*4.1. Experiment Setup*

To demonstrate the validity of the proposed model, there are some groups of experiments conducted. Figure 5 shows the experimental setup, where a 1-DOF compliant mechanism stage was actuated by a stack piezoelectric ceramic actuator (PST 150/7/60VS12, Coremorrow, Harbin, China). Its nominal displacement was 60 μm for the maximum input voltage of 150 V. This piezoelectric ceramic actuator was made of PZT (Pb-based Lanthanum-doped Zirconate Titanates), whose detailed information is shown in Table 1. The strain gauge position sensor (SGS) included in the piezoelectric ceramic actuator measured the output displacement. dSPACE-DS1104 rapid prototyping controller board equipped with a 16-bit analogue-to-digital converter (ADC) and 16-bit digital-to-analogue converter (DAC) was used to control this 1-DOF compliant mechanism. In addition, an XE-500 controller (Coremorrow, Harbin, China) equipped with an amplifier with 15 times and a signal conditioner was also adopted. A computer with Control Desk 5.0-dSPACE and MATLAB/Simulink was used to conduct all experiments and obtain all experimental data. The detailed experimental steps were carried out as follow:


**Figure 5.** Experimental setup.


#### *4.2. Experiment Results and Discussion*

In order to evaluate the MDM performance comprehensively, six groups of experiments with different frequencies or amplitudes were conducted. The first and second groups of experiments adopted high-frequency and high-amplitude excitation signals. These two groups had the same amplitudes, but different frequencies. The third and fourth groups adopted low-frequency and low-amplitude excitation signals. The above four groups used multi-frequency signals. The fifth and sixth groups adopted high-frequency and low-amplitude excitation signals. The last two groups used single-frequency signals. The piezoelectric actuator used in our experiments was made of encapsulated stacked piezoelectric ceramics. It must be noted that its input frequency was controlled under 150 Hz to avoid a high dynamic force for security protection. The maximum input voltage of the actuator was 150 V. In the product manuals, it is recommended to control the input voltage within 120 V to guarantee the service life. Therefore, in the experiments, the six groups needed to follow this rule

and its input frequencies and amplitudes were generally not high. Finally, the input frequencies including 1, 5, 10, 15, 20, 30, 40 and 50 Hz, and the amplitudes including 3, 5, 6, 8 and 10, were selected randomly following the rule above. In each group of experiments, the chosen excitation signal was used to actuate the PCA in the experimental setup and the experimental output displacements would be recorded and obtained. Subsequently, the displacements predicted by the CDM and MDM were obtained by using the MATLAB/Simulink. Lastly, the comparison results were obtained and drawn.

In the first group of experiments, the excitation signal was *<sup>u</sup>*1(*t*) = 10 sin(<sup>2</sup>π · *t*) + 8 sin(<sup>2</sup>π · 10*t*) + 6 sin(<sup>2</sup>π · 20*t*) + 24 with multi-frequency 1, 10 and 20 Hz. The excitation signal in the second group of experiments was *<sup>u</sup>*2(*t*) = 10 sin(<sup>2</sup>π · *t*) + 8 sin(<sup>2</sup>π · 15*t*) + 6 sin(<sup>2</sup>π · 40*t*) + 24 with multi-frequency 1, 15 and 40 Hz. The excitation signal in the third group of experiments was *<sup>u</sup>*3(*t*) = 10 sin(<sup>2</sup>π · 5*t*) + 6 sin(<sup>2</sup>π · 10*t*) + 16 with multi-frequency 5 and 10 Hz. In the fourth group of experiments, the excitation signal was *<sup>u</sup>*4(*t*) = 5 sin(<sup>2</sup>π · *t*) + 3 sin(<sup>2</sup>π · 5*t*) + 8 with multi-frequency 1 and 5 Hz. The fifth group of experiments took the excitation signal *<sup>u</sup>*5(*t*) = 5 sin(<sup>2</sup>π · 30*t*)+5 at a frequency of 30 Hz to actuate the piezoelectric actuator. The last group of experiments took another excitation signal *<sup>u</sup>*6(*t*) = 5 sin(<sup>2</sup>π · 50*t*) + 5 at a frequency of 50 Hz.

To further demonstrate the effectiveness of the MDM, the CDM is set as a comparison. In addition, the modeling errors between the predicted output displacements of the two models and experimental displacements were drawn. In theory, any experimental signals can be used to identify the parameters of the MDM using the nonlinear least squares method. To ge<sup>t</sup> better prediction performances of hysteresis models including both the CDM and MDM, the more complex signals were generally adopted for identification, which is acceptable. In this work, the signals of both the first and second group of experiments, which were more complex than the others, were adopted to identify the parameters of the MDM and CDM. The detailed identified parameters of MDM and CDM are shown in Table 2.


**Table 2.** Identified parameters of CDM and modified Duhem model (MDM).

Figure 6 shows the comparison of the experimental and simulation results of the CDM and MDM. Figure 6a gives the input voltage at each moment, Figure 6b shows the simulation and experimental results and Figure 6c presents the final modeling errors of the CDM and MDM. The blue dotted line represents the predicted results of the CDM. Meanwhile, the red solid line represents the predicted results of the MDM. It can clearly be seen that the simulation results of the MDM are closer to the experimental data. The modeling errors of the MDM are obviously smaller than that of the CDM. Figures 7–11 show the experimental results of the last five groups of experiments. These results further reveal that the MDM simulation results are much closer to the experimental output displacement than the CDM simulation results. The corresponding modeling errors of MDM are much smaller than that of the CDM. In addition, it should be also noted that though the second group of experiments has higher frequency and higher amplitude compared with the other three groups, the MDM still maintains better stability and accuracy compared with the CDM.

