Representation of Hysteresis with Return Point Memory: Expanding the Operator Basis

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Representation of Hysteresis with Return Point
Memory Expanding the Operator Basis

• Gary Friedman
• Department of Electrical and Computer Engineering
• Drexel University

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Hysteresis forms
M
Dave
H
Dint
Ratchets, swimming, molecular motors, etc.
Form most frequently associated with hysteresis
magnets
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Return Point (wiping-out) Memory
The internal state variables return when the
input returns to its previous extremum.
Also found in micro-models Random Field Ising
Models (with positive interactions), Sherrington
- Kirkpatrick type models, models of domain
motion in random potential,
Experimentally observed in magnetic materials,
superconductors, piezo-electric materials, shape
memory alloys, absorption
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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
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Congruency
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.
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Examples of systems with Return Point Memory, but
without Congruency
Mean-field models in physics
Interacting networks of economic agents
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
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Mapping of history into output of the model
(Martin Brokate)
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How can an approximation be devised?
Assume both, the given hysteresis transducer and
the approximation we seek are sufficiently smooth
mappings of history into the output
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Nth order approximation
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Building Nth order approximation
Matryoshka threshold set
Key point as long as operators are functions of
elementary rectangular loop operator, the system
retains Return Point Memory
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Higher order elementary operators
Second order elementary operator example
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Why use only Matryoshka threshold sets?
Non-Matryoshka operators can be reduced to
lower order Matryoshka operators
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Nth order Preisach model
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
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Nth order approximation
Due to second order Preisach model
Due to first order Preisach model
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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