Preisach

The Preisach model is a mathematical framework for describing hysteresis, a memory effect where the output depends on the history of the input. It was introduced by F. Preisach in 1935 to describe magnetization in ferromagnetic materials and has since become a general tool for rate-independent hysteretic systems, including ferroelectrics and smart actuators.

The model represents a system as a continuum of elementary two-state memory elements, called relays or hysterons. Each hysteron is characterized by a pair of thresholds (α, β) with α ≥ β. The collection of all such hysterons forms the Preisach plane. Each hysteron has a state sαβ that takes values +1 or −1 and toggles when the input u(t) crosses its thresholds: the relay outputs +1 when the input is above its upper threshold α, −1 when it falls below its lower threshold β, and retains its previous state when the input lies between β and α.

The overall output y(t) is obtained by integrating the states of all relays, weighted by a Preisach distribution μ(α, β): y(t) = ∫∫ μ(α, β) sαβ(t) dα dβ, over the region α ≥ β. The distribution μ encodes the density of hysterons with given thresholds and is often determined from experimental measurements, for example via the FORC (First-Order Reversal Curve) technique.

Applications of the Preisach model span magnetic hysteresis, ferroelectric and piezoelectric materials, and various smart-material systems. It provides a flexible, phenomenological description of history-dependent behavior and underpins many control and modeling approaches for devices exhibiting hysteresis. Extensions include dynamic variants and non-ideal relay formulations to capture rate effects or asymmetries.