## 2 Second configuration: The Improved ADRC (IADRC)
### Results of the Decentralized ADRC Control Scheme  
The proposed decentralized control scheme based on IADRC configuration for the MIMO system in Equation (42) was simulated for tracking the same reference input signals \(r_{1}\) and \(r_{2}\), as in the simulation of the first scheme. The same two configurations used in the simulations of the decoupled control scheme will be utilized here again for comparison purposes with \(u_{i}=u_{i}^{*}\), \(i\in\{1,2\}\). Specifically, the proposed control laws \(u_{1}\) and \(u_{2}\) are the saturated values of that in Equation (16), where \(v_{i}\) is given by Equation (26), i.e., they are reformulated as:  
\[\left\{ \begin{array}{l}u_{1}=sat\Big{(}k_{1,1}fal(\overline{e}_{1,1},\alpha _{1,1},\delta_{1,1})+k_{1,2}fal(\overline{e}_{1,2},\alpha_{1,2},\delta_{1,2}) -\hat{\xi}_{1,3},\delta_{1}\Big{)}\\ u_{2}=sat\Big{(}k_{2,1}fal(\overline{e}_{2,1},\alpha_{2,1},\delta_{2,1})+k_{2,2}fal(\overline{e}_{2,2},\alpha_{2,2},\delta_{2,2})+\hat{\xi}_{2,3},\delta_{2 }\Big{)}\end{array} \right. \tag{51}\]  
where \(\delta_{i}\), \(i\in\{1,2\}\) is a design parameter and \(Sat(u,\delta)\) is defined as in Equation (50). The parameters of the first and second configuration of the decentralized ADRC scheme for the two channels are listed in Table 4; Table 5, respectively. The same scenarios used in the validation of the decoupled control scheme [23] were used here to check the effectiveness of the decentralized control scheme and compare between both schemes. \begin{table}
\begin{tabular}{c c c c c} \hline \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 & \(R_{2}\) & 88.4423 \\ \hline
**LESO** & \(\omega_{0,1}\) & 68.3308 & \(\omega_{0,2}\) & 53.1690 \\  & \(b_{1,1}\) & 1.0000 & \(b_{2,2}\) & \(-1.0000\) \\ \hline  & \(\delta_{1,1}\) & 0.0010 & \(\delta_{2,1}\) & 0.1445 \\  & \(\delta_{1,2}\) & 0.2834 & \(\delta_{2,2}\) & 0.7346 \\  & \(\* **Case (1): Output tracking**  
Two reference signals, \(r_{1}(t)=\sin(t)\) and \(r_{2}(t)=\cos(t)\), were applied to the closed-loop system of Figure 5 to authenticate the output tracking capability of the proposed decentralized control scheme, using both CADRC and IADRC configurations. The performance indices for the two configurations are listed in Table 6. For these two configurations, the output response curves of the numerical simulations for the proposed decentralized scheme based on ADRC are shown in Figures 10 and 11, respectively. As illustrated in Table 6, the reduction in the values of the ITAE and ISU indices for the two channels in the IADRC is apparent, in comparison with the CADRC. Figure 11: _Cont._  
Figure 10: The output response of Equation (42) using CADRC, (**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 \(\xi_{2,3}\). Moreover, less chattering control efforts \(u_{1}\) and \(u_{2}\) were produced by the IADRC, in contrast with their counterparts in the first configuration. The tracking output response for the IADRC was better than in the CADRC, specifically during the transient period, where both configurations entirely attenuated the effect of the exogenous disturbances \(w_{1}\) and \(w_{2}\), the state couplings for each subsystem, and the time-varying input gains \(g_{1,1}\), \(g_{1,2}\), \(g_{2,1}\), and \(g_{2,2}\) on the output response of the two channels. It should be noted that the exogenous disturbances \(w_{1}\) and \(w_{2}\) are taken as \(w_{1}=1+\sin(t)\), \(w_{2}=2^{-t}\cos(t)\), which simulate a big mismatch between the nominal process and the mathematical model upon all the closed-loop simulations that have been demonstrated.