Representation of Hysteresis with Return Point Memory: Expanding

the Operator Basis

Gary FriedmanDepartment of Electrical and Computer

EngineeringDrexel University

Hysteresis forms

H

M

Dint

Dave

Form most frequently associated with hysteresis: magnets

Ratchets, swimming, molecular motors, etc.

Return Point (wiping-out) MemoryThe internal state variables return when the input returns to its previous extremum.

Wipe-out property (1500, 800 Oe) at room T(@294K)

-8.00E-03

-4.00E-03

0.00E+00

4.00E-03

8.00E-03

-3.00E+03 -2.00E+03 -1.00E+03 0.00E+00 1.00E+03 2.00E+03 3.00E+03

H (Oe)

M (

emu

)

Experimentally observed in: magnetic materials, superconductors, piezo-electric materials, shape memory alloys, absorption

Also found in micro-models: Random Field Ising Models (with positive interactions), Sherrington - Kirkpatrick type models, models of domain motion in random potential,

How can we represent any hysteresis with wipe-out memory in general?

Can we approximate any hysteresis with wipe-out memory?

Preisach model represents some hysteresis with wipe-out memory because each bistable relay has wipe-out memory. It also has the property of Congruency which is an additional restriction

u

u

1

dduuHf ˆ,ˆ

Congruencyf

u

f

u

Any higher order reversal curve is congruent to the first order reversal curve. All loops bounded between the same input values are congruent.

Higher order reversal curves could, in general, deviate from first order reversal curve. These deviations can not be accounted for in the Preisach model.

Examples of systems with Return Point Memory, but without Congruency

u

Interacting networks of economic agents

ddfuuHf ˆ,ˆ

Mean-field models in physics

Theorem: as long as interactions are positive, such systems have RPM (Jim Senthna, Karin Dahmen)

Problem: Not clear if or when model unique model parameters can be identified using macroscopic observations

Mapping of history into output of the model

I

0

0

u f I

Any hysteresis with wipe-out memory can be represented by a mapping of the interface function into the output

I

(Martin Brokate)

How can an approximation be devised?

I 1

I 2

H H Cu u u u1 21 2 I I1 2 H H Cu u u u1 21 2 I I1 2

Assume both, the given hysteresis transducer and the approximation we seek are sufficiently smooth mappings of history into the output

Nth order approximation

Let H H HN N N N ( )I I I( ) ( ) ( )

(N)

N(N)(N)NN

(N)(N)NN

(N)(N)NN

CCHHHH

HHHH

HHHHHH

IIIIII

IIII

IIIIII

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆˆˆ

)()(

)()(

)()(

N

12

1NN I ( )N

I

CCif NN

~lim

0ˆˆlim

N

NHH

Building Nth order approximation

ikki ,

niiin 1,

nnnn 1111

iku

ki1

1

111

ˆˆˆn

i

n

iiiiin

Key point: as long as operators are functions of elementary rectangular loop operator, the system retains Return Point Memory

“Matryoshka” threshold set

Higher order elementary operators

11 2

2

2211ˆˆ

12ˆ 2211

ˆ

221 1 1

2

212211122112ˆˆˆˆˆˆˆ

1

1

2

1

iiiiii

Second order elementary operator example

Why use only “Matryoshka” threshold sets?

11,

22 ,

22u

ii

11

22u

2211ˆˆ

Non-”Matryoshka” operators can be reduced to lower order “Matryoshka” operators

Nth order Preisach model

N

N NNN duuP

ˆˆ

uPNˆ

u

Loops appear only after Nth order reversal.

Reversal curves following that are congruent to Nth order reversal as long they have the same preceding set of first N reversals

Nth order approximation

Due to first order Preisach model

Due to second order Preisach model

N

kkN uPuH

1

ˆˆ uH N

ˆ

u

Conclusion• As long as the hysteretic system with RPM is a

“smooth” mapping of history, it is possible to approximate it with arbitrary accuracy on the basis of higher order rectangular hysteresis operators. It is a sort of analog to Taylor series expansion of functions;

• Nth order approximation satisfies Nth order congruency property which is much less restrictive than the first order congruency property