## Appendix A  
The design of the SMC-ESO controller: Define the position error \(z_{1}=x_{1}-x_{1\text{d}}\) and \(z_{2}=\hat{x}_{2}-\dot{x}_{1\text{d}}\), and selected the SMC surface as  
\[\sigma=z_{2}+k_{1}z_{1} \tag{11}\]  
where \(k_{1}\) is a positive design parameter.  
The time derivative of \(\dot{\sigma}\) is  
\[\begin{array}{ll}\dot{\sigma}&=\dot{x}_{2}-\ddot{x}_{1\text{d}}+k_{1}\dot{z }_{1}\\ &=\dot{x}_{2}-\dot{x}_{2}-\ddot{x}_{1\text{d}}+k_{1}\dot{z}_{1}\\ &=\theta_{1n}u+\theta_{2n}x_{2}+x_{3}-\left(\theta_{2n}\widetilde{x}_{2}+ \widetilde{x}_{3}-l_{2}\omega_{0}^{2}\widetilde{x}_{1}\right)-\ddot{x}_{1 \text{d}}+k_{1}\left(\hat{x}_{2}+\widetilde{x}_{2}-\dot{x}_{1\text{d}}\right) \end{array} \tag{12}\]  
The control \(u\) of SMC-ESO controller is designed as  
\[u=-\frac{1}{\theta_{1n}}\left(u_{eq}+u_{sw}\right) \tag{13}\]  
\[u_{eq}=\theta_{2n}\hat{x}_{2}+\hat{x}_{3}+k_{1}\left(\hat{x}_{2}-\dot{x}_{1 \text{d}}\right)-\ddot{x}_{1\text{d}} \tag{14}\]  
\[u_{sw}=\eta\text{sign}(\sigma)+k_{2}\sigma \tag{15}\]  
where \(k_{2}\) and \(\eta\) are positive design parameters.  
Adding Equations (13)-(15) into (12) gives  
\[\dot{\sigma}=l_{2}\omega_{0}^{2}\widetilde{x}_{1}+k_{1}\widetilde{x}_{2}- \eta\text{sign}(\sigma)-k_{2}\sigma \tag{16}\]  
Define the Lyapunov function as  
\[V=\frac{1}{2}\sigma^{2} \tag{17}\]  
According to Equation (16), the time derivative of \(V\) is  
\[\begin{array}{ll}\dot{V}&=\sigma\left  
where \(\alpha_{1},\alpha_{2}\) are positive constants subject to \(0<\alpha_{1}<1\) and \(\alpha_{1}<\alpha_{2}\), \(|z_{i}|^{[a_{1}]}=|z_{i}|^{a_{i}}\text{sgn}(z_{i})\), \(i=1,\ 2;k_{1}\) and \(k_{2}\) are positive design parameters.  
The time derivative of \(\sigma\) is  
\[\begin{array}{ll}\dot{\sigma}&=\dot{\tilde{x}}_{2}-\tilde{x}_{1 \text{d}}+k_{1}|z_{1}|^{[a_{1}]}+k_{2}|z_{2}|^{[a_{2}]}\\ &=\dot{x}_{2}-\tilde{x}_{2}-\tilde{x}_{1\text{d}}+k_{1}|z_{1}|^{[a_{1}]}+k_{2}| z_{2}|^{[a_{2}]}\\ &=\theta_{1n}u+\theta_{2n}x_{2}+x_{3}-\left(\theta_{2n}\tilde{x}_{2}+\tilde{x }_{3}-l_{2}\omega_{0}^{2}\tilde{x}_{1}\right)-\tilde{x}_{1\text{d}}+k_{1}|z_{1 }|^{[a_{1}]}+k_{2}|z_{2}|^{[a_{2}]}\end{array} \tag{10}\]  
The control u of NFTSMC-ESO controller is designed as  
\[u=-\frac{1}{\theta_{1n}}\big{(}u_{eq}+u_{sw}\big{)} \tag{11}\]  
\[u_{eq}=\theta_{2n}\dot{x}_{2}+\hat{x}_{3}-\tilde{x}_{1\text{d}}+k_{1}|z_{1}|^{ [a_{1}]}+k_{2}|z_{2}|^{[a_{2}]} \tag{12}\]  
\[u_{sw}=\eta\text{sign}(\sigma)+k_{2}\sigma \tag{13}\]  
where \(k_{2}\) and \(\eta\) are positive design parameters.  
Adding Equations (11)-(13) into (10) gives  
\[\dot{\sigma}=l_{2}\omega_{0}^{2}\tilde{x}_{1}-\eta\text{sign}(\sigma)-k_{2}\sigma \tag{14}\]  
Define the Lyapunov function as  
\[V=\frac{1}{2}\sigma^{2} \tag{15}\]  
From Equation (14), the time derivative of \(V\) is  
\[\begin{array}{ll}\dot{V}&=\sigma\big{[}l_{2}\omega_{0}^{2}\tilde{x}_{1}-\eta \text{sign}(\sigma)-k_{2}\sigma\big{]}\\ &=l_{2}\omega_{0}^{2}\tilde{x}_{1}\sigma-\eta|\sigma|-k_{2}\sigma^{2}\\ &\leq\frac{1}{2}\big{(}l_{2}\omega_{0}^{2}\tilde{x}_{1}\big{)}^{2}+\frac{1}{2 }\sigma^{2}-\eta|\sigma|-k_{2}\sigma^{2}\\ &=-\Big{(}k_{2}-\frac{1}{2}\Big{)}\sigma^{2}-\eta|\sigma|+\Omega_{2}\end{array} \tag{16}\]  
where \(\Omega_{2}=\frac{1}{2}l_{2}^{2}\omega_{0}^{4}\tilde{x}_{1}^{2}\).  
As explained previously, \(\Omega_{2}\) is also a function of the estimation error, which will converge to a very small value when selected a large enough \(\omega_{0}\); therefore, the stability of the closed-loop system is obtained with the proposed control law (11)-(13).