to) -

U (a + + A+(~St)U +) Uog(l~tI); i

Z(to)\Z(to + ~St) -

u (a~- + A-j-(~St)U~-) Uog(l~Stl)

/

(6.56)

CHAPTER 6 Control of Systems with Hysteresis

626

where the points a +, a7 and the sets U +, U~- are independent of ~t, and the matrices A + (St), A 7 (~t) satisfy det(A + (~t)) = O(l~tln), det(A 7 (~t)) -- O(l~tJn). The definition above parallels the definition of derivatives of an ordinary function. The usual requirement that an ordinary derivative should be a linear function of ~t is replaced by an affine 'function' of ~t, of the form a~ + A~(~t)U~. The sets U~ specify the 'shape' of the affine function, the matrices A~ (~t) specify the size of this affine function (where by 'size' we mean different sizing along different directions), and the points a~ specify the position of the affine set-valued function. In the case of derivatives of ordinary functions, the 'position' attribute of the derivatives is irrelevant, but for set-valued functions it is necessary to include this ingredient, and it is necessary to change the concept of derivative from a linear operator to an affine operator. Simple examples, which we shall present below, show that this is the minimal necessary adjustment we need to make to the concept of derivatives to make it applicable to set-valued functions. The requirement det(A~(6t)) = O(t~tl n) may seem counterintuitive, as it appears to make the set-valued derivative a higher degree, rather than an affine, function in the increment bt. The reason for this will be explained through the utilization of this concept of derivatives in the derivation of dynamic programming equations for systems with Preisach hysteresis nonlinearities. Next, we present a concept of derivatives of a set function, i.e. a realvalued function with a set-valued independent variable. DEFINITION6.4.2. A set-function V has a Frdchet-like derivative if there exist bounded measurable functions q~+, q~- such that fA

W(Z CJc~Z) - V(Z) - / 9

V(Z \ 6Z) - V(Z) - f 9

q)+ (~),~,n(d~) + 0(,~n ((~Z)),

,15Z

q)- (~),~n (d~) + o(~ n (c~Z))

,/SZ

where ,~n is an n-dimensional Lebesgue measure, and the symbol \ signifies the inclusion of ~Z into a Z modulo set of n-dimensional Lebesgue measure zero, i.e. 2 n (~Z\Z) - O. We say that V has a G~teaux-like derivative if, for variations 5Z of the form ~Z = u (ai q- Ai(~)Ui) with det(Ai(~)) - o(~n), we have i~I V(Z U c~Z) - V(Z) - ~(D+V)(Z, {ai, Ai, Ui" i E I}) q-o(~); V ( Z \ c~Z) -

V(Z) - e,(D-V)(Z, {ai, Ai, Ui " i ~ I}) + o(~)

for some functions D +, D - that depend on the indicated variables.

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

627

The two concepts of derivatives above are very different from ordinary Fr6chet and G~teaux derivatives; we use the names 'Fr6chet-like' and 'Gateaux-like' only for mnemonic purposes. There is, in general, no comparison between the two concepts of derivatives introduced in Definition 6.4.2; the existence of either of those derivatives does not imply the existence of the other. 6.4.2 CALCULATION OF SET-VALUED DERIVATIVES In the case when the set-valued function is S(t), the active set of a Preisach hysteresis operator, we can give an explicit calculation of the derivatives in the sense of the definitions of Section 6.4.1. We consider a continuous piecewise monotone signal u(.), and the corresponding evolution of the active sets on the Preisach plane with coordinates a and fl, in the part P(a0, rio) := {(a, fl):fl0 < fl < a < a0}. The symbol P(a0, rio) signifies the closure of the open set P(a0, fl0). We consider active sets S(t) such that the relative boundary of S(t) in P(ao, rio) is a piecewise smooth non-increasing curve ?(t), where the closure ?(t) extends from some point on the part {(a0, fl):flo~fl~ao}U{(a, fl0) : fl0~a~a0} ofthe boundary 0P(a0, fl0) to some point on the diagonal {(a, fl) : fl0~a = flKa0}. The point on the diagonal will be denoted by D(7(t)). We shall obtain explicit formulae for the derivatives of S(t) under smooth monotone variations of the input u(.). The curve ?(t), of the type described above, will fall into one of three categories: (i) A curve which we shall call of a-type, which contains a nontrivial (i.e. of positive length) straight line segment parallel to the a-axis with one endpoint at D(7(t)), and the maximal line segment has its other endpoint at a point Da(?(t)); we shall denote that oriented line segment by seg(D(7(t)), D~(?(t))). (ii) A curve which we shall call of fl-type, which contains a nontrivial straight line segment parallel to the fl-axis with one endpoint at D(?(t)), and the maximal line segment has its other endpoint at D~(7(t)); this oriented line segment will be denoted by seg(Dfl(7(t)), D(7(t))). (N.B. The orientation of the segments in cases (i) and (ii) is such that they are oriented in the direction of increasing a or increasing ft.) (iii) A curve that is neither of a-type nor of fl-type; in that case we shall say that 7(t) is of oblique type. For a curve of a-type, there exists a neighborhood V of D~(7(t)) such that the part (V A~/(t))\seg(D(7(t), Da(7(t))) of the curve ?(t) can be represented by an equation a = (fi(fl). Similarly, for a curve of fl-type, there exists a

CHAPTER 6 Control of Systems with Hysteresis

628

neighborhood V of D[~(y(t)) such that the part (VNy(t)) \seg (D/~(~ (t), D(7(t))) of the curve y(t) can be represented by an equation fl = @(a). The curves -- r and fl --- @(a) are both non-increasing and, for the purposes of this derivation, they are postulated to be differentiable at the points D~(y(t)) and D/~(y(t)), respectively. A curve of oblique type can be locally represented, at and near D(y(t)), by both equations a = q~(fl) and fl = ~(~). We consider an active set S(t) at time t; S(t) is a set of the type described above. The point D(~(t)) has coordinates, in the aft-coordinate system, (a, a), i.e. the present value u(t) of the input signal u(.) is u(t) = a. We shall use the following notation: R(al, ill; ~2, f12) will be the rectangular domain, on the ~fl-plane, with vertices (al, ill), (al, f12), (a2, ill), (a2, f12); T(a3, f13; a4, f14) will be the triangular domain, on the aft-plane, with vertices (0, 0), (a3, f13), (a4, f14). In the case where two of the vertices of either a triangle or a rectangle coincide, we shall identify that domain with the empty set. The gauge function g will be the diameter of a set,

g(A) = sup{dist(x, y) : x e A, y ~ A}. For an active set with a b o u n d a r y curve 7(t) of a-type, we shall denote by a~ the a-coordinate of the point Da(y(t)), and, similarly, for the case of a domain with a b o u n d a r y curve of fl-type, we shall write b/~ for the fl-coordinate of the point D~(y(t)). For a domain with a b o u n d a r y curve that is not of a-type, we identify D~(y(t)) with D(y(t)), and, by the same token, if y(t) is not of fl-type, we identify D~(y(t)) with D(y(t)). These conventions allow us to present unified formulas for the derivatives of S(t). We consider a variation vbt of the input signal u over the time interval It, t + bt) (this notation incorporates the condition that 6t > 0), with v # 0. We have: THEOREM6.4.1. Under the conditions and notational conventions stated above,

the derivatives of S(t) are given by the following formulas:

a~--

b/~

'Af(bt)=

0

0] 1 ' U+ - R(0, b/~; 1, a); 0 ] v +dt , U f - T(q~'(a), 1; 1, 0);

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

[a]a

[v'to

o]

vdt

, U~- -- T(1, 0; 1, 1);

_ = [o] ,Vjl A C,~t~-- [1 - v - ~t I ' [v t - v -0d t ] ' U2 -

a1

a 2 --

a 3 --

0

a

'

[a][v

' A3=

a

]

Ul=R(a~,0;a,1);

0

' A2 -

a

629

0

0

tO]

-v-dt

T(1, 1; 1, 6'(a));

