## 3 Full-State Constraints Control and Control Design  
To solve the FSCs problem of the system (1), the following constraint function is introduced.  
\[x_{i}=\ln\frac{\xi_{i}(t)-k_{i1}}{k_{i2}-\xi_{i}(t)},\quad i=1,\cdots,n, \tag{9}\]  
where \(k_{i1}<0\) and \(k_{i2}>0\) are constraint parameters.  
From (5), we get  
\[\dot{x}_{i}=\frac{k_{i2}-k_{i1}}{(\xi_{i}-k_{i1})(k_{i2}-\xi_{i})}\dot{\xi}_{i}. \tag{10}\]  
Substituting (1) and the inverse functions of  
where  
\[b_{n}(\xi_{n})=\frac{\partial b_{n-1}(\xi_{n})}{\partial x_{n}}g_{n}(x_{n}), \tag{24}\] \[a_{n}(\xi_{n})=\sum_{j=1}^{n-1}[\frac{\partial a_{n-1}(\xi_{n-1})}{ \partial x_{j}}F_{j}(\xi_{j+1})\] \[\quad+\frac{\partial b_{n-1}(\tilde{x}_{n})}{\partial x_{j}}F_{j} (\tilde{x}_{j+1})]+\frac{\partial b_{n-1}(\tilde{x}_{n})}{\partial x_{n}}F_{n}( \tilde{x}_{n}).\]  
According to (11)-(24), we can summarize the following system  
\[\left\{\begin{array}{l}\dot{\theta}_{i}=\theta_{i+1},\quad i=1,\cdots,n-1, \\ \dot{\theta}_{n}=a_{n}(\tilde{x}_{n})+b_{n}(\tilde{x}_{n})u.\end{array}\right. \tag{25}\]  
Next, the controller is designed based on backstepping. First, we define the following coordinate transformation  
\[\eta_{1}=\theta_{1}-Y_{0},\quad\eta_{i}=\theta_{i}-\alpha_{i-1},\quad i=2, \cdots,n, \tag{26}\]  
where \(Y_{0}=\ln\frac{y_{0}(t)-k_{11}}{k_{12}-y_{0}(t)}\), and the constraint parameters \(k_{11}\) and \(k_{12}\) need to satisfy \(k_{11}<y_{0}(t)<k_{12}\), and \(\alpha_{i-1}\) is the virtual controller to be designed. According to the process of designing the controller by backstepping, the virtual controllers and the controller \(u\) are designed as follows  
\[\left\{\begin{array}{l}\alpha_{1}=-\varepsilon_{1}\eta_{1}+\dot{Y}_{0},\\ \alpha_{i}=-\varepsilon_{i}\eta_{i}+\dot{\alpha}_{i-1},\quad i=2,\cdots,n-1, \\ u=\frac{1}{b_{n}(\tilde{x}_{n})}[-\varepsilon_{n}\eta_{n}+\dot{\alpha}_{n-1} -a_{n}(\tilde{x}_{n})],\end{array}\right. \tag{27}\]  
where \(\varepsilon_{1}>\frac{1}{2}\), \(\varepsilon_{i}>1\) (\(i=2,\cdots,n-1\)) and \(\varepsilon_{n}>\frac{1}{2}\) is the control gain parameter. According to the above design process, the definition of \(b_{n}(\tilde{x}_{n})\) satisfies \(b_{n}(\tilde{x}_{n})\
eq 0\).  
Let Lyapunov function \(V=\frac{1}{2}\Sigma_{j=1}^{n}\eta_{j}^{2}\). Substituting the designed virtual controllers and controller (23) into \(\dot{V}\), then  
\[\dot{V}_{n}= -\sum_{j=1}^{n}\varepsilon_{j}\eta_{j}^{2}+\sum_{j=1}^{n-1}\eta_{ j}\eta_{j+1} \tag{28}\] \[\leq -(\varepsilon_{1}-\frac{1}{2})\eta_{1}^{2}-\sum_{j=2}^{n-1}( \varepsilon_{j}-1)\eta_{j}^{2}-(\varepsilon_{n}-\frac{1}{2})\eta_{n}^{2}\] \[\leq -\rho V_{n},\]  
where \(\rho=\min\{(2c_{1}-1),(2c_{i}-2),(2c_{n}-1)\}\) (\(i=2,\cdots,n-1\)).  
Therefore, signals \(\eta_{1},\cdots,\eta_{n}\) are bounded, and \(\eta_{1},\cdots,\eta_{n}\) satisfy \(\lim_{t\rightarrow\infty}\eta_{1}=0\), \(\cdots,\lim_{t\rightarrow\infty}\eta_{n}=0\). According to Lemma 1, \(\lim_{t\rightarrow\infty}e(t)=0\), and the signals \(\theta_{1},\cdots,\theta_{n}\) are bounded. From the definition of \(\theta_{1},\cdots,\theta_{n}\), signals \(x_{1}\), \(F_{1}(\tilde{x}_{2}),\cdots,F_{n-1}(\tilde{x}_{n})\) are bounded.  
Since \(F_{1}(\tilde{x}_{2})\) is a smooth function, there is at least one point \(\varepsilon_{1}\) (\(\varepsilon_{1}\in(\min(0,x_{2})\), \(\max(0,x_{2}))\)) to make  
\[F_{1}(x_{1},x_{2})-F_{1}(x_{1},0)=\frac{\partial F_{1}(x_{1},x_{2})}{\partial x _{2}}|_{x_{2}=\varepsilon_{1}}(x_{2}-0). \tag{29}\]  
Then, \(x_{2}(t)\) is also bounded. Similarly, we can determine that \(x_{3}(t),\cdots,x_{n}(t)\) are bounded. According to (5), the FSCs have not been violated. Therefore, \(k_{i1}<\xi_{i}(t)<k_{i2}\), (\(i=1,\cdots,n\)).  
**Remark 2**: _According to (28), we can get_  
\[|\eta_{i}|\leq\sqrt{2|V_{n}(0)|e^{-\rho t}},\quad i=1,\cdots,n.\]  
_From the above formula, we can see that the amplitude of the closed-loop system will increase with the increase in the initial value \(V_{n}(0)\). And with the increase in the parameter \(\rho\), the convergence speed of the closed-loop system increases._