Discrete Preisach Hysteresis Algorithm Implementation

The Discrete Preisach hysteresis algorithm provides a computational framework for modeling hysteresis phenomena in various physical systems, including ferromagnetic materials, piezoelectric actuators, and smart materials. Originally developed for magnetic systems, the Preisach model has become a standard approach for capturing hysteresis due to its effective representation of memory effects and nonlinear responses. ### Core Concept The Preisach model represents hysteresis through the superposition of elementary hysteresis operators called "hysterons." Each hysteron exhibits binary behavior, switching between two states (typically +1 and -1) based on historical input extremes. The discrete implementation approximates the continuous Preisach plane using a finite grid structure, enabling practical computational implementation. ### Algorithmic Implementation Discretization of the Preisach Plane: The continuous Preisach plane (defined by threshold parameters α and β) is discretized into a finite grid where each cell corresponds to a hysteron. In code implementation, this typically involves creating a 2D array representation with appropriate resolution. Memory Effect Tracking: The algorithm maintains a history of input extrema since hysteresis depends on input variation history rather than instantaneous values. Programming implementation requires storing and updating reversal points in a data structure that preserves sequence information. Output Calculation: For each input value, the algorithm determines active hysterons by comparing current input against stored extrema. The total output is computed by summing weighted contributions from all active hysterons. Code implementation involves iterative grid scanning and accumulation loops, with hysteron weights calibrated from experimental data using optimization techniques. ### Applications The Discrete Preisach algorithm is particularly valuable in control systems where hysteresis introduces nonlinearity, such as precision positioning systems utilizing piezoelectric actuators. It also serves as an important tool in material science for characterizing ferromagnetic and ferroelectric behaviors through numerical simulation. ### Strengths and Limitations Strengths: - Effectively captures memory-dependent nonlinearity through systematic implementation - Adaptable to various hysteresis shapes through adjustable weight parameters - Straightforward numerical implementation using matrix operations and threshold comparisons Limitations: - Requires extensive experimental data for hysteron weight identification - Computational complexity increases significantly with finer grid discretization - Memory requirements grow with the number of hysterons in the discretized model This algorithm provides a structured computational approach for modeling systems where hysteresis plays a critical role, enabling improved control strategies and prediction capabilities in engineering applications through systematic numerical implementation.