## 4 State Feedback Control Design  
### Backstepping Control Design  
From (1), we have  
\[\left\{\begin{aligned} \dot{x}&=u+c,\\ \dot{u}&=\frac{X_{u|u|}}{m-X_{\dot{u}}}u|u|+\frac{1- C_{n}}{m-X_{\dot{u}}}\tau+\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)\end{aligned}\right. \tag{7}\]  
Define \(e=x-x_{d},z_{1}=x-x_{d}^{*}\) and \(z_{2}=u-\alpha\), where \(\alpha\) is a virtual controller to be designed. From the construction of \(x_{d}^{*}\), we can see that \(z_{1}(0)=0\) and \(e(t)=z_{1}(t)\) for \(t\geq t_{d}\). From (7), we obtain  
\[\dot{z}_{1}=u+c-\dot{x}_{d}=z_{2}+\alpha+c-\dot{x}_{d}^{*}. \tag{8}\]  
Then, we define a Lyapunov function as follows:  
\[V_{1}=\frac{z_{1}^{2}}{2(b_{z}^{2}-z_{1}^{2})}. \tag{9}\]  
According to (8) and (9), we have  
\[\dot{V}_{1}=\sigma z_{1}(z_{2}+\alpha+c-\dot{x}_{d}^{*}), \tag{10}\]  
where \(\sigma=\frac{b_{z}^{2}}{(b_{z}^{2}-z_{1}^{2})^{2}}\). Through designing  
\[\alpha= \dot{x}_{d}^{*}-\frac{\gamma_{1}}{2}(b_{z}^{2}-z_{1}^{2})z_{1}/b_{ z}^{2}-\frac{c_{\rho}^{2}\sigma z_{1}}{\sqrt{\xi_{\rho}^{2}\sigma^{2}z_{1}^{2}+ \zeta^{2}}}, \tag{11}\]  
with \(\iota>1\), \(\gamma_{1}>\frac{1}{2}\). Applying Lemma 1 and substituting (11) into (10), we can obtain  
\[\dot{V}_{1}= -\frac{\gamma_{1}}{2}\frac{z_{1}^{2}}{b_{z}^{2}-z_{1}^{2}}+\sigma z _{1}(z_{2}+c-\frac{c_{\rho}^{2}\sigma z_{1}}{\sqrt{\xi_{\rho}^{2}\sigma^{2}z_ {1}^{2}+\zeta^{2}}})\] \[= -\frac{\gamma_{1}}{2}\frac{z_{1}^{2}}{b_{z}^{2}-z_{1}^{2}}+\sigma z _{1}z_{2}+\sigma(z_{1}c-|z_{1}|\xi_{\rho})+\sigma|z_{1}|\xi_{\rho}-\frac{\xi_ {\rho}^{2}\sigma^{2}z_{1}^{2}}{\sqrt{\xi_{\rho}^{2}\sigma^{2}z_{1}^{2}+\zeta^ {2}}}\] \[\leq -\frac{\gamma_{1}}{2}\frac{z_{1}^{2}}{b_{z}^{2}-z_{1}^{2}}+\sigma z _{1}z_{2}+\sigma|z_{1}|\xi_{\rho}+\zeta \tag{12}\]  
From (7), we obtain  
\[\dot{z}_{2}= \dot{u}-\dot{u}\] \[= \frac{X_{u|u|}}{m-X_{\dot{u}}}u|u|+\frac{1-C_{n}}{m-X_{\dot{u}}} \tau+\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)-\dot{u} \tag{13}\]  
Let \(\varpi=\frac{1-C_{n}}{m-X_{\dot{u}}}\), \(\mathbf{\theta}=[\varpi^{-1}\frac{X_{u|u|}}{m-X_{\dot{u}}},-\frac{1-C_{n}}{m-X_{ \dot{u}}}]^{T}\), \(\mathbf{\varphi}=[u|u|,\dot{u}]^{T}\). Define \(V_{2}=\frac{1}{2\varpi}z_{2}^{2}\).  
\[\dot{V}_{2}= z_{2}[\varpi^{-1}\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)+\mathbf{\theta}^ {T}\mathbf{\varphi}+\tau] \tag{14}\]A sigmoid neural network [34] is employed to approximate \(\omega^{-1}\frac{1}{m-X_{a}}\Delta f(x,u)\) as follows:  
\[\omega^{-1}\frac{1}{m-X_{a}}\Delta f(x,u)=\mathbf{\omega}^{*T}\mathbf{\psi}(\mathbf{X})+ \epsilon(\mathbf{X}) \tag{15}\]  
where \(|\epsilon(\mathbf{X})|\leq\epsilon_{x,N}\), with \(\epsilon_{x,N}\) being an unknown constant,  
\[\mathbf{\psi}(\mathbf{X})=\frac{\xi_{x1}}{\xi_{x2}+\exp(-\mathbf{X}/\xi_{x3})}+\xi_{x4}, \tag{16}\]  
\(\mathbf{X}=[x,u,x_{d},\dot{x}_{d},\ddot{x}_{d}]\), \(\xi_{x1}-\xi_{x4}\) are the design parameters, and \(\epsilon_{N}\) is an unknown bounded constant. Combining (15) with (14) yields  
\[\dot{V}_{2}= z_{2}[\mathbf{\omega}^{*T}\mathbf{\psi}(\mathbf{X})+\epsilon(\mathbf{X})+\mathbf{ \theta}^{T}\mathbf{\varphi}+\tau], \tag{17}\]  
Based on (12) and (17), the following control law is designed as  
\[\tau= -\sigma z_{1}-\gamma_{2}z_{2}-\mathbf{\dot{\omega}}^{T}\mathbf{\psi}(\mathbf{ X})-\mathbf{\dot{\theta}}^{T}\mathbf{\varphi} \tag{18}\] \[\mathbf{\dot{\omega}}= \gamma_{3}z_{2}\mathbf{\psi}(\mathbf{X})-k_{1}\gamma_{3}\mathbf{\dot{\omega}},\] (19) \[\mathbf{\dot{\theta}}= \gamma_{4}z_{2}\mathbf{\varphi}-k_{2}\gamma_{4}\mathbf{\dot{\theta}},\] (20) \[\dot{\epsilon}_{\rho}= \gamma_{5}\sigma|z_{1}|-k_{3}\gamma_{5}\dot{\epsilon}_{\rho}, \tag{21}\]  
where \(\gamma_{2}>1\), \(\gamma_{3}>0,\gamma_{4}>0,\gamma_{5}>0,k_{1}>0\), \(k_{2}>0,k_{3}>0\), and \(\mathbf{\dot{\omega}}\) and \(\mathbf{\dot{\theta}}\) are the estimations of \(\mathbf{\omega}^{*}\) and \(\mathbf{\theta}\), respectively.