## 3 Feedforward Control Considering Non-Uniform Creep Difference and Hysteresis Nonlinearity
### Module Function Introduction
#### 3.2.2 NCD Model  
On the basis of NCD separation, it is only necessary to convert it into a voltage value to achieve dynamic compensation for the piezoelectric actuator. Through the analysis in Section 3.1, it can be seen that the variation trend in NCD in both the boost and back  
Figure 4: The relationship curve between NCD and velocity: (**a**) 0.25 Hz. (**b**) 0.5 Hz. (**c**) 1 Hz. (**d**) 2 Hz.  
phase exhibits a complex nonlinear characteristic of first increasing, then decreasing, then increasing again, and finally decreasing again. This variation trend is closely related to the dynamic response characteristics of the piezoelectric actuator, especially under high-speed motion or frequent acceleration and deceleration conditions, where the NCD variation becomes more significant. To accurately describe this nonlinear relationship, this paper adopts polynomial functions to model the boost and buck segments separately. Specifically, the relationship between NCD and voltage can be expressed by the following polynomial function:  
\[d_{\text{NCD}}=a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4} \tag{12}\]  
where \(d_{\text{NCD}}\) is the NCD value, \(v\) is the voltage value, and \(a_{i}\) are the coefficients determined by experimental data fitting. This polynomial modeling method can effectively describe the nonlinear relationship between NCD and voltage, especially the complex variation trends of NCD in the boost and back phase.  
Since polynomial functions themselves cannot be directly inverted, it is necessary to segment the NCD curve based on its extreme points. Specifically, the NCD curve can be divided into multiple monotonic intervals, and the polynomial function is inverted within each monotonic interval. This segmented inversion method effectively addresses the non-invertibility issue of polynomial functions while ensuring the accuracy of the inversion results. After segmented inversion, the obtained voltage value needs to be further divided by the sensitivity coefficient \(K\) of the piezoelectric actuator to obtain the actual compensation voltage value. The sensitivity coefficient \(K\) of the piezoelectric actuator is defined as the ratio of the steady-state displacement change \(\Delta x\) to the voltage change \(\Delta u\)--i.e.,  
\[K=\frac{\Delta x}{\Delta u} \tag{13}\]  
Therefore, the compensation voltage \(u_{\text{NCD}}\) can be calculated using the following formula:  
\[u_{\text{NCD}}=\frac{u}{K} \tag{14}\]  
where \(u\) is the voltage value obtained after segmented inversion, and \(K\) is the sensitivity coefficient of the piezoelectric actuator. Through this method, the NCD value can be converted into the actual compensation voltage value, thereby achieving precise control of the piezoelectric actuator.  
In practical applications, for NCD generated by different predetermined trajectories, it is necessary to select appropriate mathematical modeling methods based on their characteristics. Although polynomial functions can effectively describe the nonlinear relationship between NCD and voltage, in some cases, higher-order polynomials or other nonlinear functions (such as exponential functions or piecewise linear functions) may be required to improve modeling accuracy. Additionally, for certain special motion trajectories, machine learning methods (such as neural networks or support vector machines) can be combined to model NCD, further enhancing the adaptability and prediction accuracy of the model.