' U3 = T ( 1 ' 1;0,1)

where v + "- max(v, 0), v - - m a x ( - v , 0). The i m p o r t a n c e of this definition of derivatives of set-valued functions is also s h o w n in the following theorem: THEOREM 6.4.2. For piecewise monotone inputs u(.), which are also piecewise

continuously differentiable over each interval of monotonicity, if the derivatives as calculated in Theorem 6.4.1 above are specified at each time instant t, and if the active set S(O) is specified at an initial time t = 0, then the active set S(t) is uniquely determined for all t. The above t h e o r e m is an existence a n d u n i q u e n e s s t h e o r e m for a type of differential equation w i t h set-valued functions as u n k n o w n s . We consider a controlled s y s t e m of the form dx(t) dt

=fit, x(t), S(t), v(t)); x(0) - x0, S(0) - So

(6.57)

w h e r e v(t) is the control, a n d the set-valued function S(t) evolves according to the derivatives described in T h e o r e m 6.4.1 above. This type of s y s t e m includes the case of Preisach hysteresis operators w i t h inputs u(t) that satisfy du(t) dt

= v(t)

(6.58)

w h e n the Preisach operator is

(Hu)(t) - / fs(t) w(~' fl) d~ dfl

(6.59)

or, in a more general sense,

(Hu)(t) - / f s

(t)

w(t, a, fl) dadfl

(6.60)

630

CHAPTER 6 Control of Systems with Hysteresis

with a time-varying weight function w. This model does not include the case of Preisach operators with Borel measures (with possibly a singular part) instead of the weight function w. In the latter case, some modifications would have to be made to the concept of derivative of a set-valued function to include special treatment to the support of the singular part of that Borel measure. The control function u(.) itself does not need to appear explicitly in the controlled dynamical system (6.5.7), because the value u(t) is uniquely determined by the coordinates of D(~;(t)) on the Preisach plane P(ao, rio). The optimal control problem is the minimization of a functional J :=

J0

f(t, x(t), u(t), (Hu)(t)) dt + f0(x(r), (Su)(r)).

(6.61)

The dynamic programming equations, under conditions of differentiability of the Hamilton-Jacobi function W(t, ~, S), take the form

OW(t, ~, S) Qt

+ min{inf{VcW(t, ~, S)f(t, ~, S, v) v~>0

+D+'VW(t, ~, S) + F(t, ~, S, v)}, inf{VcW(t, ~, S)f(t, ~, S, v) + D-'VW(t, ~, S) + F(t, ~, S v)} = 0; v~<0

W(T, ~, S) = Fo(~, S). (6.62) The superscript "v" in the G~teaux-like derivatives D • denotes that the value of the variation v is used in the definition of the derivatives of the Hamilton-Jacobi function W with respect to the set-valued variable S. Further, the essential concept of weak solutions in terms of extremal approximations (this process is most commonly called 'viscosity solutions') can be applied once the appropriate concept of derivatives is available. This has been summarized in [21]. 6.4.3 OTHER RELATED TECHNIQUES

A second approach to the problem of defining, calculating and utilizing a concept of derivatives, is based on a concept of variations of curves. The relevant curve, for the Preisach model, will be the relative boundary of the active set. Our definition of the appropriate concept of derivatives is inspired by Hadamard's work in the calculus of variations: reference [44] contains, among lots of other things, the calculation of the variation of Green's functions for the Dirichlet and the Neumann boundary value problems of potential theory, relative to perturbations of the domain on which the boundary value problems are to be solved. (There exist modern offshoots of Hadamard's ideas, in the context of shape optimization

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

631

and sensitivity analysis, relative to domain perturbations, for partial differential equations and variational inequalities. For the purposes of the present exposition, it is more efficient to refer directly to H a d a m a r d ' s work.) H a d a m a r d defined the perturbation of a surface as the normal distance bn between two nearby surfaces; cf. articles 249-252 of [44]. Following the ideas, but not the specific technicalities, of [44], we may define the derivative of a curve with respect to a parameter in the following way: DEFINITION6.4.3. Let (7o) be a continuously differentiable curve in !R n with equation P = P(t), t o ~ t K t l , where P stands for a generic point on that curve. (We identify a point P with the vector 0--~, where 0 stands for the origin of a coordinate system.) Let a family of continuous curves (7~), 0 < ~ < ~0, be described by an equation P~ = P(t) + rl(t, ~)--~ (t), where P~ is a generic point on (7~), -it (t) is the vector field on unit normals to the curve (70) at each point P(t) of (70), and the function rl is continuous in both variables and differentiable at ~ = 0 +, with r/(t, 0 +) ~_ lim r/(t, ~) = 0. Then we define the derivative ~--,0 +

~/~(t, 0 +) = lim ~rl(t, ~) as the normal derivative of (7~) with respect to the pa~--,0 +

rameter ~ at ~ -- 0 +. More generally, if (70) is a continuous curve with equation P = P(t), t o K t ~ t l , if--d (t) is a vectorfield of vectors of constant length (for technical convenience, we may want to use a length other than unity) defined on (?0), and if a family of curves (7~), 0 < ~ < ~o is defined by P~ - P(t) + rl(t, ~)-d (t), 1 where rl has the same properties as above, then we define r/~(t, 0 +) = lira ~/(t, ~) ~--,0 +

as the derivative relative to the vector field --d (t) of (7~) with respect to the parameter ~ at ~ = 0 +.

On the Preisach plane P(a0, rio), let (~0) be a piecewise smooth nonincreasing curve, with diagonal endpoint (i.e. endpoint on the diagonal a-flofP(a0

flo)) denoted b y D(7o) W e u s e thevector field -d~ (t) _ [ l 1 I..

_!

which suffices to describe variations due to a small increase or decrease of the input u(.). The calculations of the variations for domains of different types (atype, fl-type, or oblique type can be carried out by techniques analogous to Section 6.4.2 above, and the related dynamic programming equations can be written down. This alternative definition of the derivatives of the active set may be useful for the description and analysis of more general controlled systems with a set-valued state. For the specific issue of control of Preisach hysteresis, this second concept of derivatives of set-valued functions will be presented in a future work, and is not reported here.

632

CHAPTER 6 Control of Systems with Hysteresis

6.5 H Y S T E R E S I S F R O M O P T I M A L C O N T R O L , A N D HYSTERESIS FOR OPTIMAL CONTROL In this section, we outline certain cases in which a controlled system, in its original formulation, does not contain any hysteresis elements, but hysteresis is later introduced into the system through the design of optimal control policies. Sometimes, an optimal control policy, expressed as a feedback law, leads to hysteretic behavior of the corresponding optimal control function or the corresponding optimal state trajectory. Sometimes, hysteresis is intentionally part of the design of a control policy. The general phenomenon of control systems that acquire new, nonsmooth, behavior, or some other kind of behavior that is qualitatively different from the original description of the system, is well known and well understood in many parts of optimal control theory. For instance, we may start with a system that we originally intend to describe as an ODE system with controls that are continuous in time; however, an optimal control problem may lead to a feedback law that creates discontinuities, as in the case of 'bang-bang' controls in (certain classes of) minimum time problems. An analogous situation occurs in connection with the emergence of hysteresis operators as a consequence of optimality.

6.5.1 HYSTERESISARISING FROM DISCRETE-VALUED CONTROLS IN DETERMINISTIC AND STOCHASTIC SYSTEMS A controlled ODE system with a binary control decision, say 0 or 1, can be described by dx(t) =fit, x(t), u(t)); u(t) ~ {0, 1}; x(0) = x0. dt

(6.63)

There are two possible cost functionals that are of interest for control applications: ]1 "-

f(t, x(t), u(t)) dt + fo(x(T))

(6.64)

and

J2 "- f0 T F(t, x(t), u(t)) dt + ~

k(:i, x('ci),

U('C~), U(':+)) q- FO(x(T)).

(6.65)

i In the case of h , the Hamilton-Jacobi function Wl(t, x) satisfies, in the appropriate sense of generalized solutions, the dynamic programming

6.5 HYSTERESISFROM OPTIMAL CONTROL

633

equation: C~Vl (t, x) ~t

+ min {VxV1 (t, x)f(t, x, a) + F(t, x, a) } = 0; a~{0,1}

V1 (T, x) = Fo(x).

