## 4 Main Results
### Closed-Loop Stability Analysis
** The closed-loop tracking error between the reference profile \(r_{i,l}\) of the TD and the corresponding estimated states \(\hat{\xi}_{i,l}\) of the MIMO nonlinear system is given as:  
\[\overline{\epsilon}_{i,l}=r_{i,l}-\hat{\xi}_{i,l}\,\ i\in\{1,..,\ p\},\ \ l\in\{1,2,\ldots,\gamma_{i}\} \tag{27}\]  
If Assumptions 1 and 2 hold, then the tracking error \(\overline{\epsilon}_{i,j}\) can be written as:  
\[\overline{\epsilon}_{i,l}=r_{i}^{(l-1)}-\xi_{i,l}\,\ \ \ i\in\{1,..,\ p\},l\in\{1,2,\ldots, \gamma_{i}\} \tag{28}\]  
For the nonlinear plant of Equation (15), the states \(\xi_{i,l}\) can be expressed in terms of the system output as:  
\[\xi_{i,l}=y_{i}^{(l-1)}\,i\in\{1,..,\ p\},\ \ l\in\{1,2,\ldots,\gamma_{i}\} \tag{29}\]  
Substituting Equation (29) in Equation (28) yields,  
\[\overline{\epsilon}_{i,l}=r_{i}^{(l-1)}-y_{i}^{(l-1)}\,\ i\in\{1,..,\ p\},\ \ l\in\{1,2,\ldots,\gamma_{i}\}, \tag{30}\]  
differentiating both sides of Equation (30) for \(i\in\{1,..,\ p\}\) and \(l\in\{1,2,\ldots,\gamma_{i}\}\), which results in  
\[\dot{\overline{\epsilon}}_{i,l}=r_{i}^{(l)}-y_{i}^{(l)}=\overline{\epsilon}_{ i,\ l+1} \tag{31}\]  
It yields the following dynamics for the tracking error \(\overline{\epsilon}_{i,l}\,\ i\in\{1,..,\ p\}\) and \(l\in\{1,2,\ldots,\gamma_{i}\}\) :\[\left\{ \begin{array}{c}\dot{\overline{e}}_{i,1}=\overline{e}_{i,2}\\ \dot{\overline{e}}_{i,2}=\overline{e}_{i,3}\\ \vdots\\ \dot{\overline{e}}_{i,y_{i}}=r_{i}(y_{i})-y_{i}(y_{i})=r_{i}(y_{i})-\xi_{i,y_{i }},i\in\{1,..,\ p\}\end{array} \right. \tag{32}\]  
This, together with Equation (15b), gives:  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{i,1}=\overline{e}_{i,2}\\ \dot{\overline{e}}_{i,2}=\overline{e}_{i,3}\\ \vdots\\ \dot{\overline{e}}_{i,y_{i}}=r_{i}(y_{i})-\left(F_{i}^{\prime}+b_{i,i}u_{i} \right),\quad,i\in\{1,\ldots,p\}\end{array} \right. \tag{33}\]  
Substituting Equation (25) in Equation (33), we get,  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{i,1}=\overline{e}_{i,2}\\ \dot{\overline{e}}_{i,2}=\overline{e}_{i,3}\\ \vdots\\ \dot{\overline{e}}_{i,y_{i}}=r_{i}(y_{i})-b_{i,i}v_{i}+\xi_{i,y_{i}+1}-F_{i}^ {\prime},\quad i\in\{1,\ldots,p\}\end{array} \right. \tag{34}\]  
Based on Assumption 1, it follows that  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{i,1}=\overline{e}_{i,2}\\ \dot{\overline{e}}_{i,2}=\overline{e}_{i,3}\\ \vdots\\ \dot{\overline{e}}_{i,y_{i}}=r_{i}(y_{i})-b_{i,i}v_{i},i\in\{1,\ldots,p\} \end{array} \right. \tag{35}\]  
The dynamics of the tracking error given in Equation (35), together with the control law \(v_{i}\) designed in Equation (26), are written as:  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{i,1}=\overline{e}_{i,2}\\ \dot{\overline{e}}_{i,2}=\overline{e}_{i,3}\\ \vdots\\ \dot{\overline{e}}_{i,y_{i}}=-b_{i,i}\dot{\boldsymbol{e}}_{i,1}(\overline{e }_{i,1})\overline{e}_{i,1}-b_{i,i}\dot{\boldsymbol{e}}_{i,2}(\overline{e}_{i,2})\overline{e}_{i,2}-\cdots-b_{i,i}\dot{\boldsymbol{e}}_{i,y_{i}}\!\!\! \left(\overline{e}_{i,y_{i}}\!\!\!\right)\overline{e}_{i,y_{i}}\end{array}  \right. \tag{36}\]  
with \(\ i\in\{1,..,\ p\}\). The dynamics given in Equation (36) can be represented in matrix form as:  
\[\dot{\overline{e}}_{i}=A_{i}\overline{e}_{i},\ \ i\in\{1,..,\ p\}, \tag{37}\]  
where,  
\[A_{i}=\left( \begin{array}{cccccc}0&1&0&\cdots&0&0\\ 0&0&1&\cdots&0&0\\ \vdots&\cdots&\cdots&\cdots&\vdots&\vdots\\ 0&0&0&\cdots&1&0\\ 0&0&0&\cdots&0&1\\ -\overline{\boldsymbol{e}}_{i,1}(\overline{e}_{i,1})&-\overline{\boldsymbol{e }}_{i,1}(\overline{e}_{i,2})&-\overline{\boldsymbol{e}}_{i,3}(\overline{e}_ {i,3})&\cdots&-\overline{\boldsymbol{e}}_{i,Y_{i}-1}\!\!\left(\overline{e}_{ i,Y_{i}-1}\right)&-\overline{\boldsymbol{e}}_{i,Y_{i}}\!\!\!\left(\overline{e}_{ i,Y_{i}}\!\!\!\right)\end{array} \right. \tag{38}\]\(\overline{\hat{\epsilon}}_{i,1}=b_{i,i}\hat{\epsilon}_{i,j}\), \(i\in\{1,..,\ p\}\), \(l\in\{1,2,\ldots,\gamma_{i}\}\), and \(\overline{\epsilon}_{i}=\left(\overline{\epsilon}_{i,1},\overline{\epsilon}_{ i,2},\ldots,\overline{\epsilon}_{i,\ \gamma_{i}}\right)^{T}\).