## 4 Non-Overshooting Tracking Control
\begin{array}{ll}\eta_{1}(0)\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.\\ \eta_{3}(0)\leq 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{92}\]  
According to (33) and (34), \(\eta_{1}(t)\geq 0\Rightarrow\varepsilon(t)\geq 0\). Therefore, any one of the inequalities (89)-(92) can satisfy that the tracking error satisfies \(e(t)\geq 0\). Then, inequalities (89)-(92) are NOTC conditions. **Case 5:**\(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=\varepsilon\). According to (30), we integrate this subsystem along \([0,t]\), \(\eta_{3}(t)\) is obtained as (48), and \(\eta_{2}(t)\) is obtained as (74). Then, \(\eta_{1}(t)\) can be written as  
\[\eta_{1}(t)=(\eta_{1}(0)+\eta_{2}(0)t+\eta_{3}(0)t^{2})e^{-ct}. \tag{93}\]  
We obtain the following conditions by analyzing (93)  
\[\left\{ \begin{array}{ll}\eta_{1}(0)\leq 0,&\\ \eta_{2}(0)\leq 0,&\Rightarrow\eta_{1}(t)\leq 0,\\ \eta_{3}(0)\leq 0,&\end{array} \right. \tag{94}\]  
\[\left\{ \begin{array}{ll}\eta_{3}(0)<0,&\\ 2\eta_{1}(0)\eta_{3}(0)-\eta_{2}^{2}(0)\geq 0,&\Rightarrow\eta_{1}(t)\leq 0. \end{array} \right. \tag{95}\]According to (33) and (34), \(\eta_{1}(t)\leq 0\Rightarrow e(t)\leq 0\). Therefore, either of the inequalities (94) and (95) can satisfy that the tracking error satisfies \(e(t)\leq 0\). Then, inequalities (94) and (95) are NOTC conditions. Subsequently, we obtain one of the conditions for ensuring that \(\eta_{1}(t)\geq 0\) is as follows  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq 0,\\ \eta_{2}(0)\geq 0,\\ \eta_{3}(0)\geq 0,\end{array} \right.\Rightarrow\eta_{1}(t)\geq 0, \tag{96}\]  
\[\left\{ \begin{array}{l}\eta_{3}(0)>0,\\ 2\eta_{1}(0)\eta_{3}(0)-\eta_{2}^{2}(0)\geq 0.\end{array} \right.\Rightarrow\eta_{1}(t) \geq 0, \tag{97}\]  
According to (33) and (34), \(\eta_{1}(t)\geq 0\Rightarrow e(t)\geq 0\). Therefore, inequalities (96) and (97) can satisfy that the tracking error satisfies \(e(t)\geq 0\). Then, inequalities (96) and (97) are NOTC conditions. To sum up, we obtain the following theorem. Theorem 3: For the third-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 create the following two situations according to the control gain:  
(1) If \(\varepsilon_{1}\
eq\varepsilon_{2}\
eq\varepsilon_{3}\), NOTC conditions need to satisfy one of the conditions (51)-(64);  
(2) If \(\varepsilon_{1}=\varepsilon_{2}\
eq\varepsilon_{3}\), NOTC conditions need to satisfy one of the conditions (66)-(73);  
(3) If \(\varepsilon_{1}\
eq\varepsilon_{2}=\varepsilon_{3}\), NOTC conditions need to satisfy one of the conditions (76)-(83);  
(4) If \(\varepsilon_{1}=\varepsilon_{3}\
eq\varepsilon_{2}\), NOTC conditions need to satisfy one of the conditions (85)-(92);  
(5) If \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}\), NOTC conditions need to satisfy one of the conditions (94)-(97). (4) The situation when \(n\geq 4\). If the order of the system is \(n\geq 4\), if the conditions of NOTC are analyzed by calculating the equation of \(\eta_{1}(t)\), then the calculation amount at this time is very complicated. Therefore, this article will introduce the following methods in this part. This method no longer depends on the control gain \(\varepsilon_{1},\cdots,\varepsilon_{n}\). At the same time, this method is divided into the following two parts to illustrate. **Case 1:**\(\eta_{n}(0)\leq 0\). The subsystem \(\vartheta_{n}=-c_{n}\eta_{n}\) is integrated along \([0,t]\). Then, we get \(\eta_{n}(t)\) as  
\[\eta_{n}(t)=\eta_{n}(0)e^{-\varepsilon_{n}t}. \tag{98}\]  
If \(\eta_{n}(t)\leq 0\), then  
\[\eta_{n}(0)\leq 0\Rightarrow\eta_{n}(t)\leq 0. \tag{99}\]  
According to \(\dot{\eta}_{n-1}(t)=-\varepsilon_{n}\eta_{n-1}(t)+\eta_{n}(t)\).