(6.66)

In the case of J2, the Hamilton-Jacobi function V2 (t, x, a) is parameterized also by the current value of the control, a = u(t), and satisfies, in the appropriate generalized sense, the dynamic programming equation: min

OV2(t, x, a) -Ot 4- VxV2(t, x, a)f(t, x, a) + F(t, x, a),

V2(t, x, b) - V2(t, x, a) 4- k(t, x, a, b) } = O; V2(T, x, a) = Fo(x)

(6.67)

where, in (6.67), a and b take values in 0, 1 and b ~- a. In the case of minimizing J1, if an optimal control policy exists, it can be described in feedback form in terms of continuation sets C a, a ~ {0, 1}. Each set C a consists of all values of (t, x) such that, when the value x(t) of the state trajectory (solution of (6.63)) satisfies (t, x(t)) ~ Ca, then an optimal decision is to take u(t) = a. According to this decision rule, switching from one value of the control to another is consistent with optimality if (t, x(t)) E C a N C b. The situation is a little more involved for the case of minimizing J2, but an analogous principle of optimal feedback applies to that case. Now, restricting our attention, for simplicity, to the problem of minimizing J1, we remark that, in general, the intersection C a A C b contains an open set, which creates the analog of a nonideal relay (Preisach relay) for the optimal control policy. Switchings become mandatory when the point (t, x(t)) escapes from a continuation set. The switchings that are mandatory for optimality are those that are delayed for as long as possible, i.e. those that satisfy the analog of Visintin's 'laziness principle' or minimum bounded variation for the Preisach relays (cf. [27]). In the particular case of one-dimensional state x(t), with continuation sets that are independent of t and are of the form C O ~-- ( - o o , 0~], C 1 = [fl, ff-oo) in the state-space (-oo, +oo), with ~ > fl, we get precisely a relay of the Preisach type as the optimal feedback law that describes many possible optimal switching policies (in general, an optimal policy, in this context, is not unique), with the 'laziness principle' dictating when switchings become inevitable. Thus, we have, in this case, arrived at the following situation: a simple control system, with binary control decisions, through a requirement of optimality, becomes a system with a nonideal relay: a system that was originally dx(t) dt = fit, x(t), u(t)) has dx(t) been transformed into a system -aN = f i t , x(t), (R~flx)(t)). We remark that

634

CHAPTER 6 Control of Systems with Hysteresis

this principle is not limited to systems governed by ordinary differential equations, but applies to all kinds of systems for which a generator (in the sense of evolution semigroups) can be defined, for example, systems governed by Ito stochastic differential equations (cf., for instance, [45]). 6.5.2 HYSTERESIS FOR THE CONTROL OF DYNAMICAL SYSTEMS

The situation described in Section 6.5.1 above is sometimes put in reverse. Starting with a control system and some optimality criterion, a binary decision structure is introduced, and a feedback control policy is postulated in the form of a general choice about the nature of the continuation sets. For example, one may postulate that CO = (-cx~, a], C 1 -- [/~, q - o o ) , and seek an optimality condition. Under these scenarios, an optimality condition then leads to a conclusion (among other things) that ~ > fl, and thus hysteresis has been intentionally designed for the purpose of controlling a dynamical system. We must point out that the description of the process of arriving at a hysteretic control policy is our interpretation, and the researchers who apply equivalent methods may proceed in ways that are technically different, or conceptually explained in different ways, from what we have described. For example, see the paper [46] for hysteresis arising in 'rational' (economic) institutions, or the paper [47] for hysteresis introduced for the design of optimal control of a queueing system. Further, we can consider a continuous collection of controlled systems with binary controls, each system having its own (locally) optimal policy defined by a pair of continuation sets, with the collective outcome being a weighted superposition of the 'locally' optimal outcomes; this leads naturally to a Preisach hysteresis operator. This last step in the description of an instance of 'hysteresis from control' was inspired by the paper [48] on hysteresis in economic systems.

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CHAPTER 6 Control of Systems with Hysteresis

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35. L. T. Ashchepkov, Optimal Control of Discontinuous Systems, Nauka, Novosibirsk (1987) (in Rusian). 36. B.M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-continuous Systems, Kluwer, New York (2003). 37. S. A. Belbas and V. M. Matrosov, 'Optimal control of systems in continuous-discrete time scales', Int. Congress on Nonlinear Analysis and its Applications, Moscow, International Academy of Nonlinear Sciences (1998); preprint at arxiv.com. 38. J. W. Macki, P. Nistri, and P. Zecca, 'Mathematical models for hysteresis', SIAM Review 35(1), 94-123 (1993). 39. M. Brokate, 'Optimal control of ODE systems with hysteresis nonlinearities', in Trends in Mathematical Optimization, K.-H. Hoffmann et al. Eds, Birkh~iuser-Verlag, Basel (1998), pp. 25-41. 40. V. F. Demyanov, C. Lemarechal, and J. Zowe, 'Trying to approximate a set-valued mapping', Technical Report MRC-2786, Mathematics Research Center, Madison, WI (1985). 41. G. Friedman, 'New formulation of the Stoner-Wohlfarth hysteresis model and the identification problem', J. Appl. Phys. 67(9), 5361-5363 (1990). 42. G. Friedman, 'Second-order Preisach model of scalar hystresis', Physica B 275, 173-178 (2000). 43. G. Friedman and K. Cha, 'Representation of hysteresis with wipe-out memory', J. App1. Phys. 89, 7236-7238 (2001). 44. J. Hadamard, Ler sur le Calcul des Variations, A. Hermann et ills, Paris (1910). 45. S. A. Belbas, 'The dynamic programming equations for stochastic games with discrete actions', Systems and Control Letters 11, 235-242 (1988). 46. R. Di Tella and R. Mac Culloch, 'Rational institutions yield hysteresis', Report No. B9, Zentrum f. Europ. Integrationsforschung, Bonn (2000). 47. A. N. Dudin, 'Optimal hysteresis control for an unreliable BMAP/SM/1 system with two operation modes', Automation and Remote Control 68, 1585-1596 (2002). 48. R. Cross, J. Darby, J. Ireland, and L. Piscitelli, 'Hysteresis and unemployment: a preliminary study', preprint, Strathclyde University (1998). 49. F. Bagagiolo and A. Visintin, 'Hysteresis in filtration through porous media', Zeitschrift f. Analysis u. Anwendungen 19, 977-998 (2000).

CHAPTER 7

Binary Detection and Rectangular Hysteresis Operators Can E. Korman, Department of Electrical and Computer Engineering,

The George Washington University, Washington, DC 20052, USA

Contents 7.1 7.2 7.3 7.4

Introduction General System Description and Outline Input Process and Mid-Span Repeater Receiver Integrator Output: Probability Density Function Calculation 7.5 Summary 7.6 Concluding Remarks References

639 641 644 646 662 663 664

7.1 I N T R O D U C T I O N The subject of hysteresis has been studied and is of considerable interest in a broad range of disciplines in engineering, material science, finance, economics, etc. Hysteresis arises in such fields either due to the nature of the problem or it may be deliberately incorporated for the purpose of improving system performance. Fields of science where hysteresis is due to the physical nature of the material include magnetics, superconductivity, magnetostriction, etc. Applications where hysteresis is specifically introduced to improve system performance, or minimize some cost function, include the setting of trade prices for the buying and selling of stocks, control systems, etc. For instance, in a thermostat-based temperature control system, hysteresis is generally introduced for the purpose of reducing multiple switching of equipment due to randomly fluctuating inputs near The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

640

CHAPTER7 Binary Detection and Rectangular Hysteresis Operators

a threshold. In such systems, hysteresis helps minimize the system's cost function by preventing multiple rapid switching due to the fluctuation of inputs (stock price, temperature, etc.) around a set threshold. The original impetus for this work was to investigate the effect of hysteresis on the performance of an optical communication link. In order to set the problem in the proper context, it is beneficial to briefly review the relevant requirements of an optical network. For the purpose of reliable optical communication links over long distances, optical repeaters are placed at appropriate intervals to detect the incoming optical signal, amplify it and then retransmit it. Naturally, the optimal detector in a repeater should include a filter matched to the bit rate of the optical signal. However, owing to network design considerations, the optical links may be required to carry signals with different bit rates. Considering the fact that the same repeater must handle signals with multiple bit rates, optical repeaters are required to be bit-rate transparent. The issue of bit-rate transparent repeaters has been investigated by several groups in the recent past [1-4]. The repeaters in such networks are required to detect and retransmit rectangular pulses that have different bit rates. In an optical link these signals may be detected and retransmitted by such mid-span repeaters several times. If these repeaters were designed for fixed pulse widths, the optimum design would naturally be a matched filter. Unfortunately, the design of such repeaters is complicated by the fact that matched filters are only optimal for a given fixed pulse width. This being the case, it is necessary to employ suboptimum filters in the design of such mid-span repeaters in optical network links [5]. Arguably, the simplest possible binary bit-rate-independent detector is a step threshold device that can distinguish the presence or absence of light at its input. In such a device, hysteresis can either be a natural part of the optical physical layer or, if desirable, may be introduced deliberately to improve system performance. In light of this possibility, one begs the question of whether hysteresis in a binary threshold-type detector offers any advantages over a simple threshold scheme, whether in an all-optical network link or another application. This work explores the possibility of employing a binary threshold device in a binary detection problem and outlines the mathematical framework that is necessary to calculate various system performance measures. In this sense, it is meant to be a general mathematical approach even if the original motivation came from the analysis of optical network links. Therefore, it is applicable for the analysis of diverse applications that can be modeled by the system and noise constraints stated below. Problems of this nature have been studied before but have largely focused on nonlinearities with no hysteresis. In particular, the problem

