Preisach model of hysteresis¶

A rate-independent hysteresis model that represents path-dependent output as a weighted superposition of two-threshold relay operators retaining state between threshold crossings.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11452
Domain group: Natural Sciences
Origin domain: Physics
Subdomain: Magnetic Hysteresis → Physics

Core Idea¶

The classical scalar Preisach model represents rate-independent hysteresis as a weighted parallel superposition of elementary two-threshold relays, or hysterons. Each relay switches into one state when the scalar input crosses an upper threshold and into the other when the input crosses a lower threshold. Between thresholds it retains its previous state, so present output depends on input history rather than current input alone.

Threshold pairs occupy an ordered half-plane, conventionally written α ≥ β, called the Preisach plane. A density μ(α,β) weights the contribution of each relay. In continuous form the output is an integral of the relay states over that plane; a discrete implementation approximates the integral with a weighted grid.

The model began in magnetic hysteresis but functions more broadly as a phenomenological hysteresis operator. A fitted density can reproduce major and minor loops without proving that a material literally contains independent binary domains corresponding to every relay. The classical operator is rate-independent: traversing the same input path at a different speed should not change its ideal output path. Dynamic, vector, and other modified Preisach models extend this boundary.

Structural Signature¶

Scalar input history supplies the ordered trajectory that drives switching.
Two-threshold relay provides elementary bistable memory.
Preisach plane indexes admissible upper/lower threshold pairs.
Density or discrete weights set each relay's output contribution.
Parallel superposition aggregates elementary states into macroscopic output.
Memory state records which relays remain on or off after previous crossings.

A single present input value is insufficient to determine output. The relay-state boundary created by past extrema is part of the model state.

What It Is Not¶

The Preisach model is not a memoryless nonlinear response curve. Two visits to the same input can yield different outputs after different histories. It is not one Schmitt trigger or relay; the distributed weighted superposition is constitutive.

It is not automatically a microscopic theory of magnetic domains, a guarantee of thermodynamic consistency, or a model of rate-dependent loss. Prandtl–Ishlinskii models commonly superpose play or stop operators and form a neighboring hysteresis family with different elementary operators.

Scope of Application¶

Classical and modified Preisach models are used for magnetic materials, piezoelectric and magnetostrictive actuators, shape-memory and mechanical systems, porous-media retention, soil behavior, and control compensation. Literal use requires an input–output relation with repeatable path dependence that the chosen relay family can approximate.

Identification requires histories rich enough to excite relevant threshold regions. Extrapolation beyond measured amplitudes or reversal sequences is hazardous. If loop shape changes substantially with input rate, temperature, aging, or unmodeled internal variables, the classical density alone is inadequate.

Clarity¶

A clear model states threshold orientation, relay output levels, initial state, integration domain, density units or normalization, input history, and output scaling. Because authors swap the symbols assigned to upper and lower thresholds, inequalities and switching directions must be defined rather than inferred from letters.

“Preisach fit” should also identify whether the implementation is scalar, vector, discrete, dynamic, analytic-Everett, or another variant. Good agreement on one major loop does not demonstrate correct minor-loop memory.

Manages Complexity¶

The model compresses distributed path memory into a population state over the Preisach plane. Instead of storing the entire input time series, an implementation updates the relay configuration as thresholds are crossed and sums its weighted state.

This representation can reproduce nested loops and reversal memory with simple local updates. The cost is a potentially large density-identification problem and a phenomenological parameter surface whose physical interpretation may be limited.

Abstract Reasoning¶

Define the scalar input, output, rate range, and initial or demagnetized state.
Choose a consistent upper/lower threshold convention and admissible half-plane.
Specify relay switching and retention rules.
Identify a nonnegative or otherwise justified density from sufficiently rich loop data.
Update relay states under each input crossing and integrate or sum weighted outputs.
Validate major loops, minor loops, reversal behavior, and unseen histories separately.
Add dynamic or vector structure only when residual behavior demonstrates the classical boundary has failed.

