Displacement self-sensing of piezoelectric stack actuators based on extended piezoelectric equations

Piezoelectric self-sensing technology has significant advantages in space-constrained or cost-sensitive applications such as nanoscale positioning and vibration suppression. However, existing charge-based self-sensing methods typically require introducing hysteresis or nonlinear descriptions into electromechanical coupling models, leading to issues such as complex models, numerous parameters to identify, and heavy real-time computational burdens. This paper develops a structurally simple and computationally efficient displacement self-sensing model. First, extended piezoelectric constitutive equations are proposed. The core idea is to extract the purely dielectric and purely elastic terms from the electric displacement and strain responses, whereas the remaining electromechanical coupling term and its corresponding piezoelectric effect term maintain a synchronous and linear relationship, even though the latter two terms individually exhibit nonlinear or hysteretic behavior. Then, for a piezoelectric stack actuator with one end mechanically free, the displacement self-sensing model is established by appropriately simplifying the energy-loss compensation terms. This model dynamically describes the relationships among the displacement, total charge, and driving voltage. Experiments with displacement and charge measurements are subsequently conducted, and parameter identification is performed via a nonlinear least-squares algorithm. Model validation results demonstrate that the threshold average relative error (> 5%) of displacement estimation for the proposed model does not exceed 5.7% within the ranges of 0–70 V driving voltage and 1–15 Hz excitation frequency. The proposed self-sensing model offers a clear structure, few parameters, and low computational cost, ensuring strong practical utility in engineering applications.