7.2 GENERAL SYSTEM DESCRIPTION AND OUTLINE

641

of the detector where a low-pass filter is cascaded to a hard-limiter has been investigated by several groups [6-12]. In this context a hard-limiter is nothing more than a special case of a hysteresis switch with zero width. Therefore, the study of a detector with a hysteresis switch can be viewed as a generalization of prior work done in this field. To our knowledge no analytical investigations have been performed where the detector has a hysteresis-type nonlinearity. The work presented here is largely based on our prior research in this field [9]. While the work is partly motivated by all-optical networks, where network links are required to be bit-rate transparent, the analysis presented here is quite general.

7.2 GENERAL SYSTEM D E S C R I P T I O N A N D OUTLINE The general block diagram of the system to be considered in this chapter is shown in Fig. 7.1. The system consists of a binary transmitter, a number of mid-span repeaters and additive noise sources, all terminated at a receiver matched to the bit rate of the transmitter. Each mid-span stage can be considered to be identical to the others where the respective signals are perturbed by mutually independent noise sources. For clarity, it is beneficial to consider a communication link with only one midspan repeater. This one-stage link can be modeled by a system whose block diagram is shown in Fig. 7.2. Here the system is modeled by a one-stage bit-rate-transparent repeater that consists of a rectangular hysteresis loop and a receiver that is tuned to the specific bit rate of the binary transmitter. The input source of the system is modeled by a series of rectangular pulses perturbed by additive Gaussian noise. For mathematical

oJ Bin~ Midspan Repeater

noise

Midspan Repeater

"'" ~

Receiver

noise

y

Cascade of midspan repeaters

FIGURE 7.1 General system block diagram mith multiple mid-span repeaters terminated by a receiver tuned to a specific bit rate.

642

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators Bit-Rate Independent Mid-span Repeater White Noise Hysteresis Loop

nt

~t

Non- ] xt [ hysteretic[....~ filter

I-I

~(t)

JL2

O-Binary Input Signal T >T/2 ~
1 0

I

(*)dt

0

Decision Rule

Integrator

Receiver Module

FIGURE 7.2 Blockdiagram of the system to be modeled in greater detail with a one-stage mid-span repeater consisting of a hysteresis-type nonlinearity and binary detector. convenience, the binary signal perturbed by white noise is considered to be an input first to a non-hysteretic filter whose output then drives a second filter with hysteresis. The hysteresis in the mid-span repeater is modeled by a rectangular hysteresis loop, an elementary hysteresis operator that can be employed as a building block to model more complex hysteretic systems (see, for instance [13-16]). The output of the hysteresis operator is terminated by a receiver that is tuned to the bit rate of the transmitter. The receiver is composed of an integrator, which integrates the output signal from the rectangular loop over one bit duration, and this is followed by a threshold detector that makes the binary decision. At this last stage, the integrator averages the sum of the output of the hysteresis loop and a second independent noise source, which models noise in the receiver. The analysis of the system with a rectangular hysteresis loop driven by stochastic processes poses certain mathematical difficulties. This is due to the fact that the state of the hysteresis loop depends on the history of past input extrema of the input stochastic process. Fortunately, the problem of stochastic inputs to hysteretic systems has been considered previously in the somewhat unrelated problem of aftereffect modeling and viscosity (for instance, see [17-19]). In such problems, the stochastic input models thermal perturbations in metastable systems that lead the system to relax from higher energy states to lower energy states [20]. Examples of

7.2 GENERALSYSTEM DESCRIPTION AND OUTLINE

643

such problems include magnetic viscosity, aftereffects in superconducting materials, shape memory alloys, etc. In the analysis of such aftereffect problems it was shown by the author that many of the mathematical difficulties can be overcome when the stochastic inputs to the rectangular hysteresis loop are considered to be a stationary Markov process. Based on such an assumption it was shown that key calculations can be reduced to the well-known mathematical tool of 'exit problem' of stochastic differential equation (SDE) theory [21]. This mathematical tool is also known in the mathematics and physics literature as the 'first passage problem'. The mathematical tool of 'exit problem' is employed in the analysis of binary detectors with hysteresis. Employing this tool, a method is developed to compute the probability density function (pdf) of the output of the receiver integrator in terms of exit time probability density functions of the input stochastic process [9]. Owing to the nature of the calculations, it turns out that the pdf of the integrator output can be represented in terms of generating functions (Laplace transform domain). In terms of the Laplace transforms of the exit time pdfs (expressions (7.44) and (7.45)) compact closed-form expressions of the integrator output pdf are derived in the transform domain. Subsequently, it is shown that the derived expressions can be employed to compute system performance measures, such as bit error rates. The chapter is organized as follows. The characteristics of the stochastic noise process that is driving the hysteretic system will be described first. For practical application considerations and, to some extent, mathematical convenience, it will be stressed that the analysis presented here requires the input noise to the rectangular hysteresis loop to be a stationary Markov process. In the following section, calculations will be presented to compute the probability density function of the integrator output in terms of the exit time probability density functions of the input process. It will be shown that compact expressions of the receiver integrator output can be derived in terms of the Laplace transforms of the aforementioned exit time probability density functions. The preliminary analysis will be presented for the general case of any stationary Markov input process. Subsequently, a particular case will be presented for which the additive noise process is stationary, Markov and Gaussian. This process can be modeled when the non-hysteretic filter is considered to be a first-order low-pass filter driven by white Gaussian noise (AWGN). This case will be presented as a particular example where the Laplace transforms of the relevant exit time probability density functions can be explicitly calculated in terms of parabolic cylinder functions [21]. Finally, conclusions will be presented, including a brief discussion of how this work can be employed beyond the scope outlined in this chapter.

644

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

7.3 INPUT PROCESS A N D M I D - S P A N REPEATER A binary signal ~(t) in the presence of additive noise nt is considered to be the input to the mid-span repeater. Over a one-bit duration T the binary signal takes the values of zero or one. The mid-span repeater is modeled by a cascade of two filters as shown in Fig. 7.2: the first one is a nonhysteretic filter while the second one is a filter that exhibits hysteresis. The latter is modeled by a rectangular hysteresis loop with output values of zero and one. Therefore, the output of the mid-span repeater is a binary process whose value depends on the past switching history of the rectangular loop. This point will be elaborated further in the discussion below. The output of the first (non-hysteretic) filter is denoted by xt. The second stage of the mid-span is driven by this stochastic process. The analysis presented in this work requires: 9 the exit-time (or first passage time) pdfs of this process, either analytically or numerically, from semi-infinite intervals, and 9 that this stochastic is Markov and stationary around a known mean. The analysis presented in the following section is for any stationary Markov process with arbitrary exit-time pdfs. Although this approach is fairly general, there is a class of problems where a well-understood methodology exists in order to calculate the underlying exit-time pdfs of the stochastic process: the case of the so-called diffusion process of stochastic differential equation theory. In the problem considered here xt is modeled as a diffusion process, that is, the output of the non-hysteretic first-order filter driven by the superposition of white Gaussian noise and the input signal ~(t) is given as follows: dxt c Jr b(xt) = c ( x t ) | ~ ( t ) q- r 1. dt

dWt 1

|. cU _l

(7.1)

