## 4 Non-Overshooting Tracking Control
\begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_ {2}}-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{1}-\epsilon_{3})},\\ \frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}\geq\frac{\eta_{3}(0)}{(\epsilon_{1 }-\epsilon_{2})(\epsilon_{1}-\epsilon_{3})},\\ \frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}\geq\frac{\eta_{3}(0)}{(\epsilon_ {1}-\epsilon_{2})(\epsilon_{1}-\epsilon_{3})},\\ \epsilon_{2}<\epsilon_{3},\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{52}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}- \epsilon_{2}}-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{1}- \epsilon_{3})},\\ \frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3})} \geq 0,\\ \frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}\geq\frac{\eta_{3}(0)}{(\epsilon_ {1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3})},\\ \epsilon_{3}<\epsilon_{2},\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{53}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}- \epsilon_{2}}-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{2}- \epsilon_{3})},\\ \eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}-\frac{\eta_{3}(0 )}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{1}-\epsilon_{3})},\\ \frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}\geq\frac{\eta_{3}(0)}{(\epsilon_ {1}-\epsilon_{2})(\epsilon_{2}-\epsilon_{3})},\\ \epsilon_{1}<\epsilon_{3},\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{54}\]  
\[\left\{ \begin{array}{ll}\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}\geq \frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{2}-\epsilon_{3})},\\ \eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}-\frac{\eta_{3}(0 )}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{2}-\epsilon_{3})},\\ \frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3})} \geq 0,\\ \epsilon_{3}<\epsilon_{1},\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{55}\]  
\[\left\{ \begin{array}{ll}\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})( \epsilon_{2}-\epsilon_{3})}\geq 0,\\ \eta_{2}(0)\leq\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}-\frac{\eta_{3}(0 )}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{1}-\epsilon_{3})},\\ \eta_{1}(0)\geq-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}- \epsilon_{3})},\\ \epsilon_{1}<\epsilon_{2},\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{56}\]  
\[\left\{ \begin{array}{ll}\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})( \epsilon_{2}-\epsilon_{3})}\geq 0,\\ \eta_{2}(0)\leq\frac{\eta_{3}(0)}{\epsilon_{2}-\epsilon_{3}},\\ \eta_{1}(0)\geq\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}- \epsilon_{3})},\\ \epsilon_{2}<\epsilon_{1}.\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{57}\]  
According to (33) and (34), \(\eta_{1}(t)\geq 0\Rightarrow e(t)\geq 0\). Therefore, any of inequalities (51)-(57) can satisfy that the tracking error satisfies \(e(t)\geq 0\). Then, inequalities (51)-(57) are NOTC conditions. In the same way, we can obtain the following conditions by analyzing (50)  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq\frac{\eta_{2}(0)}{\epsilon_{1}- \epsilon_{2}}-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{1}- \epsilon_{3})},\\ \frac{\eta_{2}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{2}-\epsilon_{3})} \leq\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3})},\\ \frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3})}\leq 0. \end{array} \right.\Rightarrow\eta_{1}(t)\leq 0. \tag{58}\]\[\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_{1}-\varepsilon_{3})},\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}}\leq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon_{2})(\varepsilon_{2}-\varepsilon_{3})},\\ \frac{\eta_{2}(0)}{\varepsilon_{1}-\varepsilon_{2}}\leq\frac{\eta_{3}(0)}{( \varepsilon_{1}-\varepsilon_{2})(\varepsilon_{1}-\varepsilon_{3})}, \varepsilon_{2}<\varepsilon_{3},\end{array} \right.\Rightarrow\eta_{1}(t)\leq 0.