## 4 Main Results
** The tracking error between the reference trajectory and the corresponding system estimated states is given as:  
\[\overline{e_{i}}=r_{i}-\hat{z}_{i}\,i\in\{1,2,\ldots,n\} \tag{39}\]  
With outer-loop LESO and TD as in assumptions A3 and A4 respectively, the tracking error can be described as,  
\[\overline{e_{i}}=r^{(i-1)}-x_{i}\,i\in\{1,2,\ldots,n\} \tag{40}\]  
For the system given in Equation (1), the states \(x_{i}\) are expressed in term of the plant output,  
\[|x_{i}(t)-\hat{z}_{i}(t)|\leq\frac{1}{\left(\omega_{0}^{\prime}\right)^{n+1-i} }\ast\left(\frac{2M^{\prime}\lambda^{2}_{\text{max}}(P^{\prime})}{\omega_{0}^{ \prime}\lambda_{\text{min}}(P^{\prime})}\Big{(}1-e^{-\frac{t}{2\lambda_{\text{ max}}(P^{\prime})}}\Big{)}+\sqrt{\frac{V^{\prime}(\gamma(0))}{\lambda_{\text{min}}(P)}}e^{- \frac{t}{2\lambda_{\text{max}}(P^{\prime})}}\right)\]0  
Substitute Equations (41) in (40), and the tracking error is given by  
\[|x_{i}(t)-\hat{z}_{i}(t)|\leq\frac{1}{\left(\omega_{0}^{\prime}\right)^{n+1-i} }\ast\left(\frac{2M^{\prime}\lambda^{2}_{\text{max}}(P^{\prime})}{\omega_{0}^{ \prime}\lambda_{\text{min}}(P^{\prime})}\Big{(}1-e^{-\frac{t}{2\lambda_{\text{ max}}(P^{\prime})}}\Big{)}+\sqrt{\frac{V^{\prime}(\gamma(0))}{\lambda_{\text{min}}(P)}}e^{- \frac{t}{2\lambda_{\text{max}}(P^{\prime})}}\right)\]1Differentiating Equation (42) with respect to time, gives  
\[|x_{i}(t)-\hat{z}_{i}(t)|\leq\frac{1}{\left(\omega_{0}^{\prime}\right)^{n+1-i} }\ast\left(\frac{2M^{\prime}\lambda^{2}_{\text{max}}(P^{\prime})}{\omega_{0}^{ \prime}\lambda_{\text{min}}(P^{\prime})}\Big{(}1-e^{-\frac{t}{2\lambda_{\text{ max}}(P^{\prime})}}\Big{)}+\sqrt{\frac{V^{\prime}(\gamma(0))}{\lambda_{\text{min}}(P)}}e^{- \frac{t}{2\lambda_{\text{max}}(P^{\prime})}}\right)\]2  
It follows that the tracking error dynamics \(\overline{e_{i}}\,i\in\{1,2,\ldots,n\}\) are given below  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{1}=\overline{e}_{2},\\ \dot{\overline{e}}_{2}=\overline{e}_{3},\\ \dot{\overline{e}}_{n}=r^{(n)}-y^{(n)}=r^{(n)}-\dot{x}_{n}\end{array} \right. \tag{43}\]  
This together with \(n\)-\(th\) equation of Equation (27) gives,  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{1}=\overline{e}_{2},\\ \dot{\overline{e}}_{2}=\overline{e}_{3},\\ \dot{\overline{e}}_{n}=r^{(n)}-(x_{n+1}+v^{\prime}(t))\end{array} \right. \tag{44}\]  
From Equation (37), we get  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{1}=\overline{e}_{2},\\ \dot{\overline{e}}_{2}=\overline{e}_{3},\\ \dot{\overline{e}}_{n}=r^{(n)}-x_{n+1}-v+\dot{z}_{n+1}\end{array} \right. \tag{45}\]  
It follows from Equations (35) and (45) that  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{1}=\overline{e}_{2},\\ \dot{\overline{e}}_{2}=\overline{e}_{3},\\ \dot{\overline{e}}_{n}=r^{(n)}-v\end{array} \right. \tag{46}\]  
The tracking error dynamics given in Equation (46) associated with the control law \(v\) designed in Equation (38) produces the following closed-loop error dynamics if we assume that the \(n\)-th derivative of the reference signal \(r^{(n)}\) equal to zero  
\[\left\{ \begin{array}{c}\dot{\overline{e}}_{1}=\overline{e}_{2},\\ \dot{\overline{e}}_{2}=\overline{e}_{3},\\ \dot{\overline{e}}_{n}=-\|_{1}(\overline{e}_{1})\overline{e}_{1}-\|_{2}( \overline{e}_{2})\overline{e}_{2}-\ldots-\|_{n}(\overline{e}_{n})\overline{e }_{n}\end{array} \right. \tag{47}\]  
The dynamics given in Equation (47) can by represented as:  
\[\dot{\overline{e}}=A\overline{e}\]where  
\[A=\left( \begin{array}{cccccc}0&1&0&\ldots&0&0\\ 0&0&1&\ldots&0&0\\ \vdots&\vdots&\vdots&\ldots&\vdots&\vdots\\ 0&0&0&\ldots&0&1\\ -\|_{1}(\widetilde{e}_{1})&-\|_{2}(\widetilde{e}_{2})&-\|_{3}(\widetilde{e}_{3} )&-\|_{n-1}(\widetilde{e}_{n-1})&-\|_{n}(\widetilde{e}_{n})\end{array} \right)\]  
and \(\widetilde{e}=(\widetilde{e}_{1},\widetilde{e}_{2},\ldots,\widetilde{e}_{n}) ^{T}\).