An Analytical Generalized Prandtl–Ishlinskii Model Inversion for Hysteresis Compensation in Micropositioning Control

TLDR

Micropositioning smart actuators suffer from strong, often asymmetric hysteresis that degrades positioning accuracy. This work formulates an analytical inverse of a generalized Prandtl–Ishlinskii model to compensate such hysteresis. The authors extend the classical model to a continuous generalized form, derive its inverse using the classical inverse and envelope functions, and validate it against measured data and simulations for magnetostrictive, SMA, and piezo actuators. Simulations and preliminary experiments show the inverse model effectively mitigates asymmetric and saturated hysteresis, enabling real‑time feedforward compensation for magnetostrictive, SMA, and piezo micropositioning stages.

Abstract

Smart actuators employed in micropositioning are known to exhibit strong hysteresis nonlinearities, which may be asymmetric and could adversely affect the positioning accuracy. In this paper, the analytical inverse of a generalized Prandtl-Ishlinskii model is formulated to compensate for hysteresis nonlinearities of smart actuators. The generalized model was modified to ensure its continuity, and its validity in characterizing different hysteresis properties is briefly demonstrated by comparing the model responses with the measured data for the magnetostrictive, shape memory alloys (SMA), and piezo micropositioning actuators. Since the proposed generalized model is a mere extension of the analytically invertible classical Prandtl-Ishlinskii model, an inverse of the generalized model is formulated using the inverse of the classical model together with those of the envelope functions of the generalized play operator. The effectiveness of the inverse of the generalized model in compensating for the symmetric and asymmetric saturated hysteresis effects is subsequently investigated through simulations for a magnetostrictive and a SMA actuators, and through preliminary experiments performed on a piezo micropositioning stage. The simulation results suggest that the inverse of the generalized Prandtl-Ishlinskii model can be conveniently applied as a feedforward compensator to effectively mitigate the effects of the asymmetric and saturated hysteresis in magnetostrictive and SMA actuators. The experimental results further revealed that the proposed generalized analytical inverse model can be conveniently implemented as a real-time feedforward compensator to compensate for hysteresis nonlinearities of a piezo micropositioining stage.