## 4 Non-Overshooting Tracking Control  
In this section, we will calculate the expressions of tracking error through a CLS. Then, the non-overshooting control conditions with system orders of \(n=1\), \(n=2\), \(n=3\), and \(n\geq 4\) are obtained, respectively. Combining (25)-(27), we can obtain the following CLS. \[\left\{ \begin{array}{l}\dot{\eta}_{1}=-\varepsilon_{1}\eta_{1}+\eta_{2},\\ \dot{\eta}_{i}=-\varepsilon_{i}\eta_{i}+\eta_{i+1},\\ \dot{\eta}_{n}=-\varepsilon_{n}\eta_{n},\end{array} \right.\quad i=2,\cdots,n -1, \tag{30}\]  
We will analyze the control problem with NOTC conditions according to the following situations. (1) **The situation when \(n=1\)**. The CLS (30) is a first-order system \(\dot{\eta}_{1}=-\varepsilon_{1}\eta_{1}\). This subsystem is integrated along \([0,t]\), and \(\eta_{1}(t)\) is obtained as  
\[\eta_{1}(t)=\eta_{1}(0)e^{-\varepsilon_{1}t}, \tag{31}\]  
where \(\eta_{1}(0)\) is the initial value of \(\eta_{1}(t)\). Then, we obtain  
\[\eta_{1}(0)\geq 0\Rightarrow\eta_{1}(t)\geq 0, \tag{32}\]  
From (2), (9), (26) and \(Y_{0}=\ln\frac{y_{0}(t)-k_{11}}{k_{12}-y_{0}(t)}\), we have  
\[ \begin{split}\eta_{1}(t)\geq 0\Rightarrow&\theta_{1}(t)-Y_{ 0}(t)\geq 0\\ \Rightarrow&\ln\frac{\xi_{1}(t)-k_{11}}{k_{12}-\xi_{ 1}(t)}-\ln\frac{y_{0}(t)-k_{11}}{k_{12}-y_{0}(t)}\geq 0\\ \Rightarrow&\big{(}\xi_{1}(t)-k_{11})(k_{12}-y_{0}(t ))\\ \Rightarrow&\big{(}k_{12}-k_{11}\big{)}(y(t)-y_{0}(t ))\geq 0.\end{split}  \tag{33}\]  
Since \(k_{12}-k_{11}>0\), then  
\[y(t)-y_{0}(t)\geq 0\Rightarrow e(t)\geq 0. \tag{34}\]  
Combining (32)-(34), we get  
\[\eta_{1}(0)\geq 0\Rightarrow e(t)\geq 0. \tag{35}\]  
Similarly, the following result is also true  
\[\eta_{1}(0)\leq 0\Rightarrow e(t)\leq 0. \tag{36}\]  
To sum up, we can obtain the following theorem. **Theorem 1**. : _For the first-order system (1), we can make all the signals of the CLS bounded by designing the controller (27), and the tracking error satisfies \(\lim_{t\to\infty}e(t)=0\). At the same time, the NOTC needs to satisfy one of the conditions (35) and (36)._  
(2) **The situation when \(n=2\)**. The CLS is a second-order system \(\dot{\eta}_{1}=-\epsilon_{1}\eta_{1}+\eta_{2}\) and \(\dot{\eta}_{2}=-\epsilon_{2}\eta_{2}\). We will divide it into two cases to obtain the condition of non-overshooting. **Case 1:**\(\epsilon_{1}\
eq\epsilon_{2}\). We first solve the integral of \(\eta_{2}\) along \([0,t]\) is \(\eta_{2}(t)=\eta_{2}(0)e^{-\epsilon_{2}t}\), where \(\eta_{2}(0)\) is the initial value of \(\eta_{2}(t)\). Then, substituting \(\eta_{2}(t)\) into \(\dot{\eta}_{1}=-\epsilon_{1}\eta_{1}+\eta_{2}\) to get  
\[\dot{\eta}_{1}=-\epsilon_{1}\eta_{1}+\eta_{2}(0)e^{-\epsilon_{2}t}. \tag{37}\]  
This subsystem is integrated along \([0,t]\), and \(\eta_{1}(t)\) is obtained as  
\[\eta_{1}(t)=[\eta_{1}(0)-\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}]e^{- \epsilon_{1}t}+\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}}e^{-\epsilon_{2} t}, \tag{38}\]  
By analyzing (38), we can get the following conditions  
\[\left\{ \begin{array}{l}\eta_{1}(0)\leq 0,\eta_{2}(0)\geq 0,\\ \epsilon_{1}\leq\frac{\eta_{2}(0)}{\eta_{1}(0)}+\epsilon_{2},\epsilon_{2}\
eq -\frac{\eta_{2}(0)}{\eta_{1}(0)},\end{array} \right.\Rightarrow\eta_{1}(t) \leq 0, \tag{39}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq 0,\epsilon_{1}<\epsilon_{2},\\ \eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}},\end{array}  \right.\Rightarrow\eta_{1}(t)\geq 0. \tag{40}\]  
According to (33) and (34), the following two conditions with non-overshooting can be obtained. \[\left\{ \begin{array}{l}\eta_{1}(0)\leq 0,\eta_{2}(0)\geq 0,\\ \epsilon_{1}\leq\frac{\eta_{2}(0)}{\eta_{1}(0)}+\epsilon_{2},\epsilon_{2}\
eq -\frac{\eta_{2}(0)}{\eta_{1}(0)},\end{array} \right.\Rightarrow e(t)\leq 0, \tag{41}\]  
\[\left\{ \begin{array}{l}\eta_{1}(0)\geq 0,\epsilon_{1}<\epsilon_{2},\\ \eta_{1}(0)\geq\frac{\eta_{2}(0)}{\epsilon_{1}-\epsilon_{2}},\end{array}  \right.\Rightarrow e(t)\geq 0. \tag{42}\]  
**Case 2:**\(\epsilon_{1}=\epsilon_{2}=\epsilon\).