Mathematical Model
Representation of complex, path-dependent non-linearities using a collection of simple bistable operators provides an effective tool for simulating material hysteresis. The Preisach hysteresis model simulates the behavior of ferromagnetic or piezoelectric materials by integrating the responses of these individual operators over a continuous distribution. This approach is widely used to model non-linear actuator and sensor behavior.
Sensor Calibration
High-precision displacement systems based on smart material actuators must be calibrated to correct for major path-dependent positioning errors. By applying the Preisach hysteresis model, the control system calculates the appropriate input voltage required to reach a specific physical position. This calibration minimizes the tracking error of the system during cyclic operations.
Density Function
Characterizing the model requires defining a distribution function that describes the weight of each bistable operator. This density function is determined from experimental data collected during a series of nested first-order transition curves. Accurate determination of this function ensures that the Preisach hysteresis model remains valid across the entire operating range of the actuator.
Numerical Calculation
Implementing this model in real-time embedded systems requires discretization of the double integral of the operator distribution. This discrete Preisach hysteresis model is run on digital signal processors to provide fast feedforward compensation for high-speed micro-positioners. The computation must be optimized to fit within the sample period of the sensor loop, which is often less than a millisecond.
This speed is necessary to prevent phase delays that could destabilize the closed-loop control system during rapid moves. Regular checks against experimental outputs are performed to update the model coefficients as the hardware ages.