In the above stochastic differential equation, b(x) and c(x) characterize the first-order filter, r~n(X) is the measure of noise and Wt denotes the standard Wiener process. Solutions of this SDE are diffusion processes and are continuous in time. Most notably, there is a well-known particular case of this problem when the system is modeled by a first-order low-pass filter of bandwidth b. The stochastic output process of this low-pass filter xt satisfies the following first-order constant coefficient stochastic differential equation: dWt dxt r -F bxt -- bk~(t) Jr ~n - ~ - ], (7.2) d---T where cr2 - N0/2 and No denotes the spectral density of the driving white noise process. From the definition of the input data signal ~(t), the mean

7.3 INPUT PROCESS AND MID-SPAN REPEATER

645

A

7x Switch from 1 to 0

--t-0

( ~

>X

Switch from 0 to 1 FIGURE 7.3 Hysteresis nonlinearity employed in the mid-span repeater: a rectangular hysteresis loop with switching-up value a and switching-down value/3.

value of Xt is a series of pulses with rise and fall time constants of 1/b and the zero mean stochastic part is the well-known Ornstein-Uhlenbeck process. It is well known that such a stochastic process is Gaussian, Markov and stationary, according to the Doob theorem. For the Ornstein-Uhlenbeck process, well-known analytical (closed-form) solutions exist for the exittime pdfs in the transform domain. This aspect will be elaborated further in the subsequent discussions. The hysteretic filter for which the above stochastic process is an input is modeled by the rectangular hysteresis loop shown in Fig. 7.3. This rectangular hysteresis loop is characterized by the output values of one and zero, switching-up value a and switching-down value ft. Therefore, the output of this hysteresis loop, ~a~xt, is a binary stochastic process whose value depends on the random exit times of the input stochastic process from semi-infinite intervals (-cx~, a) and (fl, +cx~). Since a conventional threshold-type detector is a special case of a rectangular loop with equal switching up and down values, the analysis presented in this paper enables one to ascertain any performance gain due to hysteresis. The analysis of the hysteresis detector presented here is for arbitrary switching up and down values. However, for the particular binary signal ~(t) considered here, rectangular hysteresis loops centered on one half would be of particular interest. The output to the binary receiver is the sum of the output of the rectangular hysteresis loop and a second source of noise, ~t, that is modeled to be white Gaussian with the spectral amplitude Mo/2. This additional noise source is assumed to be uncorrelated to nt. The receiver integrates the sum of the output of the rectangular hysteresis loop and the second

646

CHAPTER7 BinaryDetection and Rectangular Hysteresis Operators

uncorrelated noise source over one bit duration T. The result of this integration is as follows: RT = [YT q- ZT], (7.3) where, YT--

f0T~aflxtdt

and

ET--

f0T~tdt.

(7.4)

As shown in Fig. 7.2, once this integration is performed over one bit interval a threshold detector makes the decision: zero if the integrator output RT is less than T/2 and one if the integrator output is greater than T/2. Therefore, in order to determine the performance of the system, such as probability of error, the probability density function of the integrator output RT needs to be computed. Since nt and ~t are uncorrelated stochastic processes, YT and ET are independent random variables, as well. Therefore, the pdf of RT can be determined from the convolution of the pdfs of the random variables TT and ET: ~'DRT "- ~'DyT * ~'D~T. (7.5) The probability density function of the latter is straightforward since, from (7.4), this random variable is normal with zero mean and variance TMo/2. Therefore, the remaining problem in the evaluation of the system performance is the computation of the pdf of YT.

7.4 RECEIVER I N T E G R A T O R OUTPUT: PROBABILITY DENSITY FUNCTION CALCULATION Let us consider the general problem of an arbitrary stochastic process as input to a rectangular hysteresis loop shown in Fig. 7.3. Since the output of this rectangular loop is only one or zero at any given time, it is clear from expression (7.4) that O
Y T < U +du},

if O<,u<~T, elsewhere.

(7.6)

In order to calculate this probability density function, it is useful to know the probability distribution function of the state of the rectangular hysteresis loop. Let us define the probability distribution of the state of the rectangular hysteresis loop employing the following notation:

q;(x, 7; t)dx - Pr{x

< Xt

< x + dx, ~flxt = 7 }.

(7.7)

7.4 RECEIVERINTEGRATOR OUTPUT

647

Let (x0, ~ x 0 ) denote the state of the rectangular hysteresis loop at the start of the current bit interval 0~
+~ 8vyT(u) - f

oo

1 dx E

q0(x, 7; 0) ~VrTl(Xo,~xo ) (UlX, 7)"

(7.8)

7=0

Here, ~DyTI(Xo,~a~Xo) denotes the conditional pdf of YT when the initial state of the rectangular hysteresis loop is (x0, ~/~x0). In order to compute the conditional pdf of YT for a given initial state (x0, ~a~x0), one must find the pdf of all input sample trajectories starting from a given initial state under the constraint that the output of the rectangular hysteresis loop is equal to one for a total duration of u and zero for a duration of T - u during one bit interval of duration T. This is the most general definition of the problem. In order to solve the problem, several basic observations can be made. First, the aforementioned constraint on the input sample trajectories may be satisfied for an arbitrary number switchings of the rectangular hysteresis loop. Second, the spaces of sample trajectories of different numbers of switchings are disjoint events. Employing these two observations, it is possible to calculate the conditional pdf of YT for a given initial state in terms of the following sum of probabilities:

oo ~DyTI(Xo,~;aflXo)(U) -- ~

n=O

" (n) ~,9yTl(xo,~aflxo ) (U),

(7.9)

where

(n) rrl(xo,?~xo) (u)du -- Pr { u < YT < u + du, n switchings } of ~/~/~in [0, t] I (xo, ~/~/~xo) "

(7.10)

Therefore, from expressions (7.6)-(7.10), it can be observed that the computation of 69yT is reduced to the computation of ~'J(n) Yzl(xo,7a~xo)" (u)du, the probability of u < YT < u + du for a given initial state ~al~x0with n switchings of the rectangular hysteresis loop. In the subsequent discussions it will be shown that this probability, together with the initial distribution, can be computed by employing the machinery of the 'exit problem' and the Kolmogorov-Feller equation of SDE theory.

648

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

7.4.1 DEFINITION OF THE EXIT PROBLEM The sample trajectories of the rectangular loop output for the two possible initial values of ~/~x0 - +1 and ~/~x0 = 0 are shown in Fig. 7.4. For example, w h e n ~a~x0 = 0, the output remains at zero until the input starting at x0 exits the semi-infinite interval ( - ~ , a]. At the time instant of switching, x = a and the output switches to one. The output of the rectangular hysteresis loop remains in this state until the input exits from the semi-infinite interval [fl, + ~ ) . At this time instant of switching, x = fl and the output switches d o w n to zero. The output of the loop remains at this zero state until the input stochastic process exits the semi-infinite interval ( - c ~ , a]. From this description, it is clear that all subsequent switchings of the rectangular hysteresis loop correspond to the last two exit problems and the switching process repeats this sequence of events. The switching events for the rectangular hysteresis loop occur at a sequence of r a n d o m time instants and corresponding exit times. Consider an input stochastic process x t that results in n~>1 switchings of the rectangular loop at times 0 < tl < t2 < ... < tn < T . For such a trajectory, the following exit times can be defined: Zl ----tl, "Ci =--ti -- t i - 1 , Zn+l - T -

2<.i<~n,

(7.11)

tn.

If an input trajectory does not switch in the current bit interval then the first exit time Zl > T and n = 0 by definition. The set of exit times defined in (7.11) are summarized in Table 7.1.

A

Initial State ofl

~')r 13Xt ~

101' '171

>1'

~2

~4 I >t T

A

Initial State of O t>

___ <

"r

>]<

'~3

~]<

~4

>i

I' >t T

FIGURE 7.4 Possible trajectories of the rectangular hysteresis loop output for the initial conditions ~ x 0 = I and ~ x 0 = 0, respectively.

7.4 RECEIVER INTEGRATOR OUTPUT TABLE 7.1

"ci

649

List of Exit Times

~;aflx 0 = 0

~flx 0 = 1

D e p e n d e n t on (x 0, ~c~flx0)?

x 0 --, ~x fl +-- ~ fl --+ ~

fl ,-- x 0 ]~ --+ ~ fl +-- 0r

YES NO NO

i= 1 i/>2, even i/>3, o d d

Based on the above definition of exit times, the r a n d o m variable YT can be expressed as follows:

07

if ~aflx0 - 0 and n - 0,

y]n+l Z.i i=2, even '

if ~/~x0 - 0 and n>~1,

T~

if ~/~x0 - I and n - 0,

y]n+l i=1, odd "Ci'

if ~/~x0 - I and n~>l.

