## 4 Non-Overshooting Tracking Control
\begin{array}{l}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{\varepsilon_{1}- \varepsilon}-\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon)^{2}},\eta_{3} (0)\leq 0,\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon}\geq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon)^{2}},\varepsilon>\varepsilon_{1},\\ \end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{82}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq 0,\eta_{2}(0)\geq\frac{\eta_{3}(0)}{ \varepsilon_{1}-\varepsilon},\\ \eta_{3}(0)\geq 0,\varepsilon<\varepsilon_{1},\\ \end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{83}\]  
According to (33) and (34), \(\eta_{1}(t)\geq 0\Rightarrow e(t)\geq 0\). Therefore, any of the inequalities (80)-(83) can satisfy that the tracking error satisfies \(e(t)\geq 0\). Then, inequalities (80)-(83) are NOTC conditions. **Case 4:**\(\varepsilon_{1}=\varepsilon_{3}=\varepsilon\
eq\varepsilon_{2}\). According to (30), we integrate this subsystem along \([0,t]\), \(\eta_{3}(t)\) and \(\eta_{2}(t)\) are obtained as (48), and \(\eta_{1}(t)\) can be written as  
\[ \begin{array}{l}\eta_{1}(t)=[\eta_{1}(0)-\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{2}}-\frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{2}) ^{2}}+\frac{\eta_{3}(0)}{\varepsilon_{2}-\varepsilon}t]e^{-\varepsilon t}\\ \hskip 14.226378 pt+[\frac{\eta_{2}(0)}{\varepsilon-\varepsilon_{2}}+ \frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{2})^{2}}]e^{-\varepsilon_{2}t}. \end{array}  \tag{84}\]  
We obtain the following conditions by analyzing (84)  
\[\left\{ \begin{array}{l}\frac{\eta_{3}(0)}{\eta_{2}-\eta}\leq 0,\frac{\eta_{1}(0)}{( \eta_{1}-\eta_{2})^{2}}\leq\frac{\eta_{2}(0)}{\varepsilon_{2}-\varepsilon},\\ \eta_{1}(0)\leq\frac{\eta_{2}(0)}{\varepsilon-\varepsilon_{2}}+\frac{\eta_{3} (0)}{(\varepsilon-\varepsilon_{2})^{2}},\end{array} \right.\Rightarrow\eta_{1}(t )\leq 0. \tag{85}\]\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq 0,\eta_{2}(0)\leq\frac{\eta_{3}(0)}{ \varepsilon-\varepsilon_{2}},&\Rightarrow\eta_{1}(t)\leq 0.\\ \eta_{3}(0)\geq 0,\varepsilon<\varepsilon_{2},&\Rightarrow\eta_{1}(t)\geq 0. \end{array} \right. \tag{86}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq 0,\eta_{2}(0)\leq\frac{\eta_{3}(0)}{ \varepsilon-\varepsilon_{2}},&\Rightarrow\eta_{1}(t)\leq 0.\\ \eta_{3}(0)\geq 0,\varepsilon_{2}<\varepsilon,&\Rightarrow\eta_{1}(t)\geq 0. \end{array} \right. \tag{87}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{2}}+\frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{2 })^{2}},&\Rightarrow\eta_{3}(0)\leq 0,\\ \frac{\eta_{2}(0)}{\varepsilon-\varepsilon_{2}}+\frac{\eta_{3}(0)}{( \varepsilon-\varepsilon_{2})^{2}}\leq 0,&\varepsilon<\varepsilon_{2}. \end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{88}\]  
According to (33) and (34), \(\eta_{1}(t)\leq 0\Rightarrow\varepsilon(t)\leq 0\). Therefore, any one of the inequalities (85)-(88) can satisfy that the tracking error satisfies \(e(t)\leq 0\). Then, inequalities (85)-(88) are NOTC conditions. In the same way, we can obtain the following conditions by analyzing (84)  
\[\left\{ \begin{array}{ll}\frac{\eta_{3}(0)}{\varepsilon_{2}- \varepsilon}\geq 0,\frac{\eta_{3}(0)}{\varepsilon-\varepsilon_{2}}+\frac{ \eta_{2}(0)}{\varepsilon_{2}-\varepsilon}\geq 0,&\Rightarrow\eta_{1}(t)\geq 0. \end{array} \right. \tag{89}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq 0,\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{2}}+\frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{2 })^{2}}\leq 0,&\Rightarrow\eta_{1}(t)\geq 0.\end{array} \right. \tag{90}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{2}}+\frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{2 })^{2}},&\Rightarrow\eta_{1}(t)\geq 0.\\ \frac{\eta_{2}(0)}{\varepsilon-\varepsilon_{2}}+\frac{\eta_{3}(0)}{( \varepsilon-\varepsilon_{2})^{2}}\geq 0,&\Rightarrow\eta_{1}(t)\geq 0.\end{array} \right. \tag{91}\]  
\[\left\{