*4.3. Discussion*

The classical Bouc–Wen model is mainly used to characterize the rate-independent hysteresis, but cannot characterize the rate-dependent hysteresis precisely. Compared with the classical Bouc–Wen model, the generalized Bouc–Wen model integrates relaxation functions. *k*(*u*(*t*), . *u*(*t*)) and *<sup>α</sup>*(*u*(*t*), . *u*(*t*)) based on the traditional fixed model parameters *k* and *α*. The relaxation functions are determined based on experimental characteristics instead of random imaginations and make model parameters closely related to the rate of input . *<sup>u</sup>*(*t*). Therefore, the proposed model can theoretically describe the rate-dependent hysteresis behaviors more precisely and the experimental and simulation results demonstrate its effectiveness. In addition, the experimental and simulation results also show that the generalized Bouc–Wen model can describe the rate-independent hysteresis behaviors more precisely than the classical Bouc–Wen model. So it can be concluded that the generalized Bouc–Wen model can characterize both rate-dependent and rate-independent hysteresis behaviors.

The enhanced Bouc–Wen model in our previous work [22] is closely related to the input frequency *f* and cannot be used to describe the rate-dependent hysteresis behaviors when the frequency is unknown. However, the generalized Bouc–Wen model is closely related to the rate of input . *u*(*t*) and can be widely used to describe the rate-dependent hysteresis behaviors without limitations. This is the main advantage of the developed model.

Compared with other existing models such as the Prandtl–Ishlinskii model and Preisach model, which are rate-independent models, the proposed model is a rate-dependent model. In addition, the proposed model has differential equations and the ability to capture an analytical form, which can provide more convenience for hysteresis compensation control. So the developed model has a broader application prospect in hysteresis modeling and compensation controls.
