## 5 RBFNN Design and Analysis of Position Output-Constrained Robust Adaptive Controller
### Control Algorithm Design  
In order to achieve the above control objectives, on the basis of the backstepping method and Lyapunov integration analysis and design, the adaptive control law and parameter learning update law were designed to ensure the adaptability and robustness of the designed controller.  
The position tracking error and its derivative signal are defined as follows:  
\[e_{1}=x_{1}-x_{d}, \tag{22}\]  
\[e_{2}=x_{2}-r, \tag{23}\]  
where \(r\) is the virtual control law to be designed, and \(x_{d}\) is the known train position information, i.e., the desired position.  
**Assumption 3**.: _The desired position \(x_{d}\) and its first-order derivatives \(\dot{x}_{d}\) and second-order derivatives \(\ddot{x}_{d}\) are known and bounded._  
The two-side differential of \(e_{1}\) is obtained as follows:  
\[\dot{e}_{1}=\dot{x}_{1}-\dot{x}_{d}=x_{2}-\dot{x}_{d}. \tag{24}\]  
The two-side differential of \(\dot{e}_{2}\) is obtained as follows:  
\[\begin{array}{cc}\dot{e}_{2}&=\dot{x}_{2}-\dot{r}\\ &=u+g+f(x)-\dot{r}\\ &=u-\frac{a_{0}}{m}-\frac{b_{0}}{m}x_{2}-\frac{c_{0}}{m}x_{2}^{2}+f(x)-\dot{r}.\end{array} \tag{25}\]  
By differentiating the two sides of the symmetric Barrier Lyapunov function \(V_{1}\) of Equation (20), it can be obtained that  
\[\dot{V}_{1}=\frac{e_{1}\,\dot{e}_{1}}{k^{2}-e_{1}^{2}}=\frac{e_{1}(x_{2}-\dot{ x}_{d})}{k^{2}-e_{1}^{2}}=\frac{e_{1}(e_{2}+r-\dot{x}_{d})}{k^{2}-e_{1}^{2}}. \tag{26}\]  
The design virtual control law is expressed as  
\[r=-\alpha e_{1}+\dot{x}_{d}, \tag{27}\]  
where \(\alpha>0\) represents the control parameters to be designed.  
The two-side differential of \(r\) is obtained as follows:  
\[\dot{r}=-\alpha\dot{e}_{1}+\ddot{x}_{d}. \tag{28}\]By substituting Equation (27) into Equation (26), we can get  
\[\dot{V}_{1}=-\frac{\alpha e_{1}^{2}}{k^{2}-e_{1}^{2}}+\frac{e_{1}e_{2}}{k^{2}-e_{ 1}^{2}}. \tag{29}\]  
The Lyapunov function is defined as  
\[V_{2}=V_{1}+\frac{1}{2}e_{2}^{2}. \tag{30}\]  
The two-side differential of \(V_{2}\) is obtained as follows:  
\[\dot{V}_{2}=\dot{V}_{1}+e_{2}\dot{e}_{2}=-\frac{\alpha e_{1}^{2}}{k^{2}-e_{1}^ {2}}+\frac{e_{1}e_{2}}{k^{2}-e_{1}^{2}}+e_{2}\dot{e}_{2}. \tag{31}\]  
By substituting Equation (25) into Equation (31), we can get  
\[\dot{V}_{2}=-\frac{\alpha e_{1}^{2}}{k^{2}-e_{1}^{2}}+\frac{e_{1}e_{2}}{k^{2}-e _{1}^{2}}+e_{2}[u-\frac{a_{0}}{m}-\frac{b_{0}}{m}x_{2}-\frac{c_{0}}{m}x_{2}^{2 }+f(\mathbf{x})-\dot{r}]. \tag{32}\]  
**Assumption 4**.: _The control input \(u\) is time-varying and bounded; that is, there is an unknown constant \(u^{*}\) satisfying \(|u|\leq u^{*}\)._  
The control law and adaptive law are designed as follows:  
\[u=\hat{a}+\hat{b}x_{2}+\hat{c}x_{2}^{2}-\hat{f}(\mathbf{x})+\dot{r}-\beta e_{2}- \frac{e_{1}}{k^{2}-e_{1}^{2}}, \tag{33}\]  
\[\dot{\hat{a}}=-\lambda_{1}(e_{2}+\sigma_{1}\hat{a}), \tag{34}\]  
\[\dot{\hat{b}}=-\lambda_{2}(x_{2}e_{2}+\sigma_{2}\hat{b}), \tag{35}\]  
\[\dot{\hat{c}}=-\lambda_{3}(x_{2}^{2}e_{2}+\sigma_{3}\hat{c}), \tag{36}\]  
\[\dot{\hat{W}}=\chi e_{2}\mathbf{h}(\mathbf{x})-\chi\mathbf{\hat{W}}, \tag{37}\]  
where \(\hat{a}\), \(\hat{b}\), and \(\hat{c}\) are the estimated values of \(\frac{a_{0}}{m}=a^{*}\), \(\frac{b_{0}}{m}=b^{*}\), and \(\frac{c_{0}}{m}=c^{*}\), respectively, \(\widetilde{a}=\hat{a}-a^{*}\), \(\widetilde{b}=\hat{b}-b^{*}\), and \(\widetilde{c}=\hat{c}-c^{*}\) are the estimated error values, \(\beta>0\) and \(\chi>0\) are design parameters, \(\hat{f}(\mathbf{x})\) is the estimated value of \(f(\mathbf{x})\), and \(\lambda_{i}\) and \(\sigma_{i}\), \(i=1,2,3\) are adaptive parameters.  
**Assumption 5**.: _The parameters satisfy \(|a^{*}|\leq a^{\dagger}\),\(|b^{*}|\leq b^{\dagger}\), and \(|c^{*}|\leq c^{\ddagger}\), where \(a^{\dagger}\),\(b^{\ddagger}\), and \(c^{\ddagger}\) are unknown and bounded values._