*4.2. Results*

To demonstrate the performance of the generalized Bouc–Wen (GB–W) model for the piezoelectric actuator, hysteresis behaviors in the piezoelectric actuator were measured and the classical Bouc–Wen (CB–W) model was constructed for comparison. In this case, we conducted two groups of experiments to demonstrate the effectiveness of characterizing rate-dependent and rate-independent hysteresis, respectively.

In the first group of experiments, we measured the outputs of the piezoelectric actuators under excitation voltage signals *u*(*t*) = 5 sin(<sup>2</sup>*π f t*) + 5(*f* = 5, 10, 20, 40, 60, 80, 90, 100, 110) to demonstrate the effectiveness of the GB–W model to characterize rate-dependent hysteresis behaviors. The measured data of excitation signal at 110 Hz is initially adopted to identify the parameters of both generalized and classical Bouc–Wen models. The identified parameters of the generalized Bouc–Wen model are *p* = 0.2107, *q* = 1.189e × <sup>10</sup>−5, *ε*= −0.1331, *δ* = 5.2622 × <sup>10</sup>−4, *β* = 5.3743 and *γ* = 6.4698 Meanwhile, the corresponding parameters of the classical Bouc–Wen model *k* = 0.2141, *α* = −0.3534, *β* = 3.1034 and *γ* = 3.7229. It must be noted that the input frequency should be controlled under 150 Hz to avoid a high dynamic force of encapsulated stack piezoelectric ceramics for security protection. Therefore, the nine groups of experiments need to follow this rule.

Figure 8 shows comparisons of the measured and simulation results predicted by the generalized and classical Bouc–Wen model. The black, green and red lines represent the experimental data, and the generalized Bouc–Wen model, respectively. It can be found that the simulation results predicted by the generalized Bouc–Wen model agree better with the experimental data than that of the classical Bouc–Wen model. Figure 9 shows the corresponding modeling errors of these models. It is clearly shown that the modeling errors of the generalized Bouc–Wen model is much smaller than that of the classical Bouc–Wen model.

**Figure 8.** Comparisons of the measured and simulation results predicted by the GB–W model and classical CB–W model.

**Figure 9.** Rate-dependent modeling errors of the GB–W model and classical CB–W model.

In the second group of experiments, two different waveforms of input excitation signals with the amplitude of 10 and 20, respectively, were adopted to actuate the 1-DOF compliant mechanism. These experiments are used to demonstrate the effectiveness of the GB–W model to characterize rate-independent hysteresis behaviors. The parameters of the GB–W model and CB–W model remained the same with the first group of experiments. Figures 10 and 11 show comparisons of the measured and simulation results predicted by the GB–W model and CB–W model. It can be found that the predicted results by the GB–W model agree better with the measured than that by the CB–W model.

**Figure 10.** Time histories of: (**a**) A waveform of input excitation signal with the amplitude of 10 and (**b**) the measured and simulation results predicted by the GB–W model and CB–W model.

To qualify modeling errors reasonably, the root-mean-square error *erms* (RMSE) and relative root-mean-square error *δ* (RRMSE) are introduced in this paper as follows:

$$c\_{rms} = \sqrt{\frac{1}{T} \int\_0^T |y(t) - y\_d(t)|^2 dt} \tag{12}$$

$$\delta = \frac{c\_{\rm rms}}{\max(y\_d(t))} \times 100\% \tag{13}$$

where *y*(*t*) and *yd*(*t*) are the simulated and measured displacements, respectively, *T* is the total time.

**Figure 11.** Time histories of: (**a**) A waveform of input excitation signal with the amplitude of 20 and (**b**) the measured and simulation results predicted by the GB–W model and CB–W model.

The detailed modeling errors are shown in Tables 3 and 4. The max displacement is 2.105 μm in the first group of experiments. According to it, in the first group of experiments, RMSE and RRMSE of the GB–W model under excitation signal at 5 Hz are 0.0742 μm and 3.52%, which are reduced by 81.5% compared with that of the CB–W model. When the frequency increases to 100 Hz and 110 Hz, the modeling errors (RMSE and RRMSE) of the GB–W model are reduced by nearly 42.1% and 56.47%, respectively. The results clearly reveal that the generalized Bouc–Wen model can predict the output of rate-dependent hysteresis curves of piezoelectric actuators more precisely. In the second group of experiments, the max measured displacements under two waveforms of input excitation signal are 2.215 μm and 4.676 μm, respectively. RMSE and RRMSE of the GB–W model under a waveform of input excitation signal with the amplitude of 10 are 0.1315 μm and 5.94%, respectively, which are reduced by 70.8% compared with that of the CB–W model. In the other experiment with the amplitude of 20, the rate-independent modeling errors of the GB–W model are still smaller and reduced by 28.9% compared with that of the CB–W model. The results above clearly reveal that the GB–W model can also predict the output of rate-independent hysteresis curves of piezoelectric actuators more precisely. According to the above analyses, it is reasonable to believe that the GB–W model is effective and can characterize both rate-dependent and rate-independent hysteresis behaviors more precisely than the CB–W model.


**Table 3.** Rate-dependent modeling errors of the GB–W model and CB–W model.

**Table 4.** Rate-independent modeling errors of the GB–W model and CB–W model.