**Figure 6.** Exp1: Comparison of the experimental and simulation results of the CDM and MDM under *<sup>u</sup>*1(*t*) = 10 sin(<sup>2</sup>π · *t*) + 8 sin(<sup>2</sup>π · 10*t*) + 6 sin(<sup>2</sup>π · 20*t*) + 24: (**a**) Time histories of input voltage, (**b**) time histories of output displacements and (**c**) time histories of errors of the CDM and MDM.

**Figure 7.** Exp2: Comparison of the experimental and simulation results of the CDM and MDM under *<sup>u</sup>*2(*t*) = 10 sin(<sup>2</sup>π · *t*) + 8 sin(<sup>2</sup>π · 15*t*) + 6 sin(<sup>2</sup>π · 40*t*) + 24: (**a**) Time histories of input voltage, (**b**) time histories of output displacements and (**c**) time histories of errors of the CDM and MDM.

**Figure 8.** Exp3: Comparison of the experimental and simulation results of the CDM and MDM under *<sup>u</sup>*3(*t*) = 10 sin(<sup>2</sup>π · 5*t*) + 6 sin(<sup>2</sup>π · 10*t*) + 16: (**a**) Time histories of input voltage, (**b**) time histories of output displacements and (**c**) time histories of errors of the CDM and MDM.

**Figure 9.** Exp4: Comparison of the experimental and simulation results of the CDM and MDM under *<sup>u</sup>*4(*t*) = 5 sin(<sup>2</sup>π · *t*) + 3 sin(<sup>2</sup>π · 5*t*) + 8: (**a**) Time histories of input voltage, (**b**) time histories of output displacements and (**c**) time histories of errors of the CDM and MDM.

**Figure 10.** Exp5: Comparison of the experimental and simulation results of the CDM and MDM under *<sup>u</sup>*5(*t*) = 5 sin(<sup>2</sup>π · 30*t*)+5: (**a**) Time histories of input voltage, (**b**) time histories of output displacements and (**c**) time histories of errors of the CDM and MDM.

**Figure 11.** Exp6: Comparison of the experimental and simulation results of the CDM and MDM under *<sup>u</sup>*5(*t*) = 5 sin(<sup>2</sup>π · 50*t*)+5: (**a**) Time histories of input voltage, (**b**) time histories of output displacements and (**c**) time histories of errors of the CDM and MDM.

To evaluate further the modeling performance of the MDM, the root mean square error *Erms*, the relative root mean square error ξ and optimization ratio ϕ between the CDM and MDM were employed in comparing the errors of two models as follows:

$$E\_{\rm rms} = \sqrt{\frac{\sum\_{i=1}^{n} \left[ \mathcal{Y}\_{\rm exp}(i) - \mathcal{Y}\_{\rm pre}(i) \right]^2}{n}} \tag{10}$$

$$\xi = \frac{E\_{\rm rms}}{\max\left[Y\_{\rm exp}(i)\right]} \times 100\% \tag{11}$$

$$\rho = \frac{\left| E\_{rms}^{CDM} - E\_{rms}^{MDM} \right|}{E\_{rms}^{CDM}} \times 100\% \tag{12}$$

where *n* is the total number of the sample and *i* is the *i*-th value in the sample, *Yexp* is measured from experiments, *Ypre* represents the displacements predicted by the hysteresis models, and *ECDM rms* and *EMDM rms* represent the root mean square error of the CDM and MDM, respectively. The details modeling errors are shown in Table 3.


**Table 3.** The simulation errors of the CDM and MDM.

It can be seen from Table 3 that in the fifth group of experiments (Exp5) at the single-frequency of 30 Hz, *Erms* and ξ of the MDM were 0.2899 μm and 13.95%,respectively, while those of the CDM were 0.3313 μm and 15.94%, respectively. Compared with the CDM, the MDM can predict more precisely the output displacements and the optimized ratio was 12.50%. In the last group (Exp6) at the single-frequency of 50 Hz, the optimized ratio was up to 14.04%. In the fourth group of experiments (Exp4) with low frequency and amplitude, *Erms* and ξ of the MDM were 0.0916 μm and 1.2% respectively, while those of the CDM were 0.1426 μm and 3.6%, respectively. Compared with the CDM, the optimized ratio was 35.76%. With the increasing of frequency and amplitude, *Erms* and ξ of the CDM in the third group of experiments (Exp3) increased to 0.1181 μm and 1.56%, respectively. The corresponding optimization ratio was up to 45.2%. Compared with the two groups above, the first and second groups of experiments (Exp1 and Exp2) had higher amplitude and frequency. The optimization ratios of Exp1 and Exp2 were 34.89% and 31.43%, respectively. Compared with the other three groups, the second group of experiments (Exp2) had the biggest amplitude and frequency, whose *Erms* and ξ of MDM were 0.5845 μm and 4.6%, respectively. It was found that the modeling errors of both the CDM and MDM increase with the increasing of frequency and amplitude.

These experimental and simulation results clearly reveal that the MDM can describe more precisely rate-dependent hysteresis behaviors at high-frequency and high-amplitude excitations compared with the CDM.
