## 4 Non-Overshooting Tracking Control
\tag{59}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{3}(0)}{\varepsilon_{1}- \varepsilon_{2}}-\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{2})( \varepsilon_{1}-\varepsilon_{3})},\\ \frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{3})(\varepsilon_{2}- \varepsilon_{3})}\leq 0,\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}}\leq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon_{2})(\varepsilon_{1}-\varepsilon_{3})}, \varepsilon_{3}<\varepsilon_{2},\end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{60}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{2}(0)}{\varepsilon_{1}- \varepsilon_{2}}-\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{2})( \varepsilon_{2}-\varepsilon_{3})},\\ \eta_{1}(0)\leq\frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}}-\frac{ \eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{2})(\varepsilon_{1}-\varepsilon_{3 })},\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}}\leq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon_{2})(\varepsilon_{2}-\varepsilon_{3})}, \varepsilon_{1}<\varepsilon_{3},\end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{61}\]  
\[\left\{ \begin{array}{ll}\frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}} \leq\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{2})(\varepsilon_{2}- \varepsilon_{3})},\\ \eta_{1}(0)\leq\frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}}-\frac{ \eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{2})(\varepsilon_{2}-\varepsilon_{3 })},\\ \frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{3})(\varepsilon_{2}- \varepsilon_{3})}\leq 0,\varepsilon_{3}<\varepsilon_{1},\end{array} \right.\Rightarrow\eta_{1}(t) \leq 0. \tag{62}\]  
\[\left\{ \begin{array}{ll}\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{3})( \varepsilon_{2}-\varepsilon_{3})}\leq 0,\\ \eta_{2}(0)\geq\frac{\eta_{3}(0)}{(\varepsilon_{2}-\varepsilon_{3})},\\ \eta_{1}(0)\leq\frac{\eta_{3}(0)}{(\varepsilon_{1}-\varepsilon_{3})( \varepsilon_{2}-\varepsilon_{3})},\varepsilon_{2}<\varepsilon_{1},\end{array}  \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{63}\]  
Similarly, according to (33) and (34), \(\eta_{1}(t)\leq 0\Rightarrow e(t)\leq 0\). Therefore, any of the inequalities (58)-(64) can satisfy that the tracking error satisfies \(e(t)\leq 0\). Then, inequalities (58)-(64) are NOTC conditions. **Case 2:**\(\varepsilon_{1}=\varepsilon_{2}=\varepsilon\
eq\varepsilon_{3}\). In this case, we can determine the expression of \(\varepsilon_{3}(t)\) and \(\varepsilon_{2}(t)\) as shown in (48). According to \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon\), we integrate \(\dot{\eta}_{1}=-\varepsilon\eta_{1}+\eta_{2}\) along \([0,t]\), then \(\eta_{1}(t)\) is  
\[\eta_{1}(t)= [\eta_{1}(0)-\frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{3})^{2 }}+(\eta_{2}(0)-\frac{\eta_{3}(0)}{\varepsilon-\varepsilon_{3}})t]e^{- \varepsilon t} \tag{65}\] \[+\frac{\eta_{3}(0)}{(\varepsilon-\varepsilon_{3})^{2}}e^{- \varepsilon_{3}t}.\]  
we obtain the following conditions by analyzing (65)  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{3}(0)}{(\varepsilon- \varepsilon_{3})^{2}},\\ \eta_{2}(0)\leq\frac{\eta_{3}(0)}{\varepsilon-\varepsilon_{3}},\eta_{3}(0) \leq 0,\end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{66}\]\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq 0,\eta_{2}(0)\leq\frac{\eta_{3}(0)}{ \varepsilon-\varepsilon_{3}},&\Rightarrow\eta_{1}(t)\leq 0.\\ \eta_{3}(0)\geq 0,\varepsilon<\varepsilon_{3},&\Rightarrow\eta_{1}(t)\geq 0.\\ \end{array} \right. \tag{67}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq 0,\eta_{2}(0)\leq\frac{\eta_{3}(0)}{ \varepsilon-\varepsilon_{3}},&\Rightarrow\eta_{1}(t)\leq 0.\\ \end{array} \right. \tag{68}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{2}(0)}{ \varepsilon-\varepsilon_{3}},\eta_{2}(0)\leq\frac{\eta_{2}(0)}{\varepsilon- \varepsilon_{3}},&\Rightarrow\eta_{1}(t)\leq 0.\\ \end{array} \right. \tag{69}\]  
According to (33) and (34), \(\eta_{1}(t)\leq 0\Rightarrow e(t)\leq 0\). Therefore, any of the inequalities (66)-(69) can indicate that the tracking error satisfies \(e(t)\leq 0\). Then, inequalities (66)-(69) are NOTC conditions.