## 3 Feedforward Control Considering Non-Uniform Creep Difference and Hysteresis Nonlinearity
### Module Function Introduction
#### 3.2.3 Clockwise Operator and Clockwise Operator Model  
Apart from the NCD caused by speed change, piezoelectric actuators also have inherent hysteresis nonlinearity. To address this issue, this paper directly compensates for the displacement-voltage characteristics using the clockwise operator model, avoiding the problems of multiple control links and difficult inversion in the traditional feedforward control that requires modeling first and then inversion. The clockwise operator, as one of the fundamental operators of the clockwise operator model, describes the lead characteristics in the displacement-voltage nonlinearity of piezoelectric actuators. Since general piezoelectric actuators can only withstand positive voltage, this paper adopts a unilateral clockwise operator. The expression for the unilateral clockwise operator is as follows:  
\[y(t)=\max(x(t)-r,\min(x(t),y(t+1))) \tag{15}\]  
\[y(n)=\max(x(n)-r,\min(x(n),y(n))) \tag{16}\]  
where \(x(t)\) or \(x(n)\) is the input of the unilateral clockwise operator at time \(t\) or \(n\); \(y(t)\) or \(y(n)\) is the output of the unilateral clockwise operator at time \(t\) or \(n\); and \(r\) is the threshold of the unilateral clockwise operator. Figure 5a is a schematic diagram of the unilateral clockwise operator, illustrating the relationship between the input \(x(t)\) and the output \(y(t)\). When the threshold \(r=6\), the input and output are \(x(t)\) and \(y(t)\), respectively; when the threshold \(r=3\), the input and output are \(x_{1}(t)\) and \(y_{1}(t)\), respectively. Using MATLAB-2022a to compute the unilateral clockwise operator, the unilateral clockwise operator plots for the same input but different thresholds are shown in Figure 5b. The unilateral clockwise operator output continues to increase at the same rate as the input until it reaches a point where the input distance is less than the threshold. At this point, the clockwise operator output begins to remain constant. When the input value reaches its maximum, the unilateral clockwise operator output starts to decrease at the same rate until the input returns to the threshold. At this stage, the unilateral clockwise operator output continues to remain unchanged, ultimately forming a closed loop. The clockwise operator model is derived by performing a weighted summation of different unilateral clockwise operators. Its specific expression is as follows:  
\[y(t)=H[x(t),y(t+1)]=\sum_{k=1}^{n}\omega_{i}^{*}(t)\max(x(t)-r,\min(x(t),y(t+1))) \tag{17}\]  
where \(H[\cdot]\) represents the lead relationship. \(x(t)\) and \(y(t)\) are the input and output of the advanced prediction model, respectively. The weight vector is \(\omega^{*}(t)=(\omega_{1}^{*}(t),\dots,\omega_{n}^{*}(t))^{T}\). The threshold vector is \(r=(r_{1},\dots,r_{n})^{T}\). The final-state vector of unilateral advanced operators is \(y[n]=(y_{1}[n],\dots,y_{n}[n])^{T}\). Figure 5: Diagram and simulation of clockwise operator: (**a**) Diagram of clockwise operator. (**b**) Simulation of clockwise operator. The problem of obtaining a suitable clockwise operator model for system nonlinearity depends on selecting an appropriate set of thresholds and then determining the coefficients that best fit the nonlinear curve associated with this set. This can be achieved using parameter estimation algorithms such as Particle Swarm Optimization (PSO). However, to obtain a more accurate model, a large number of unilateral clockwise operators are required, each with a coefficient that needs to be fitted. This results in a vast optimization space, which in many cases leads to slow convergence. Fortunately, the coefficient values can be derived by studying the geometry of the clockwise operator model. Considering an ordered set \(\{r_{i}\}\), where \(r_{i}<r_{i+1}\), representing the thresholds of the unilateral clockwise operator that constitute the clockwise operator model.