## 3 Feedforward Control Considering Non-Uniform Creep Difference and Hysteresis Nonlinearity
### Hybrid Control Strategy Combining Nonlinear Compensation and Trajectory Shaping  
Under the S-shaped trajectory driving voltage, the nonlinear characteristics of piezoelectric actuators manifest as the combined effect of their inherent hysteresis nonlinearity and NCD caused by speed changes. This complex nonlinear characteristic can lead to significant deviations between the actual motion trajectory of the actuator and the predetermined trajectory, seriously affecting the performance of high-precision positioning systems. To address this issue, this paper proposes a hybrid control strategy that combines nonlinear compensation and trajectory shaping. The control block diagram is shown in Figure 6. The core of this strategy lies in the step-by-step processing of the nonlinear characteristics of  
Figure 6: Block diagram of composite nonlinear feedforward control strategy. piezoelectric actuators and the use of model-driven compensation methods to effectively suppress complex nonlinearity. Firstly, the hysteresis characteristics of the piezoelectric actuator are calibrated through experiments, and its hysteresis model is established. On this basis, the motion data of the piezoelectric actuator at different speeds are analyzed using an NCD processor to separate the NCD values caused by speed changes. The NCD processor decouples the dynamic response of the input signal, extracts creep components related to velocity, and quantifies them into specific NCD values. Subsequently, a corresponding compensation voltage is generated based on the NCD model, which can effectively counteract the non-uniform creep effect caused by speed changes. At the same time, by bringing in the output of the clockwise operator model, hysteresis nonlinear compensation and NCD compensation are combined to generate a synthesized voltage applied to the piezoelectric actuator. It is worth noting that through the analysis of the displacement voltage characteristics of piezoelectric actuators, it is known that they have asymmetry, while the output of the general clockwise operator model is a symmetrical model. To improve the accuracy of the model, this paper models the displacement-voltage characteristics of piezoelectric actuators in segments based on the law of velocity deformation. In the modeling process, the complete clockwise operator was not used, but the clockwise operator was selected based on the principle of concave convex consistency. This means that in nonlinear curves, the convex curve of the boosting part and the concave curve of the lowering part use the corresponding unilateral operator. The specific operation steps are as follows:  
1. First Segment (Rising Curve from Zero Displacement to the Inflection Point); The inflection point is determined based on the fuzzy selection of voltage rate. When the voltage rises from 0 to the minimum rate, the relationship between voltage and displacement is described by a unilateral operator, as shown by curve 1 in Figure 7a. 2. Second Segment (Rising Curve from the Inflection Point to the Maximum Displacement); In this segment, the relationship between voltage and displacement is described by another unilateral operator, as shown by curve 2 in Figure 7a. During the displacement increase, the voltage rate also increases, which exhibits a trend opposite to the falling unilateral process of the clockwise operator. Therefore, a reverse falling unilateral clockwise operator is used for special modeling in this segment. 3. Third Segment (Falling Curve from Maximum Displacement to Zero Displacement). The relationship between voltage and displacement during this phase is described by another unilateral operator, as shown by curve 3 in Figure 7a. The weight and threshold selection method is consistent with the general clockwise operator model. The final synthesis result of the segmented clockwise operator model modeling is shown in Figure 7b.