## 2 Second configuration: The Improved ADRC (IADRC)
### Results of the Decoupled ADRC Control Scheme [23]  
The input gain matrix \(B=\left( \begin{array}{cc}1&1\\ 1&-1\end{array} \right)\), then \(B^{-1}=\left( \begin{array}{cc}\frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&\frac{-1}{2}\end{array} \right)\). The proposed decoupler element used in this scheme given in Equation (7) is found as:  
\[\left( \begin{array}{c}u_{1}\\ u_{2}\end{array} \right)=\left( \begin{array}{cc}\frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&\frac{-1}{2}\end{array} \right)\left( \begin{array}{c}u_{1}^{*}\\ u_{2}^{*}\end{array} \right)=\left( \begin{array}{c}\frac{u_{1}^{*}}{2}+\frac {u_{2}^{*}}{2}\\ \frac{u_{1}^{*}}{2}-\frac{u_{2}^{*}}{2}\end{array} \right) \tag{48}\]  
Then, the suggested feedback control laws \(u_{1}\) and \(u_{2}\) are formulated, as in Reference [23]:  
\[\left\{ \begin{array}{l}u_{1}=Sat\bigg{(}\frac{u_{1}^{*}}{2}+\frac{u_{2}^{*} }{2},\delta_{1}\bigg{)}\\ u_{2}=Sat\bigg{(}\frac{u_{1}^{*}}{2}+\frac{u_{2}^{*}}{2},\delta_{2}\bigg{)} \end{array} \right. \tag{49}\]  
where \(\delta_{i}\), \(i\in\{1,2\}\) is a design parameter and the function \(Sat(u,\delta)\) is defined as:  
\[sat(u,\delta)=\left\{ \begin{array}{cc}\delta&u\geq\delta\\ u&-\delta<u<\delta\\ -\delta&u\leq-\delta\end{array} \right. \tag{50}\]  
The virtual control signals \(u_{i}^{*},\ i\in\{1,2\}\) in Equation (49) and indicated in Figure 3 is derived from the _fal_-based control law given in Equation (44). The desired transient trajectories \((r_{1,1},r_{1,2})^{T}\) and \((r_{2,1},r_{2,2})^{T}\) are generated from the reference signals \(r_{1}\) and \(r_{2}\) via the tracking differentiators described by Equation (43). The parameters of the first and second configurations for the two channels are listed in Table 1; Table 2, respectively, where these coefficients are obtained by tuning their values using a genetic algorithm (GA) by minimizing a multi-objective performance index, which is a weighted combination of the Integration of the Time Absolute Error (ITAE) defined as \(\text{ITAE}=\int_{0}^{t_{f}}t\big{|}y-r\big{|}dt\), and the ISU of both channels, where ISU is defined as \(\text{ISU}=\int_{0}^{t_{f}}v^{2}\ dt\), and \(t_{f}\) is the final simulation time. Two scenarios were conducted to test the effectiveness of the decoupling control scheme. These are:  
\(\bullet\) **Case (1): Output Tracking** \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**First Channel Parameters**} & \multicolumn{2}{c}{**Second Channel Parameters**} \\  & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline
**TD** & \(R_{1}\) & 92.2713 & \(R_{2}\) & 88.4424 \\ \hline
**LESO** & \(\omega_{0,1}\) & 68.3308 & \(\omega_{0,2}\) & 53.1690 \\ \hline  & \(\delta_{1,1}\) & 0.0010 & \(\delta_{2,1}\) & 0.14456 \\  & \(\delta_{1,2}\) & 0.2834 & \(\delta_{2,2}\) & 0.73456 \\  & \(\alpha_{1,1}\) & 0.1629 & \(\alpha_{2,1}\) & 0.02730 \\ _fal_-**based control law** & \(\alpha_{1,2}\) & 0.7946 & \(\alpha_{2,2}\) & 0.93745 \\  & \(k_{1,1}\) & 12.8015 & \(k_{2,1}\) & 18.3095 \\  & \(k_{1,2}\) & 11.2999 & \(k_{2,2}\) & 19.52670 \\  & \(\delta_{1}\) & 40 & \(\delta_{2}\) & 40 \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters of the first configuration (conventional ADRC (CADRC)) [23]. Figure 6: The output response of Equation (42) using CADRC configuration, (**a**) output \(y_{1}\), (**b**) output \(y_{2}\), (**c**) control signals \(u_{1}\) and \(u_{2}\), and (**d**) estimated generalized disturbances \(\xi_{1,3}\) and \(\xi_{2,3}\)[23]. Figure 7: _Cont._  
As illustrated in Table 3, the reduction was very evident in the values of the ITAE and ISU indices of the two channels for the second configuration, except for \(\text{ISU}_{1}\), where it slightly reduced from its value in the CADRC. This has been reflected in the control efforts \(u_{1}\) and \(u_{2}\), shown in Figures 6c and 7c, where \(u_{1}\) and \(u_{2}\) for the IADRC witnessed less activity than the CADRC.