## 4 Main Results
### The Improved ADRC (Iadrc)  
The structure of the IADRC that was used in the proposed control scheme for MIMO uncertain nonlinear systems was the same as that of the conventional ADRC (CADRC), with the same units of  
Figure 4: Generalized disturbance representation for the decentralized ADRC scheme.  
Figure 5: The proposed decentralized ADRC control scheme, where each ADRC block involves a SISO connection of TD, NLSEF, and ESO, as in Figure 1.  
conventional TD and an NLSEF, except for the conventional linear ESO (LESO), which was replaced by a novel nonlinear higher order ESO (NHOESO). The dynamical structure of the TD is considered as [62]:  
\[\left\{\begin{array}{c}\dot{r}_{i,j}=r_{i,j+1},\quad j\in[1,2,\ldots,\gamma_{i }-1]\\ \dot{r}_{i,\gamma_{i}}=-R_{i}sign\Big{(}r_{i,1}-r_{i}+\frac{r_{i,2}|r_{i,2}|}{2 R_{i}}\Big{)},\ i\in\{1,..,\ p\}\end{array}\right. \tag{17}\]  
where \(R_{i}\), \(i\in\{1,..,\ p\}\) is a design parameter and an application-dependent parameter, set accordingly to speed up or slow down the transient profile, \(r_{i,j}\), \(j\in\{1,2,\ldots,\gamma_{i}-1\}\) and \(i\in\{1,..,\ p\}\) is the output of the TD. The NLSEF has the following nonlinear error function [62]:  
\[fal_{i,j}\Big{(}\overline{e}_{i,l},\alpha_{i,j},\delta_{i,j}\Big{)}=\left\{ \begin{array}{cc}\overline{\epsilon}_{i,l}&\left|\overline{e}_{i,l}\right| \leq\delta_{i,l}\\ \overline{\epsilon}_{i,l}\Big{|}^{\alpha_{i}}sign\Big{(}\overline{e}_{i,l} \Big{)}&\left|\overline{e}_{i,l}\right|\geq\delta_{i,l}\end{array}\right. \tag{18}\]  
where \(l\in\{1,2,\ldots,\gamma_{i}\}\), \(i\in\{1,..,\ p\}\), \(\alpha\), and \(\delta\) are design parameters and, usually, \(\delta_{i,l}\) indicates how wide the linear part is and \(0<\alpha_{i,l}<1\). With proper selection of these parameters, the closed-loop tracking error \(\overline{e}_{i,l}\) could diminish approaching \(0\)) in a very short time. The novel NHOESO is proposed as follows:  
\[\left\{\begin{array}{c}\dot{\xi}_{i,j}=\xi_{i,j+1}+\alpha_{i,l}\omega_{o,i}^ {j}\boldsymbol{\beta}_{i}\Big{(}y_{i}-\xi_{i,1}\Big{)},\quad j\in\{1,2,\ldots, \gamma_{i}-1\}\\ \dot{\xi}_{i,\gamma_{i}}=\xi_{i,\gamma_{i}+1}+\alpha_{i,y}\omega_{o,i}^{w_{i} }\boldsymbol{\beta}_{i}\Big{(}y_{i}-\xi_{i,1}\Big{)}+u_{i}^{*}\\ \dot{\xi}_{i,\gamma_{i}+1}=\dot{\xi}_{i,\gamma_{i}+2}+\alpha_{i,y_{i}+1}\omega _{o,i}^{y_{i}+1}\boldsymbol{\beta}_{i}\Big{(}y_{i}-\dot{\xi}_{i,1}\Big{)},\\ \dot{\xi}_{i,\gamma_{i}+2}=\alpha_{i,y_{i}+2}\omega_{o,i}^{y_{i}+1}\boldsymbol {\beta}_{i}\Big{(}y_{i}-\dot{\xi}_{i,j}\Big{)},\quad i\in\{1,..,\ p\}\end{array}\right. \tag{19}\]  
where the vectors \(\big{(}\xi_{i,1},\ldots,\dot{\xi}_{i,\gamma_{i}}\big{)}^{T}\) and \(i\in\{1,..,\ p\}\) are the observed model states and \(\dot{\xi}_{i,\gamma_{i}+1},\ i\in\{1,..,\ p\}\) is the estimated generalized disturbance, \(\omega_{o,i}\) is the bandwidth of the NHOESO of the \(i\)-th subsystem, \(a_{i,s}\), and \(,\ i\in\{1,..,\ p\}\) and \(s\in\{1,..,\ \gamma_{i}+2\}\) are the design parameters of the NHOESO. They were selected such that the following matrix was Hurwitz.  
\[E=\left[\begin{array}{cccc}-a_{i,1}&1&0&\cdots&0\\ -a_{i,2}&0&1&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ -a_{i,\gamma_{i}+1}&0&0&\ddots&1\\ -a_{i,\gamma_{i}+2}&0&0&\cdots&\end{array}\right]_{(\gamma_{i}+2)(\gamma_{i}+ 2)} \tag{20}\]  
On the other hand, the nonlinear function \(\boldsymbol{\vartheta}_{i}:\mathbb{R}\rightarrow\mathbb{R},i\in\{1,..,\ p\}\) was designed as in Reference [63]:  
\[\boldsymbol{\vartheta}_{i}(e_{i})=K_{i,\alpha}|e_{i}|^{\alpha_{i}}sign(e_{i})+ K_{i,\beta}|e_{i}|^{\beta_{i}}e_{i} \tag{21}\]  
where \(K_{i,\alpha},K_{i,\beta},\alpha_{i}\) and \(\beta_{i}\) are positive design parameters and \(e_{i}\) is the estimation error, defined as \(e_{i}=y_{i}-\dot{\xi}_{i,1}\).