## 4 Experiments and Discussion
### Separation Results from Model Identification Results  
The NCD separation method is used to separate the NCD values in the nonlinear features under the S-shaped trajectory driving voltage, as shown in Figure 10.  
Figure 11 clearly shows the specific NCD separated from the experimental data. As shown in Figure 11, the extracted NCD values exhibit consistent trends at different frequencies, which is highly consistent with theoretical expectations and previous analysis results. This consistency not only validates the reliability of the NCD separation method, but also highlights the robustness and adaptability of the proposed method under different operating conditions. Specifically, under driving signals of different frequencies, the separation results of NCD values show similar regular changes, indicating that this method can effectively capture the NCD effect caused by speed changes and is not significantly affected by frequency changes.  
In addition, this consistency further demonstrates the universality of the NCD separation method. No matter how the frequency of the driving signal changes, the extraction process of the NCD value can maintain high accuracy and stability, which provides a reliable data basis for subsequent nonlinear compensation. At the same time, the experimental results also indicate that the proposed method can effectively address the various working conditions that piezoelectric actuators may encounter in practical applications,  
Figure 9: The functional test platform.  
We used the least-squares optimization algorithm to identify the parameters in Formula (12) [31]. The parameters are shown in Table 3:  
\begin{tabular}{c c c c c c} \hline \hline
**Frequency** & \(\mathbf{a_{0}}\) & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) & \(\mathbf{a_{4}}\) \\ \hline
0.25 Hz & \(9.9565\times 10^{-8}\) & \(-2.3243\times 10^{-5}\) & 0.0016 & \(-0.0305\) & 0.0694 \\
0.5 Hz & \(1.1026\times 10^{-7}\) & \(-2.5842\times 10^{-5}\) & 0.0019 & \(-0.0403\) & 0.0385 \\
1 Hz & \(1.0219\times 10^{-7}\) & \(-2.3784\times 10^{-5}\) & 0.0017 & \(-0.0336\) & 0.0489 \\
2 Hz & \(1.2875\times 10^{-7}\) & \(-3.1169\times 10^{-5}\) & 0.0024 & \(-0.0661\) & 0.0279 \\ \hline \hline \end{tabular}  
The parameters obtained using the same optimization method are shown in Table 4:  
\begin{tabular}{c c c c c} \hline \hline
**Frequency** & \(\mathbf{a_{0}}\) & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) & \(\mathbf{a_{4}}\) \\ \hline
0.25 Hz & \(-7.7840\times 10^{-8}\) & \(1.7520\times 10^{-5}\) & \(-0.0012\) & 0.0249 & 0.1321 \\
0.5 Hz & \(-9.9332\times 10^{-8}\) & \(2.3277\times 10^{-5}\) & \(-0.0017\) & 0.0449 & 0.1484 \\
1 Hz & \(-8.2943\times 10^{-8}\) & \(1.8969\times 10^{-5}\) & \(-0.0013\) & 0.0314 & 0.1194 \\
2 Hz & \(-1.1929\times 10^{-7}\) & \(2.2908\times 10^{-5}\) & \(-0.0023\) & 0.0661 & 0.2259 \\ \hline \hline \end{tabular}  
Figures 14 and 15 show the inverses of the NCD models. Since the NCD model cannot be directly inverted, it can be segmented based on its extremum points. The curves 1-4 in the figure represent the inverse of each segment.  
Figure 14: _Cont._  
Figure 13: NCD model in back phase: (**a**) 0.25 Hz. (**b**) 0.5 Hz. (**c**) 1 Hz. (**d**) 2 Hz.