## 4 Non-Overshooting Tracking Control
In the same way, we can obtain the following conditions by analyzing (65)  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{( \varepsilon-\varepsilon_{3})^{2}},\eta_{2}(0)\geq\frac{\eta_{3}(0)}{ \varepsilon-\varepsilon_{3}},&\Rightarrow\eta_{1}(t)\geq 0.\\ \eta_{3}(0)\geq 0,&\end{array} \right. \tag{70}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq 0,\eta_{2}(0)\geq\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{3}},&\Rightarrow\eta_{1}(t)\geq 0.\\ \end{array} \right. \tag{71}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{3}},\eta_{2}(0)\geq\frac{\eta_{2}(0)}{\varepsilon- \varepsilon_{3}},&\Rightarrow\eta_{1}(t)\geq 0.\\ \end{array} \right. \tag{72}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq 0,\eta_{2}(0)\geq\frac{\eta_{3}(0)}{ \varepsilon-\varepsilon_{3}},&\Rightarrow\eta_{1}(t)\geq 0.\\ \eta_{3}(0)\geq 0,\varepsilon_{3}<\varepsilon.&\end{array} \right. \tag{73}\]  
According to (33) and (34), \(\eta_{1}(t)\geq 0\Rightarrow e(t)\geq 0\). Therefore, any of the inequalities (70)-(73) can indicate that the tracking error satisfies \(e(t)\geq 0\). Then, inequalities (70)-(73) are NOTC conditions. **Case 3:**\(\varepsilon_{1}\
eq\varepsilon_{2}=\varepsilon_{3}=\varepsilon\). According to (30), we integrate this subsystem along \([0,t]\), \(\eta_{3}(t)\) is obtained as (48), and \(\eta_{2}(t)\) and \(\eta_{1}(t)\) can be written as  
\[\eta_{2}(t)=(\eta_{2}(0)+\eta_{3}(0)t)e^{-\varepsilon t}, \tag{74}\]  
\[\eta_{1}(t)= [\eta_{1}(0)-\frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}+ \frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon)^{2}}]e^{-\varepsilon_{1}t} \tag{75}\] \[+[\frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}-\frac{\eta_{3} (0)}{(\varepsilon_{1}-\varepsilon)^{2}}+\frac{\eta_{3}(0)}{\varepsilon_{1}- \varepsilon}t]e^{-\varepsilon t}.\]  
We obtain the following conditions by analyzing (75)  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{2}(0)}{ \varepsilon_{1}-\varepsilon}-\frac{\eta_{3}(0)}{(\varepsilon_{1}- \varepsilon)^{2}},&\Rightarrow\eta_{1}(t)\leq 0.\\ \end{array} \right. \tag{76}\]\[\left\{ \begin{array}{l}\eta_{1}(0)\leq 0,\eta_{3}(0)\geq 0,\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}\geq\eta_{1}(0)+\frac{\eta_{3} (0)}{(\varepsilon_{1}-\varepsilon)^{2}},\varepsilon>\varepsilon_{1},\\ \end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{77}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\leq\frac{\eta_{2}(0)}{\varepsilon_{1}- \varepsilon}-\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon)^{2}},\eta_{3} (0)\geq 0,\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}\leq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon)^{2}},\varepsilon>\varepsilon_{1},\\ \end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{78}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\leq 0,\eta_{2}(0)\leq\frac{\eta_{3}(0)}{ \varepsilon_{1}-\varepsilon},\\ \eta_{3}(0)\leq 0,\varepsilon<\varepsilon_{1},\\ \end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{79}\]  
According to (33) and (34), \(\eta_{1}(t)\leq 0\Rightarrow e(t)\leq 0\). Therefore, any of the inequalities (76)-(79) can satisfy that the tracking error satisfies \(e(t)\leq 0\). Then, inequalities (76)-(79) are NOTC conditions. In the same way, we can obtain the following conditions by analyzing (75)  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{\varepsilon_{1}- \varepsilon}-\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon)^{2}},\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}\geq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon)^{2}},\varepsilon_{1}-\varepsilon_{1}\geq 0,\\ \end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{80}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq 0,\eta_{3}(0)\geq 0,\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}\leq\eta_{1}(0)+\frac{\eta_{3 }(0)}{(\varepsilon_{1}-\varepsilon)^{2}},\varepsilon>\varepsilon_{1},\\ \end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{81}\]  
\[\left\{