Formalizing the Madelung Rule for Hysteresis Modeling in Piezoelectric Actuators

Piezoelectric actuators (PEAs) are significantly limited in positioning accuracy within the field of precision engineering due to their inherent hysteresis nonlinearity. To address this issue, a previously reported hysteresis description algorithm based on the Madelung rule successfully improved modeling accuracy. However, this algorithm is primarily presented in the form of flowcharts, which entails limitations such as semantic ambiguity and a lack of closed-form mathematical expressions, thereby hindering its deep integration with modern control theory. This paper aims to formalize the Madelung-rulebased description algorithm by deriving and establishing a unified Madelung analytical mathematical formula. Through a symbolic system, a rigorous and unified description of the algorithmic logic is achieved, eliminating the ambiguity inherent in flowchart representations. Furthermore, experimental validation was conducted using complex, non-periodic signals to verify the model's efficacy. Results indicate that the formalized Madelung model accurately predicts asymmetric hysteresis with a Root-Mean-Square Error (RMSE) of 0.063 μ m and a maximum absolute error within 0.176 μ m. This research overcomes the limitations of the Madelung rule, which previously remained confined to a logical level, and establishes a solid theoretical and empirical foundation for high-precision piezoelectric compensation control.