# A Parameter Self-Tuning Decoupling Controller Based on an Improved ADRC for Tension Systems
## 3 Decoupled Controller Design for Global Tension System
### GA-Based Self-Tuning of ADRC Parameters  
The better robustness of the active disturbance rejection controller(ADRC) compared to the conventional proportional-integral-differential(PID) controller makes it a good choice for applicators, and current applications range from power electronics, motion control, and tension control [19]. The global tension system coupling model uses five ADRCs with the same structure for tension control. This ADRC consists of four main components, namely the most rapid tracking differentiator(TD), the first-order nonlinear error feedback control(NLESF), the first-order dilated state observer(ESO), and compensating the system based on the perturbation estimation, with the discrete formulas as shown below:  
\[\left\{ \begin{array}{l}e_{T}(k)=x_{1}(k)-T_{0}(k)\\ fh(k)=\text{fhan}(e_{T}(k),x_{2}(k),r_{0},h_{0})\\ x_{1}(k+1)=x_{1}(k)+hx_{2}(k)\\ x_{2}(k+1)=x_{2}(k)+hfh(k)\\ e_{E}(k)=z_{1}(k)-T(k)\\ z_{1}(k+1)=z_{1}(k)+h[z_{2}(k)-\beta_{1}e_{E}(k)+b_{0}u(k)]\\ z_{2}(k+1)=z_{2}(k)+h[-\beta_{2}fal(e_{E}(k),\alpha_{1},\delta_{1})]\\ e_{N}(k+1)=x_{1}(k+1)-z_{1}(k+1)\\ u(k+1)=K_{p}fal(e_{N}(k+1),\alpha_{2},\delta_{2})-z_{2}(k+1)/b_{0}\end{array} \right. \tag{12}\]  
where \(r_{0}\) is the time scale that responds to the response speed of the system, \(h_{0}\) is the filtering factor that determines the effect of filtering against noise [20]; and \(\alpha_{1}\) and \(\alpha_{2}\) are the nonlinear factors [21], which are quantities reflecting the speed of convergence of the function, \(0<\alpha<1\); \(\delta_{1}\), \(\delta_{2}\) are the filtering factors [21], which are quantities that determine the length of the linear segment of the function; \(b_{0}\) is a quantity related to the system itself, which is a quantity that responds to the strength of the compensation of the system error; \(\beta_{1}\), \(\beta_{2}\), and \(K_{p}\) are the quantities of the function gain in ESO and NLESF; \(h\), on the other hand, is related to the simulation accuracy. The above 11 parameters, except for the three parameters of \(\beta_{1}\), \(\beta_{2}\), and \(K_{p}\), can be selected empirically, and Table 3 shows the parametric quantities selected empirically, while the three parameters of \(\beta_{1}\), \(\beta_{2}\), and \(K_{p}\) are directly related to the control effect due to the intrinsic connection between each other, and are directly related to the control effect. The traditional empirical selection is characterized by trial and error and has certain limitations. Genetic algorithm (GA) has been successfully applied to a wide range of problems due to its simplicity, global perspective, and intrinsic parallel processing [22]. Therefore, the optimization tuning of the three parameters is carried out using GA, and the parametric quantities that rely on the empirical method of initialization are shown in Table 4.  
GA typically begins its search with a set of random solutions encoded in a binary string structure or floating-point numbers. Each solution is assigned a value related to the objective function. The population of solutions is then modified into a new population by applying three operators similar to natural genetic operators--reproduction, crossover, and mutation. The GA works iteratively by applying these three operators successively in each \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & \(K_{p}\) & \(\beta_{1}\) & \(\beta_{2}\) \\ \hline
**Value** & 100 & 50 & 6500 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: GAoptimization parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & \(r_{0}\) & \(h_{0}\) & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\delta_{1}\) & \(\delta_{2}\) & \(b_{0}\) & \(h\) \\ \hline
**Value** & 600 & 0.6 & 0.5 & 0.5 & 0.01 & 0.01 & 1 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Empirical Selection of Parameters. generation until the termination criterion is satisfied [22]. The specific setup of the GA in this paper is as follows:  
1.