Rate-Dependent Modeling of Piezoelectric Actuators for Nano Manipulation Based on Fractional Hammerstein Model

Piezoelectric actuators (PEAs), as a smart material with excellent characteristics, are increasingly used in high-precision and high-speed nano-positioning systems. Different from the usual positioning control or fixed frequency tracking control, the more accurate rate-dependent PEA nonlinear model is needed in random signal dynamic tracking control systems such as active vibration control. In response to this problem, this paper proposes a Hammerstein model based on fractional order rate correlation. The improved Bouc-Wen model is used to describe the asymmetric hysteresis characteristics of PEA, and the fractional order model is used to describe the dynamic characteristics of PEA. The nonlinear rate-dependent hysteresis model can be used to accurately describe the dynamic characteristics of PEA. Compared with the integer order model or linear autoregressive model to describe the dynamic characteristics of the PEA Hammerstein model, the modeling accuracy is higher. Moreover, an artificial bee colony algorithm (DE-ABC) based on differential evolution was proposed to identify model parameters. By adding the mutation strategy and chaos search of the genetic algorithm into the previous ABC, the convergence speed of the algorithm is faster and the identification accuracy is higher, and the simultaneous identification of order and coefficient of the fractional model is realized. Finally, by comparing the simulation and experimental data of multiple sets of sinusoidal excitation with different frequencies, the effectiveness of the proposed modeling method and the accuracy and rapidity of the identification algorithm are verified. The results show that, in the wide frequency range of 1–100 Hz, the proposed method can obtain more accurate rate-correlation models than the Bouc-Wen model, the Hammerstein model based on integer order or the linear autoregressive model to describe dynamic characteristics. The maximum error (Max error) is 0.0915 μm, and the maximum mean square error (RMSE) is 0.0244.