(7.12)

YT --

This expression shows that I'T can be expressed as a sum of even and odd sequences of exit times. Since the underlying input process that results in these exit times is assumed to be Markov, this set of exit times consists of mutually independent r a n d o m variables. Therefore, the probability density function of their sums can be calculated by convolution of the pdfs of all the r a n d o m variables in a given sum. Consequently, in order to compute the pdf of Y T , the pdfs of the above exit times need to be determined. In order to compute these exit-time pdfs, it will be convenient to define the following exit-time probabilities: V + (r, x) - Pr

~a/~ does not switch in time interval(t', t'

+~)l}

I Xt, -- X, ~o~fl Xt' -- 1

(7.13)

V - ( r , x) - Pr

~B does not switch in time interval(t', t' I Xt, - - X ,

+~)1

~

~ f l X t, - - 0

(7.14) Furthermore, the corresponding pdfs of these exit times, v +, can be defined as follows: V +(r, x) - fO~ v +0:,l x)d~:,I ~:>~0. (7.15)

650

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

Based on the above definitions, the probabilities for the exit times summarized in Table 7.1 can be stated as follows: P~.--xo ('0 = V + ('c, xo), Pxo~a('c) = V - ('c, xo),

P~,__~('c) = V + ('c, ~), P ~ a ( ' c ) = V - ('c, fl).

(7.16)

From expressions (7.15) and (7.16), the corresponding exit-time pdfs can be calculated from the following expressions: P~.--xo (~') = v+ (~'' xo) , P x o ~ ( ~ ) = v-(~, xo),

p~._~(~) = v + (~, ~), p~__.~(~) - v-(~, fl).

(7.17)

7.4.2 EXIT-TIME PROBABILITY IN THE LAPLACE TRANSFORM DOMAIN

As eluded to in the previous discussion, the computation of the integrator output probability density function requires the convolutions of exit-time probability density functions. Consequently, it is mathematically convenient to express such convolutions in the Laplace transform domain, which enables one to simplify some of the resulting expressions. Therefore, at this stage it is also useful to state the definitions of the corresponding probability functions in the Laplace transform domain: V•

x) -

V•

f0 ~176

x)e-S~d~,

(7.18)

and (7.19)

0 + (s, x) = 1 - sV • (s, x). 7.4.3 CASE OF THE ORNSTEIN-UHLENBECK PROCESS

The so-called Ornstein-Uhlenbeck process is employed in this section as a source of noise to the hysteresis detector filter. As is well known, this is an asymptotically stationary Gaussian stochastic process. In general, for the stochastic process given in (7.1), it is well known that the exit time probabilities satisfy the backward Kolmogorov equation (for instance, see [21]). In particular, for the Omstein-Uhlenbeck stochastic process given in (7.2) there is a closed-form solution to the corresponding backward Kolmogorov equation in the Laplace transform domain. During a time interval when the binary input signal is constant, ~(t) = z, the exit-time probabilities are the solutions of the following partial differential equation: OV+ - b ( x - z) ~ v • (b~ cgz = -~x q 2

c92V+ cgx2

(7.20) '

7.4 RECEIVER INTEGRATOR OUTPUT

651

subject to the following boundary and initial conditions: V + (~, x)Ix=/~= O, V + (~, x)IT=o- 1, fl < x < ~ , V-(r, X ) ] x = - 0,

V-(r,

x)]~=0= 1, - o e < x < c~.

(7.21) (7.22)

In the time domain this differential equation does not lead to a closed-form solution directly. In order to obtain closed-form solutions, it is possible to rewrite the above differential equation by employing Laplace transforms. Applying the Laplace transform to expressions (7.20)-(7.22) results in the following boundary value problem: (bo'n)2 d2V + 2 dx 2

~+

d~ +

b(x -

z) dx - s

19+ (s, c+) = 0,

(s, x) = -1, 1

V + (s, -t-ec) = -,

(7.23/ (7.24)

S

Here, c+ denote the exit points for the process Xt that are equal t o / / a n d c~, respectively. The solution of such a boundary value problem is well known (for instance, see [21]) and can be expressed by the following expression: l)+(s, x) - -sl [1 - e

(x-z)2-(c+-z)2~)-sIb(~-'~)1 422

(7.25)

lg_s/b ( ~ f -~) where lg_s/b is the parabolic cylinder function and 2 - br~n/~/~. One can calculate the probabilities that characterize the exit times summarized in Table 7.1 based on the solutions of the backward Kolmogorov equation. In the following, it will be shown that PYT can be expressed in terms of the convolutions of exit-time probabilities. Therefore, by enabling such convolutions to be expressed as products, it will be shown that the closed-form solutions of the exit-time probabilities in terms of their Laplace transforms are very useful. 7.4.4 PROBABILITY DENSITY FUNCTION FOR A KNOWN

INITIAL STATE The next step in the calculation is to compute the conditional pdf of TT for a given initial state in terms of the aforementioned exit-time probabilities. According to expression (7.9), this probability density function can be computed from the probability of all trajectories that result in n~>0 number of switchings of the rectangular hysteresis loop. Therefore, one needs to compute the probabilities of all sample paths that start from (x0, 9~/~x0) and result in the output of the hysteresis loop being equal to one for a total

652

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

duration of (u, u + du) during one bit interval T. For clarity of the presentation, the cases for the two different initial conditions of the rectangular loop output are analyzed separately.

Initial State of ~ x 0 - 0 The required probabilities can be determined separately for all possible input trajectories that result in n number of switchings. For n = 0 (no switching), the output of the rectangular loop remains zero during the bit duration. Therefore, it is clear that:

t;j(o) YTI(Xo,O)(u)

(7.26)

- Pxo__,.(T) 6(u)

For n = 1, the output of the rectangular loop is equal to zero during T1 and then remains one at least until the end of the bit interval. Therefore, one obtains: 8(1) YTI(xo,O)(u)du =

Pxo--*o~( T - u) P~,_~(u)du.

(7.27)

For n = 2, the output of the rectangular loop is zero during 171, one during z2 and remains zero at least until the end of the bit interval. This results in the following convolution in terms of exit-time probability density functions:

$(2) (u)du YTI(X0,0) -- ~0T-u d'ClPx0_,a(l:l) = [Px0--*~ * P~_.,~](T-

p&_~(u)du P~__,~(T- u -

1:1)

u)p~,_~(u)du.

(7.28)

For n = 3, the output of the rectangular loop is one during z2 and T4, and zero elsewhere. This results in the following two convolutions:

. (3) (u)du~JYT[(X0,0)

/0 9p f l ~ ( T -

d'ClPxo__,a(/:l)

dz2p//,__o~(T2) /0 u u - zl)du Pp,_.~(u- 1:2)

= [Px0~ * p~_,~](T-

u)[p/~,__~* P~](u)du.

(7.29)

The computations above can easily be generalized for an arbitrary number of switchings n. This can be accomplished by constraining all exit times for which the rectangular hysteresis loop is equal to one to be of duration u and constraining all exit times for which the rectangular hysteresis loop is equal to zero to be of duration T - u. Consequently, it can be shown that the following set of iterated convolutions holds for all odd and even

7.4 RECEIVER INTEGRATOR OUTPUT

653

numbers of switchings, respectively: . (n) ~OYT [(x0,0) (U) --

[

Pxo--*~ *

E'.

fl--~a * ' "

1]

* p~__,

(T-

w~ n-1 2

u)

(7.30) n--1 2

and,

rTl(X0,0)

xo~

B-*~ ~r n--2 2

9[p/~,__a,.--,..

pfl,_~](u),n=2,4,6

.....

(7.31)

n

Initial State of ~ x o

- 1

Similar to the 9~]~x0 - 0 case, the required probabilities can be determined separately for all possible input trajectories that result in n n u m b e r of switchings. For n - 0, the output of the rectangular loop remains one during the bit duration. Therefore, it is clear that:

~j(0) (u) - P/~<--xo (T) $(u - T) " YTI(Xo,1)

(7.32)

The derivations for higher n u m b e r of switchings can be done in a very similar manner to the ~a/~x0 = 0 case and result in the following iterated convolutions:

Yvl(x0,1)(u) -

P~,-xo *

~_~ ,...

