Modeling Piezoelectric Actuator Hysteresis with Singularity Free Prandtl-Ishlinskii Model

Actuators like piezoelectric, magnetostrictive and shape memory alloy are gaining importance in applications involving high frequency, high precision and yet compactness is needed. The PI operator, while being able to model the hysteresis behaviour of a piezoelectric actuator well, has one major inadequacy: the inverse of the operator does not exist when the slope of the hysteretic curve is not positive definite, i.e. singularity occurs when the PI weights les 0. There is an inevitable trade-off between modeling accuracy and inversion stability. The modeling of the hysteretic loop gets better with the number of backlash operators used in the modeling. However, as the piecewise continuous interval represented by each backlash operator shrinks, there is a greater chance for the reciprocal of the PI weights to be ill conditioned, especially at the hysteretic curve turning points. Similar ill conditioned situations also arise when the actuators are used to actuate heavy loads or when operating at high frequency. This paper proposes extension to the PI operator to map the hysteresis data through a linear transformation onto another domain, where the inversion would be better behaved. The inverse weights are evaluated in this domain and are subsequently used to compute the inverse hysteresis model, which is to be used in the feed-forward controller, before the inverse model is transformed back to the original domain.