Quasi-static self-sensing piezoelectric actuator position control with complex permittivity-enhanced hybrid neural network

This paper proposes a high-precision self-sensing method for piezoelectric actuators using a hybrid neural network that integrates complex permittivity information. The proposed method addresses the limitations of conventional permittivity-based self-sensing, which typically exhibits approximately 1% remaining hysteresis. A knowledge-based polynomial model is first employed to capture the primary displacement–permittivity relationship. To enhance accuracy, a neural network is then introduced to compensate for residual nonlinearity using the input voltage, permittivity, and leakage current as its inputs. Experimental validation was conducted on a one-degree-of-freedom (1-DOF) push–pull stage with a stroke of approximately 9 μ m. The results demonstrate that the proposed framework achieves a root mean square (RMS) prediction error of 9 nm accross the entire stroke. Furthermore, the trained estimator is deployed in a closed-loop proportional–integral(PI) controller for completely sensorless step positioning, maintaining steady-state errors within ± 25 nm. These results represent a significant improvement over conventional permittivity-based and other existing self-sensing approaches, confirming the effectiveness of integrating piezoelectric self-sensing with machine learning for high-precision displacement estimation and real-time control.