,

(u)

n-1 2

(7.33) n--1 2

654

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

and, rTl(X0,1 ) ( U ) - -

Pfl~---X0 *

~+--0~* ' ' " * tOt+--

*

(U)

n-2 ---2-

9

/~Z~,..., ,

,,

p//_,~ ( T -

u), n = 2 , 4 , 6 . . . . .

(7.34)

y

tl

Reformulation in the Transform Domain From expressions (7.26) and (7.30)-(7.34) it can be seen that the conditional probabilities for an arbitrary number of switchings can be expressed in terms of iterated convolutions of exit-time probabilities p/~__,~and p/~,__a. Therefore, according to (7.9) and the above formulas, the conditional pdf for a given initial state is an infinite sum of iterated convolutions and has the following form: oo

~rTI(Xo,~a~sXo) (U) -~ E g (n) (T - u)h (n) (u)

(7.35)

n--O

where the functions g and h denote the iterated convolutions of exit-time probability density functions. Clearly, computations based on the above expression are not practical, since this is an infinite sum of iterated convolutions. However, it turns out that this difficulty can be overcome by employing the two-dimensional Laplace transform of the above conditional probability density function. Consequently, the above sum of iterated convolutions takes the following form in the transform domain: oo

~(x0,~/~x0)($1, $2) -- E ~(n)($2) ~/(n)(S1 if- S2) n=0

(7.36)

where

~'D(Xo,~)aflXo) ($1,

$2)

= f / ~'DyTl(XO,~)o~flXO ) (u)e-(SlU+S2T)du dT

(7.37)

is the two-dimensional Laplace transform of the conditional probability density function and ~ and h denote the Laplace transforms of the respective functions. Applying the two-dimensional Laplace transformation to formulas (7.26) and (7.30)-(7.34) results in the following

7.4 RECEIVER INTEGRATOR OUTPUT

655

expressions: ~(0) (xo,O)(S1, $2) --/~x0__~(S2)

(7.38)

(xo,o)(sl,s2)-

(s2)

_ 1 (S1 9 [ ~ f l , _ _ = ]n--1 T pfl,__=

[

~(n)

(xo,O)(S1, S2) =

9

(7.39)

if- S2), n = 1, 3, 5 .....

n-2

~ ]

/9x0___~c~[/9//__~0rT Pfl--,c~ (s2)

H fl,--o~

(81+ s2),

(7.40)

n = 2, 4, 6 ....

and ~(o) (xo,1) (S1, S2) -- Pfl+--xo (s1 "4- $2) ,

(xo,1)(S1,S2)--

9

(7.41)

/9//,__xo[/9//,__ct]~-! (Slff-S2)

[ ill,ft._.O~ _]>n~l~P#__,~ ] (s2),

(7.42)

n - 1, 3, 5 .....

~(n) [ n~2 ] (x0,1)(SI' S2) -- /gfl~_xo[/gfl,__0r P//,--0r (s1 if- s2) 9[/gfl___~a]~~ ~ (S2), n -- 2, 4, 6 .....

(7.43)

By e m p l o y i n g expressions (7.38)-(7.43), it is n o w possible to express the Laplace transform of the conditional probability density function given in (7.36) in a more compact form. For the case of the initial state ~flxo - 0, it can be s h o w n that:

~(x0,0) (S1, S2) -- Px0__~c~(s2) /gxo__,c~(s2) q- [i~fl,__~(Sl -}-82)if- Pfl__,o~(sa)Dfl,__o~(s1 -ff 82)] +

1

-- Dfl__,o~(S2)~)fl,__o~(S1 -Jr-S2)

" (7.44)

656

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

Similarly, for the case of the initial state ~ x 0 - 1, one obtains: ~(x0,1) ($1, $2) --

Pfl*--xo($1 -}- $2)

(Sl q- 82)if- -[Pfl '*~162 if- Pfl*--~ +

if-s2)[gfl-~a(s2)la

1 -- p//__,~(S2)p]/~__~(S1 q- $2)

(7.45) As a result, it can be seen that in the transform domain these expressions provide a closed-form solution of the conditional probability density function of YT for a given initial state (x0, ~px0). In this form, the Laplace transform of the corresponding conditional probability density functions can easily be computed for any process, xt, for which the corresponding exit-time probability density functions are known, either analytically or numerically. The closed-form solutions of exit-time probabilities given in (7.25) can be used directly in the particular case of the Ornstein-Uhlenbeck process. 7.4.5 INITIAL STATE DISTRIBUTION

As can be seen from expression (7.8), the probability density function of YT is given in terms of its conditional pdf for a given initial state and the probability distribution of this initial state ~. Therefore, the next step in the computation of the probability density function of YT is to determine the initial distribution by employing the machinery of the 'exit problem' developed above. In this analysis it will be assumed that the input stochastic process to the rectangular hysteresis loop is continuous in time. Indeed, this assumption is naturally satisfied for the diffusion process defined by the SDE given in (7.1). Based on this property, it can easily be shown that the joint probability density function satisfies the following initial condition: ~(x, 7; 0+) = ~o(x, 7; 0-). (7.46) For clarity, it will be assumed that the system reaches a steady state by the end of each bit interval. Therefore, the joint probability density function of the state of the rectangular hysteresis loop at the end of the previous bit interval, q0(x, 7; 0-), may be approximated to be the steady-state probability density function for the prior bit. Since the steady-state joint distribution function is of interest here, one can begin the derivation from an arbitrary initial state (xt0, ~flXto) at some time to << 0. Without loss of generality, Xto < a and ~o~Xto - 0 can be chosen as an initial state. Next, consider some time interval [to, t] that is within the duration of the previous bit interval and let n denote the number of times the

7.4 RECEIVERINTEGRATOR OUTPUT

657

rectangular hysteresis loop switches in this time interval. Since the space of sample paths of the stochastic process can be divided into sets of disjoint events where sample paths have different numbers of switchings it is easy to show that oo

q~(X, 7~ t) -- E

q~(n)(X, 7~ t),

(7.47)

n--0

where

q~(n)(x, 7; t ) d x =_ P r

x < x t < x nt- d x , ~;o~fl x t - 7, } n switchings of ~ in [to, t] "

(7.48)

Furthermore, it is also clear that all sample paths with zero or even numbers of switchings result in the hysteresis loop being in the '0' state, and all sample paths with odd numbers of switchings result in the hysteresis loop being in the '1' state. Consequently, the joint probabilities can be expressed as follows: (x)

(p(x, 7 - O; t) -

E q~(n) (x, ? - 0; t), n=0, even oo

~(x, ? - 1" t ) -

E

q~(n)(X, ~; -- 1; t).

(7.49)

n=l, odd

Green's Function Formulation It is clear that in order to compute the above expressions it is necessary to compute the probability density function of the input xt conditioned on the premise that it does exit the semi-infinite intervals [fl, +cx~) and (-cx~, ~]. Towards this end, let ~b+ and ~b- denote these conditional probability density functions, respectively. Naturally, these functions depend on the characteristics of the input stochastic process xt. In the case of a diffusion process (7.1), there is a well-established mathematical procedure to calculate these conditional probability density functions. For the particular case when the system is modeled by a first-order low-pass filter given in (7.2), the aforementioned conditional probability density functions satisfy the following differential equation: C9C~+

c9"c -

-b(x

- z) c9c~+

- ~ x -}

(bo_n)2 ~2(~+ 2 c~x2 ,

(7.50)

658

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

subject to the boundary and initial conditions: O+(z, x; X')lx=/~= 0,

(7.51)

~b+(~, x;

(7.52)

X !

)[3=0 = c~(x - x ! ), / / < x, x ! < ~ ,

~b- (z, x; x')Ix=~ - 0, ~b-(~, x; x !) I3=0 =

(7.53)

6(x -

x !), - ~

< x, x ! < :r

(7.54)

where x' denotes the value of the input at the beginning of the interval. Based on this definition, it is clear that 4~+ is Green's function of the backward Kolmogorov equation. Similar to the 'exit problem' formulation, its solution is greatly facilitated by employing the Laplace transform. Applying the Laplace transform to equations (7.50)-(7.54), one obtains:

(bc~n)2 d2~ + 2 dx 2

b(x-z)dgP+

s dP+(s, x) = - c ~ ( x - x 1,

~+(s, c+; x') = 0, where ~+(s, x; x') :

Loo ~+ (~, x; x')e-s'~dr.

