## 3 Feedforward Control Considering Non-Uniform Creep Difference and Hysteresis Nonlinearity
### Module Function Introduction
#### 3.2.1 NCD Processor  
The NCD processor is a dynamic compensation module specifically designed for piezoelectric actuators. Its core function is to output the NCD caused by speed changes by analyzing the velocity characteristics of a predetermined S-shaped trajectory driving voltage. This processor takes the instantaneous velocity of the S-shaped trajectory driving voltage as an input parameter, and the output NCD can be used for feedforward compensation to counteract dynamic positioning errors caused by velocity non-uniformity. By decomposing the nonlinear characteristics of the S-shaped trajectory into independently processable hysteresis and creep components, the NCD processor significantly improves the trajectory tracking accuracy of piezoelectric systems. The relationship curve between NCD and velocity is shown in Figure 4.  
In practical applications, the NCD processor demonstrates clear regularity in the NCD separated from the S-shaped trajectory driving voltage of different frequencies. Experimental results show that as the velocity increases, the NCD exhibits a decreasing trend, which is consistent with the basic theory discussed earlier. Specifically, during the acceleration and deceleration phases of the boost phase, the NCD curves show a high degree of overlap, indicating that the piezoelectric actuator's response is relatively consistent in these phases. However, during the acceleration and deceleration phases of the back phase, the NCD curves exhibit significant differences. This phenomenon is primarily due to the structural complexity of the piezoelectric actuator. In addition to the piezoelectric ceramic, the piezoelectric actuator also includes adhesive layers, electrode layers, and alumina protective layers. These materials inherently possess nonlinear elastic properties, especially the adhesive layer, which exhibits significant viscoelastic hysteresis and stress bufferingeffects during the back phase, leading to a larger NCD gap between the acceleration and deceleration phases. This phenomenon is consistent with recent research [30], further validating the theoretical foundation of the NCD processor.  
In this analysis, macroscopic effects caused by materials and manufacturing processes are neglected, making the proposed capacitance change theory and NCD separation method reasonable. Through this simplification, the NCD processor can effectively separate the hysteresis and creep components in the S-shaped trajectory, thereby achieving precise control of the piezoelectric actuator. Additionally, the design of the NCD processor takes into account the dynamic response characteristics of the piezoelectric system, particularly the nonlinear effects under high-speed motion. Through feedforward compensation, the NCD processor can significantly reduce dynamic positioning errors caused by speed changes, further enhancing the overall performance of the system.  
In summary, with its unique dynamic compensation mechanism, the NCD processor not only effectively decomposes the nonlinear characteristics in the S-shaped trajectory but also maintains high trajectory tracking accuracy under different velocity conditions. This technology provides a new solution for the application of piezoelectric actuators in high-precision positioning and control fields, offering broad application prospects.