Knowledge Transfer¶

The operator transfers among domains when a scalar input drives repeatable, approximately rate-independent hysteresis that can be decomposed into weighted relay memory. Thresholds and density acquire domain units, but the update structure remains the same.

Transfer fails when a fitted loop is merely visually similar or when dynamics, vector direction, stochastic switching, drift, or energetic constraints dominate. The graph presently offers no validated hysteresis-operator genus, so the entry retains root placement.

Cross-Domain Echoes¶

See how this entry connects to another domain.

The same reading can mean two different states

Examples¶

Canonical¶

A magnetic field rises and falls. Relays switch as their upper and lower thresholds are crossed, and the identified density sums their states into a magnetization loop. After a reversal, relays between the new and prior extrema retain a memory boundary.

Mapped back: input → field history; relay → elementary switch; plane → threshold pairs; density → identified magnetic weights; superposition → magnetization; memory → on/off population.

Applied / In Practice¶

A control engineer discretizes the Preisach plane for an actuator, identifies weights from major and first-order reversal data, and updates relay states online to predict output under command reversals. Rate-dependent residuals are tracked separately rather than absorbed silently.

Mapped back: input → command; grid → finite threshold pairs; weights → identified coefficients; state → online relay map; output → predicted displacement; boundary → dynamic residual.

Structural Tensions¶

Phenomenological fit versus physical interpretation. A density can reproduce loops without representing literal microstructure. Diagnostic: Is μ an operator parameter or a claimed population of physical domains?

Expressive memory versus identification burden. A fine plane captures rich loops but requires informative excitation and regularization. Diagnostic: Which threshold regions are constrained by the data?

Rate independence versus dynamic behavior. Classical relays remember path but not traversal speed. Diagnostic: Does the response change when the same input path is run at another rate?

Structural–Framed Character¶

Preisach Model of Hysteresis is strongly structural as an operator construction. Its domain framing lies in the physical input, output, density identification, admissibility constraints, and validation regime.

The model is descriptive and predictive within a calibration envelope. Its relay representation should not be upgraded into ontology without independent physical evidence.

Structural Core vs. Domain Accent¶

The core is input history → threshold-relay states → weighted parallel sum. Magnetism and other application domains supply field variables, observable responses, physical constraints, and fitting data.

Remove state retention and the model becomes memoryless. Remove distributed weights and only one relay remains. Add essential rate dependence without modifying the operator and the classical identity no longer fits the phenomenon.

This entry is a kind of Formal Model.

Approved unparented root. No live hysteresis-operator node is a verified immediate parent.
Memory is realized by retained relay states.
Threshold governs state transitions.
Superposition aggregates weighted elementary outputs.
History dependence distinguishes output from a single-valued static function.

Relationships to Other Abstractions¶

Current abstraction Preisach model of hysteresis Domain-specific

Parents (1) — more general patterns this builds on

Preisach model of hysteresis is a kind of Formal Model Domain-specific

It is a mathematical operator model of hysteresis.

Hierarchy path (1) — routes to 1 parentless root

Preisach model of hysteresis → Formal Model → Representation → Abstraction

Neighborhood in Abstraction Space¶

Preisach model of hysteresis sits in a sparse region of the domain-specific corpus (97^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Memoryless nonlinearity: one output for each current input.
Single relay or Schmitt trigger: one elementary component, not the distributed model.
Prandtl–Ishlinskii model: typically superposes play or stop operators.
Dynamic Preisach variant: adds rate-dependent state or parameters.
Literal domain population: physical interpretation not guaranteed by fit.

References¶

D. Carbone and D. C. Jiles, “Review of Play and Preisach Models for Hysteresis in Magnetic Materials,” Materials 16 (2023): https://pmc.ncbi.nlm.nih.gov/articles/PMC10051722/
Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Preisach_model_of_hysteresis
Ferenc Preisach, 1935 original model (bibliographic provenance retained in the discovery revision).

The repair normalizes threshold orientation, separates the classical scalar operator from variants, and avoids treating relay density as literal microstructure.