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

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