**5. Conclusions**

Because of its simple structure and analytical inverse, the PI model are used widely in hysteretic nonlinear modeling and compensation. However, the PI model can only describe symmetric hysteresis. In this paper, based on PI model, we provide a novel PMPI model to describe and compensate asymmetric hysteresis of PEAs. First, the PMPI model is introduced, which includes the M-Play operator and memoryless polynomial. Then, the congruency property of PMPI model is analyzed and verified. The minor loop mathematical model is also established. Next, the correctness of the PMPI model is proved by simulation. It should be noted that when considering the initial loading curve, the PMPI model can accurately characterize asymmetric hysteresis. Compared with the PI model and the Gu-PI model, the error of PMPI model is reduced by 83.3%. In the end, based on the PMPI model, the I-M compensator is designed for hysteresis compensation. The stability of I-M compensator is analyzed. The experimental results show that the I-M controller has superior tracking performance.

Although the PMPI model has satisfactory results for asymmetric hysteresis, it is not suitable for rate-dependent and load-dependent hysteresis. In the future, further research is to expand the PMPI model to rate-dependent and load-dependent applications.

**Author Contributions:** Conceptualization, W.W. and J.W.; methodology, J.W.; software, R.W.; validation, K.L. and R.W.; formal analysis, Z.S.; investigation, K.L.; resources, W.W.; data curation, Z.C.; writing—original draft preparation, W.W. and J.W.; writing—review and editing, Z.C. and Z.S.; visualization, J.W.; supervision, W.W.; project administration, W.W.; funding acquisition, W.W. and B.J.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ16E050001, and the National Natural Science Foundation of China under Grant No. U1709206, No.51275465.

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