## 4 Non-Overshooting Tracking Control
According to \(\epsilon_{1}=\epsilon_{2}=\epsilon\) and (30), we solve \(\dot{\eta}_{1}=-\epsilon\eta_{1}+\eta_{2}\) as follows  
\[\eta_{1}(t)=[\eta_{1}(0)+\eta_{2}(0)t]e^{-\epsilon t}. \tag{43}\]  
By analyzing Equation (43), we can obtain the following conditions  
\[\left\{ \begin{array}{l}\eta_{1}(0)\leq 0,\\ \eta_{2}(0)\leq 0,\end{array} \right.\Rightarrow\eta_{1}(t)\leq 0, \tag{44}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq 0,\\ \eta_{2}(0)\geq 0,\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{45}\]According to (33) and (34), the following two conditions with non-overshooting can be obtained. \[\left\{ \begin{array}{ll}\eta_{1}(0)\leq 0,\\ \eta_{2}(0)\leq 0,\end{array} \right.\Rightarrow e(t)\leq 0, \tag{46}\]  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\geq 0,\\ \eta_{2}(0)\geq 0,\end{array} \right.\Rightarrow e(t)\geq 0. \tag{47}\]  
To sum up, we can obtain the following theorem. **Theorem 2**. : _For the second-order system (1), we constrain the controller (27) so that all signals of the CLS are bounded, and the tracking error satisfies \(\lim_{t\rightarrow\infty}e(t)=0\), and the states constraint is not violated. At the same time, NOTC can result in the following two situations, according to the control gain:_  
1. _If_ \(\epsilon_{1}\
eq\epsilon_{2}\)_, then NOTC can satisfy (_41_) or (_42_);_
2. _If_ \(\epsilon_{1}=\epsilon_{2}\)_, then NOTC can satisfy (_46_) or (_47_)._
3. **The situation when** \(n=3\)_._  
The CLS is a third-order system; that is, \(\dot{\eta}_{1}=-\epsilon_{1}\eta_{1}+\eta_{2}\) and \(\dot{\eta}_{2}=-\epsilon_{2}\eta_{2}+\eta_{3}\) and \(\dot{\eta}_{3}=-\epsilon_{3}\eta_{3}\). We will divide it into several cases to obtain the conditions of NOTC. **Case 1:**\(\epsilon_{1}\
eq\epsilon_{2}\
eq\epsilon_{3}\). Then, we obtain the expressions of \(\eta_{3}(t)\) and \(\eta_{2}(t)\) as follows  
\[\left\{ \begin{array}{ll}\eta_{3}(t)=\eta_{3}(0)e^{-\epsilon_{3}t},\\ \eta_{2}(t)=[\eta_{2}(0)-\frac{\eta_{3}(0)}{\epsilon_{2}-\epsilon_{3}}]e^{- \epsilon_{2}t}+\frac{\eta_{3}(0)}{\epsilon_{2}-\epsilon_{3}}e^{-\epsilon_{3}t},\end{array} \right. \tag{48}\]  
where \(\eta_{3}(0)\) is the initial value of \(\eta_{3}(t)\). Then, substituting (48) into \(\dot{\eta}_{1}=-\epsilon_{1}\eta_{1}+\eta_{2}\) to get  
\[\dot{\eta}_{1}=-\epsilon_{1}\eta_{1}+[\eta_{2}(0)-\frac{\eta_{3}(0)}{\epsilon _{2}-\epsilon_{3}}]e^{-\epsilon_{2}t}+\frac{\eta_{3}(0)}{\epsilon_{2}- \epsilon_{3}}e^{-\epsilon_{3}t}. \tag{49}\]  
This subsystem is integrated along \([0,t]\), and \(\eta_{1}(t)\) is obtained as  
\[ \begin{split}\eta_{1}(t)=&[\eta_{1}(0)-\frac{\eta _{2}(0)}{\epsilon_{1}-\epsilon_{2}}+\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon _{2})(\epsilon_{2}-\epsilon_{3})}\\ &-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}- \epsilon_{3})}]e^{-\epsilon_{1}t}\\ &+[\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}-\frac{\eta_{3} (0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{2}-\epsilon_{3})}]e^{-\epsilon_{2} t}\\ &+\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}- \epsilon_{3})}e^{-\epsilon_{3}t},\end{split}  \tag{50}\]  
we obtain the following conditions by analyzing (50). \[\left\{ \begin{array}{ll}\eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}- \epsilon_{2}}-\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{1}- \epsilon_{3})},\\ \frac{\eta_{2}(0)}{(\epsilon_{1}-\epsilon_{2})(\epsilon_{2}-\epsilon_{3})} \geq\frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3}) },&\Rightarrow\eta_{1}(t)\geq 0.\\ \frac{\eta_{3}(0)}{(\epsilon_{1}-\epsilon_{3})(\epsilon_{2}-\epsilon_{3})} \geq 0,&\Rightarrow\eta_{1}(t)\geq 0.\end{array} \right. \tag{51}\]\[\left\{