(7.55) (7.56)

(7.57)

Here, c+ are the exit points for the process xt, which are equal to fl and a, respectively. The solution of the above differential equation is the summation of the homogeneous and particular terms. The homogeneous solution is given in terms of parabolic cylinder functions (for instance, see [18,19,21]). The particular solution (in the time domain) is simply a Gaussian pdf with mean z 4- (x' - z)exp [-b~], and variance a2/2b[1 - exp (-2b~)]. Consequently, the overall expression for the pdf can easily be found by combining the homogeneous and particular solutions and imposing the boundary condition (7.56). In the Laplace transform domain, this results in the following expression:

~•

~ 4x; x') = ~p (s, x;

X'

~'4- ,X'[ (x-z)24"-2(~+-z)a'l~)-s/b(X~--~z)] --(~p (S, C+" ) e .D_slb(~ )

(7.58)

In the above expression, ~p denotes the Laplace transform of the particular solution, whereas the expression in the square bracket is the Laplace transform of the homogeneous solution.

659

7.4 RECEIVER INTEGRATOR OUTPUT

Probability for a Given N u m b e r of Switchings The next step is to compute the probability for n number of switchings of the rectangular hysteresis loop in the time interval [to, t]. In this section, only the case of ?~Xto - 0 will be considered. It is clear that the derivation for ~/~Xto - I can be done in a similar manner. For n = 0, the output of the rectangular loop does not switch and remains at zero. Consequently, it is clear that: (p(0) (x, 0;

t ) d x - Pr

{ x<xt<xq-dx,~flxt-O]

}

no switching in [to, t]

9Pr{no switchings in [to, t] }.

(7.59)

By employing the definitions of Green's function and exit-time probabilities, one obtains the following expression: (p(~

- to, x; Xto)Pxto_,a(t - to)

0" t) = r

- [q~-(-,

x" Xto)Pxto__,~](t- to).

(7.60)

For the case of n = 1, the rectangular loop switches after some random exit time rl and remains in the '+1' state, while the input exits the interval (-cx~, a] and remains in the interval [fl, cx~) at least until time t. Therefore, the joint probability for this case can be expressed as follows: (p(1) (x, 1; t)

-

fo t-to d'clV-('cl, 9r

Xto)V+(t - to

-

"Cl, c~)

( t - to - "cl, x; a)

= Pxto--*a * [P/~,--a~b+( " x; c0](t - to).

(7.61)

For the case of n = 2, the rectangular loop switches to one and back to zero after some random exit times 'cl and "c2, respectively. The input remains in the ( - o c , ~] interval and the output remains in the zero state thereafter. The probability of this event can be computed as follows: (p(2) (x, 0; t)=

~0 t-tO

dr1

~0 t-tO-r1

9[ V - ( - , f l ) r

dr2v- ('cl,

Xto)v+ ('c2, 0r

x; f l ) ] ( t - to - 'cl

= Pxto--,a * Pfl.--~ * [ P f l ~ r

-- "Ca)

x; f l ) ] ( t - to).

(7.62)

Similar to the above formulations, for n = 3 the rectangular loop switches at three random exit times. After the third switching the input remains in the interval [fl, ec) and the output remains in the '+1' state. The

660

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

probability of this event is as follows: r

(X, 1; t)=

~ot-toft-to-zlft-to-~l dzl

dz2

dO

-z2 dO

d1:3

9v- (1;1, Xto)V + (1;2, ~)v- (1;3, ]~) 9Iv + (., ~)~+ (., x; or

to - 1;1 - 1;2 - 1;3)

= Pxto__,~ * Pfl,_~ * P f l ~

9 [P/~,__a~b+ (., x; a ) ] ( t - to).

(7.63)

From the above formulation, it is clear that all even and odd terms can be obtained recursively by the following convolutions: r

(X, ~;; t) = [p/~,__~9 p/~a] 9 r

~/; t), n~2.

(7.64)

In the Laplace transform domain, this results in the following recursive relationship:

~(n) (X, ~; S) -- [pfl~_ccpfl__,~]~p(n-2) (X, ~/; S), n~2.

(7.65)

Calculation of the Joint Probability Density Function in the Transform Domain In summary the joint probability density function of the state variables (the input and output of the rectangular hysteresis loop) can be expressed by the summations given in expression (7.49). Furthermore, each term in these summations is given in terms of the iterated convolutions (7.60)(7.64). However, it is quite cumbersome to evaluate the joint probability density function in the time domain. This difficulty can be rectified by using Laplace transformations where these summations can be expressed in a more compact form. Indeed, by employing the recursive relationship (7.65) and applying the geometric summation to all common even and odd terms, a closed-form expression can be obtained in the Laplace domain. The summation of the even terms results in the following: ~(X, 0; S) --

[pxtO__,~(s) * ~

(S, X; Xto)

]

PXto_+ ~ (S)/9/J~___~ (S) + [ P I ~ ( s ) 9 ~-(s, x; fl)]l 7 ~]~,_~(s-----~~5,----~(s)'

(7.66)

whereas the odd summation results in: q~(x, 1 ; s ) = [P/~,__~(s). ~+(s, x; a ) ] ' l -

79~.__~(s)~~(s) 9

(7.67)

7.4 RECEIVERINTEGRATOR OUTPUT

661

Consequently, the Laplace transform of the joint pdf can be expressed in a compact form and can be computed in a straightforward manner in terms of the closed-form solutions for the exit-time pdfs. Since it is assumed that the system reaches steady state by the end of a bit interval, one is actually interested in the asymptotic value of the joint probability density function. Therefore, by employing the final-value theorem, one can compute the joint probability density function of the initial state from expressions (7.66) and (7.67), as follows: (p(x, 3;; 0+) - (p(x, 3;; 0-) ~ lim sO(x, 3;;s), s---,0

(7.68)

7.4.6 THE PROBABILITY DENSITY FUNCTION OF INTEGRATOR

OUTPUT It is now possible to compute the probability density function of TT and RT in the transform domain. By applying the two-dimensional Laplace transform defined in (7.37) to expression (7.8), it can be found that:

~VYT(Sl, S2) --

oo

dx0 (p(X0, 0; 0)~(x0,0)(S1, $2) +oe

+

ffl

dx0 q0(x0, 1; 0)~(x0,1)(S1, S2).

(7.69)

Here, the two-dimensional probability density function of YT in the transform domain is defined as follows"

~VyT(Sl,S2) ~ f f ~yz(U)e-(SlU+S2T)du dT.

(7.70)

Hence, we have shown that the probability density function of YT can be computed in the transform domain by combining expressions (7.44), (7.45) and (7.66)-(7.69). As described earlier, the input to the binary receiver is the addition of two independent random variables: YT and a Gaussian random variable ET. Therefore, according to Eqns. (7.3) and (7.5), the relevant probability density function for the decision rule is simply the convolution of the probability density functions of these two random variables. In other words, the two-dimensional Laplace transform of the probability density function of RT is given by the product of the two-dimensional Laplace transform ~Syr (Sl, s2) with the one-dimensional Laplace transform of the probability density function of ET"

~RT(S1,$2) -- ~yT(S1,

S2)~,T(S1).

(7.71)

662

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

Consequently, the probability density function of the integrator output RT can be found by the following two-dimensional inverse Laplace transform of this product:

g~RT(U)-- //~TT(S1, S2)~ET(S1)e(SlU+S2T)ds1 ds2.

(7.72)

It is worthwhile to note here that by only taking the one-dimensional inverse Laplace transform in terms of the variable s2 and substituting Sl = jco results in the so-called generating function of the random variable RT.

7.5 S U M M A R Y In summary, it has been shown that the probability density function of the integrator output ~VRr can be calculated from Eqns. (7.44), (7.45), (7.66)-(7.69), (7.71) and (7.72). Once this function is determined, it is possible to calculate various system performance measures for this system. For instance, the bit error probability of the system can be calculated as follows:

Pr {Error} = Pr {~ -- 0}

/T+OO /2 ~RTI~=o(U) du+Pr {~ -

f T/2

1} e-c~ ~