Mathematical Nature
Nonlinear differential equations define the relationship between input and output in systems where the response depends on the direction of change. The duhem differential model provides a phenomenological framework for describing rate independent hysteresis in piezoelectric actuators and magnetic sensors but does not account for rate dependent losses at high frequencies.
Functional Logic
Hysteresis loops are represented by a piecewise continuous function that changes its derivative based on the sign of the input velocity. Within the duhem differential model, the output is the sum of a reversible component and an irreversible component governed by the history of the signal. The slope of the curve is adjusted dynamically to match the observed damping and memory effects in the physical hardware.
Parameter Identification
Curve fitting against experimental data allows for the determination of the slope functions and the coercive parameters that define the width of the hysteresis loop. While some models require complex integration, the duhem differential model is expressed in a form suitable for real time control and signal compensation. Accuracy depends on the selection of functions that satisfy the Madelung rules for consistency in minor loops and the exclusion of non physical crossing of trajectories.
System Integration
Digital controllers use the inverse of the response function to linearize the output of non ideal sensors. This duhem differential model allows for the prediction of lag and residual displacement in micro positioning stages. Stable operation of the model requires a sampling frequency significantly higher than the frequency of the input signal.
