**5. Conclusions**

The rate-dependence and asymmetry of the PEA's hysteresis increase the difficulty in the hysteresis modeling and compensation. Further, the PEA's hysteresis is susceptible to the system's configurations, making the hysteresis compensation of PEAs very case-sensitive. In this paper, a single-neuron adaptive hysteresis compensation method is proposed. The supervised learning and Hebb learning rules are adopted to dynamically adjust the weights of the neurons according to the error between the actual and desired trajectories and their first-order and second-order differences. As a branch of neural network control, the single-neuron adaptive control simplifies the training process of neural network control while retaining the advantages of neural network control. The learning efficiency and convergence are improved. Positioning control results show that the proposed method can reduce the steady-state tracking error to the noise level, and the transient state performance can be guaranteed. The experimental results of tracking sinusoidal and triangular trajectories with frequencies up to 50 Hz show that the proposed method can successfully compensate the rate-dependent hysteresis of the PEA. The steady-state tracking error can be maintained in a small range, showing grea<sup>t</sup> robustness and adaptability against the rate-dependence. Future work will focus on further improving the tracking performance for higher-frequency trajectories.

**Author Contributions:** Y.Q. conceived and designed the experiments; H.D. performed the experiments and sorted out the experimental results; Y.Q. analyzed the data; Y.Q. and H.D. wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported in-part by the National Natural Science Foundation of China under Grants 61873133, U1813210 and 61633012, and in part by the Fundamental Research Funds for the Central Universities, Nankai University under Grants 63191717 and 63191739.

**Conflicts of Interest:** The authors declare no conflict of interest.
