## 5 Numerical Simulations  
To show the performance of the proposed decentralized control scheme based on an IADRC for MIMO systems, consider the following MIMO uncertain nonlinear system:  
\[\left\{\begin{array}{c}\dot{\xi}_{1,1}=\xi_{1,2},\\ \dot{\xi}_{1,2}=f_{1}(\xi,\eta,w_{1})+g_{1,1}(t)u_{1}+g_{1,2}(t)u_{2},\\ y_{1}=\xi_{1,1},\\ \dot{\xi}_{2,1}=\xi_{2,2},\\ \left\{\dot{\xi}_{2,2}=f_{2}(\xi,\eta,w_{2})+g_{2,1}(t)u_{1}+g_{2,2}(t)u_{2}, \right.\\ y_{2}=\xi_{2,1},\\ \dot{\eta}=\xi_{1,2}+\xi_{2,1}+\sin(\eta)+\sin(t)\end{array}\right. \tag{42}\]  
where \(y_{1}\) and \(y_{2}\) are the outputs, \(u_{1}\) and \(\ u_{2}\) are inputs, \(\xi=\left\{\xi_{1,1},\ \xi_{1,2},\xi_{2,1},\xi_{2,2}\right\}\in\mathbb{R}^{4}\) is the external state vector, and \(\eta\in\mathbb{R}\) is the internal state of Equation (42). \(y_{1}\), \(y_{2}\), \(u_{1}\), \(u_{2}\), \(w_{1}\), and \(w_{2}\) belong to \(\mathbb{R}\), and the unknown functions are:  
\[\left\{\begin{array}{c}f_{1}=\xi_{1,1}+\xi_{2,1}+\eta+sin(\xi_{1,2}+\xi_{2,2 })w_{1},\\ f_{2}=\xi_{1,2}+\xi_{2,2}+\eta+sin(\xi_{1,1}+\xi_{2,1})w_{2},\\ g_{1,1}(t)=1+\frac{1}{10}sin(t),g_{1,2}(t)=1+\frac{1}{10}cos(t),\\ g_{2,1}(t)=1+\frac{1}{10}2^{-t},g_{2,2}(t)=-1\end{array}\right.\]  
Suppose that the exogenous disturbances \(w_{1}\) and \(w_{2}\) and the reference signals \(r_{1}\) and \(r_{2}\) are as follows: \(w_{1}=1+\sin(t)\), \(w_{2}=2^{-t}\cos(t),r_{1}=\sin(t),r_{2}=\cos(t)\). The initial values of the model are taken as follows: \((\xi_{1,1},\xi_{1,2},\xi_{2,1},\xi_{2,2},\eta)=(0.5,\ 0.5,\ 1,\ 1,0)\).  
Two ADRC Configurations will be used in the simulations for the proposed decentralized control scheme. The difference between them is the type of ESO used to estimate the generalized disturbance and the nonlinear system's states. These configurations are:  
**1. First configuration: The Conventional ADRC (CADRC)**  
The CADRC consists of the following units:
Conventional TD, described by Reference [62]:  
\[\left\{\begin{array}{c}\dot{r}_{i,1}=r_{i,2},\\ \dot{r}_{i,2}=-R_{i}sign\Big{(}r_{i,1}-r_{i}+\frac{r_{i,2}|r_{i,2}|}{2R_{i}} \Big{)},i\in\{1,2\}\end{array}\right. \tag{43}\]  
where \(R_{i}\), \(i\in\{1,2\}\) is an application-dependent design parameter and it is set accordingly to speed up or slow down the transient profile.  
_fall_-based control law, given as:  
\[\left\{\begin{array}{c}u_{1}^{*}=k_{1,1}fal(\overline{c}_{1,1},\alpha_{1,1 },\delta_{1,1})+k_{1,2}fal(\overline{e}_{1,2},\alpha_{1,2},\delta_{1,2})- \dot{\xi}_{1,3},\\ u_{2}^{*}=k_{2,1}fal(\overline{e}_{2,1},\alpha_{2,1},\delta_{2,1})+k_{2,2}fal( \overline{e}_{2,2},\alpha_{2,2},\delta_{2,2})-\dot{\xi}_{2,3}.\end{array}\right. \tag{44}\]  
where \(\overline{e}_{i,l}=r_{i,l}-\dot{\xi}_{i,l}\) with \(i,l\in\{1,2\}\) is the tracking error and \(k_{i,l}\), \(\alpha_{i,l}\) and \(\delta_{i,l}\)\(i,l\in\{1,2\}\) are design parameters.  
The LESO, given as follows [62]:  
\[\left\{\begin{array}{c}\dot{\xi}_{i,1}=\dot{\xi}_{i,2}+3\omega_{o,i}\Big{(}y _{i}-\dot{\xi}_{i,1}\Big{)},\\ \dot{\xi}_{i,2}=\dot{\xi}_{i,3}+u_{i}^{*}+3\omega_{o,i}^{2}\big{(}y_{i}-\dot{ \xi}_{i,1}\Big{)},\\ \dot{\xi}_{i,3}=\omega_{o,i}^{3}\big{(}y_{i}-\dot{\xi}_{i,1}\big{)},i\in\{1,2\} \end{array}\right. \tag{45}\]  
where the vectors \(\big{(}\dot{\xi}_{i,1},\dot{\xi}_{i,2}\big{)}^{T}\) and \(i\in\{1,2\}\) are the observed model states, \(\dot{\xi}_{i,3}\) and \(i\in\{1,2\}\) are the estimated generalized disturbance, and \(\omega_{o,i}\), \(i\in\{1,2\}\) is the bandwidth of the NHOESO of the \(i\)-th subsystem.