## 3 Control Model for Automatic Adjustable Preload System
### The Modeling of Hysteresis  
The hysteretic model is an indispensable part of the preload force control that is necessary in order to compensate the hysteresis error. In addition, the P-I model has linear expression and low complexity, so it is relatively easy to obtain the parameters and the inverted model. In this paper, the P-I model is, consequently, used as the control model, and the play operator is regarded as the elementary operator. The operator is defined as follows: Set \(t\in[t_{0},t_{n}]\), (\(t_{0}\leq t_{1}\cdots t_{i-1}\leq t\leq t_{i}\cdots\leq t_{n-1}\leq t_{n}\)), when \(t\in[t_{0},t_{n}]\), \(x(t)\) changes monotonically. The play operator can be expressed as:  
\[y(t_{i})=F(v(t),y(t_{i-1}),r)=max\{v(t)-r,min\{v(t)+r,y(t_{i-1})\}\} \tag{14}\]  
where \(y(t_{i})\), \(v(t)\), \(y(t_{i-1})\) and \(r\) are the output of the operator at time \(t_{i}\), the input, and the output at the previous moment and the threshold value, respectively. In addition, the initial operator \(y_{0}\) is equal to 0. The P-I hysteresis model can be built, based on the play operators with different threshold values, and its expression is as follows:  
\[ \begin{array}{c}O(t)=\sum\limits_{k=1}^{n}w_{k}F_{k}(v(t),y_{k}(t_{i-1}),r_ {k})=\sum\limits_{k=1}^{n}max\{v(t)-r_{k},min\{v(t)+r_{k},y_{k}(t_{i-1})\}\}\\ k=1,2,\cdots,n-1,n\end{array}  \tag{15}\]  
In Equation (15), \(w_{k}\) and \(r_{k}\) are the weight value and threshold value of the k-th play operator, and \(0=r_{1}<\cdots r_{n}<+\infty\), \(n\) is the number of operators. The threshold value of the play operator is generally evenly distributed, so the value of the threshold can be determined by the range of the input voltage. The threshold value can be defined as:  
\[r_{k}=\frac{k-1}{n}v(t)_{max} \tag{16}\]  
The value of \(n\) is 40, and \(v(t)_{max}\) is the maximum input voltage, which is 100 V.  
To obtain the parameters of the hysteretic model, first, the expression of the measured hysteresis curve is fitted; then, the curve is divided at each threshold value, and the corresponding weight value of each segment is deducted. Meanwhile, the slope of each segment is the sum of the weight values of all previous operators, with formula is given by:  
\[\left\{ \begin{array}{c}w_{1}=\frac{O_{v=r_{2}}-O_{v=r_{1}}}{P_{2}-r_{1}}\\ w_{2}+w_{1}=\frac{O_{v=r_{2}}-O_{v=r_{2}}}{r_{3}-r_{2}}\\ \vdots\\ w_{k}+w_{k-1}+\cdots+w_{2}+w_{1}=\frac{O_{v=r_{k+1}}-O_{v=k}}{r_{k+1}-r_{k}}\\ k=1,2,3,\cdots n-1,n\\ r_{k+1}=v(t)_{max}\end{array} \right. \tag{17}\]  
To simplify the calculation process of the model parameters, the input and output values are normalized. The cubic polynomial is used to fit the ascending section of the hysteresis curve. With the help of the Python algorithm, the expression of the curve can be obtained. In the same way, the descending segment can be generated. A comparison of the simulated hysteresis loop using the P-I model and the measured loop is shown in Figure 5. Furthermore, the curve expression is derived from Equation (18), where \(w_{pk}\) represents the weight of the ascending segment, and \(w_{qk}\) represents the weight of the descending segment. When the input voltage increases, \(p\) takes the value 1, and \(q\) takes the value 0. When the input voltage decreases, \(p\) takes the value 0 and \(q\) takes the value 1. \[ \begin{array}{c}O(t)=\sum\limits_{k=1}^{n}\Big{(}pw_{pk}+qw_{qk}\Big{)}F_{k}(x (t),y_{k}(t_{i-1}),r_{k})\\ =\sum\limits_{k=1}^{n}\Big{(}pw_{pk}+qw_{qk}\Big{)}max\{v(t)-r_{k},min\{v(t)+r_ {k},y_{k}(t_{i-1})\}\}\\ \left\{\begin{array}{l}p=1,q=0\;v(t)\geq v(t_{i-1})\\ p=0,q=1\;v(t)<v(t_{i-1})\end{array}\right.\end{array}  \tag{18}\]  
The inverted P-I model is symmetrical with the original model. In addition, the symmetry axis is a straight line \(y=v\). The inverted model equation can be written as Equation (19), where \(v(t)\) is the desired force, and \(O^{-1}(t)\) refers to the corresponding input voltage.