# 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
Population initialization: Population range, number, number of iterations, and coding method are shown in Table 5.  
3. Population fitness evaluation: For the tension control problem, the GA objective is to optimize tuning the three parameters of \(\beta_{1}\), \(\beta_{2}\), and \(K_{p}\) in the ADRC to control the tension error to the minimum--minimization problem [22], so the inverse of the objective function is selected as the fitness function, and the objective function is selected as the integral of absolute error (IAE) [20], taking into account the characteristics of the steady-state control of tension, the objective function is selected based on IAE, the introduction of tension overshooting (Mp) and the IAE together with the objective function, and Equation (13) expresses the selected fitness function and the objective function,\(w_{1}\), \(w_{1}\) are the weighting parameters. \[\left\{ \begin{array}{l}f(x_{1},x_{2},\ldots,x_{n})=\frac{1}{1+\left\{x_{1},x_{2},x_{m},x_{n}\right\}}\\ J(x_{1},x_{2},\ldots,x_{n})=\int_{0}^{\infty}\left[w_{1}|e(t)|+w_{2}M_{p}(t) \right]dt\end{array} \right. \tag{13}\]  
The Equation (13) provides a discrete expression for the tension error \(e(t)\) is given below:  
\[\left\{ \begin{array}{l}e(k)=z_{1}(k)-T_{\mathrm{s}}(k)\\ z_{1}(k+1)=z_{1}(k)+h[z_{2}(k)-\beta_{1}e_{\mathrm{E}}(k)+b_{0}u(k)]\\ z_{2}(k+1)=z_{2}(k)+h[-\beta_{2}e_{\mathrm{E}}(k)]\\ u(k+1)=K_{\mathrm{p}}e_{\mathrm{L}}(k+1)-z_{2}(k+1)/b_{0}\end{array} \right. \tag{14}\]  
The parameter self-tuning decoupling controller based on improved ADRC for the tension system is designed based on the static decoupling of the global coupling model of the tension system and parameter tuning of the ADRC using GA. The controller has two phases of tension control that follow the same process. In phase one, the tension error \(e(t)\) is judged every 1 s, starting from 0.5 s. If \(e(t)\) is greater than 0.5 at the detection moment, the parameters of \(\beta_{1i}\), \(\beta_{2i}\), and \(K_{ip}\) are optimized and adjusted using GA, and the parameters remain unchanged in the opposite way. In phase two, the ADRC controllers use the three parameters optimized tuning in phase one and output the virtual control quantity \(\mathbf{u}(t)\) and then output the actual control quantity \(\mathbf{\omega}(t)\) after the static decoupling model. The \(\mathbf{\omega}(t)\) is then input into the global tension system to achieve steady-state control of the global tension system. The schematic representation of the controller is shown in Figure 3, and the working principle of this controller is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Way** & Elite Choice & Hybrid crossover & Gaussian approximate mutation \\ \hline
**Probability** & 0.01 & 0.2 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Table of population renewal settings. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameters** & **Population** & **Number of Iterations** & **Coding Method** \\ \hline
**\(\mathbf{K_{p}}\)** & \(\mathbf{\beta_{1}}\) & \(\mathbf{\beta_{2}}\) & & \\ \hline
50 \(\sim\) 1000 & 50 \(\sim\) 2000 & 103 \(\sim\) 107 & 100 & 100 & Floating-point \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Table of initialization settings for populations.