## 4 Stability Verification  
We chose the Lyapunov function to validate the effectiveness of the FTAC algorithm that combined the new ADRC and the spatio-temporal RBF neural network:  
\[V=\tfrac{1}{2}s^{2}+\tfrac{1}{2}\chi_{1}^{-1}\tilde{\xi}_{1}^{2}+\tfrac{1}{2} (1-\tau)\chi_{2}^{-1}\tilde{\xi}_{2}^{2}+\tfrac{1}{2}\varphi_{1}^{-1}\tilde{ \xi}_{1}^{2}+\tfrac{1}{2}\varphi_{2}^{-1}\tilde{\xi}_{2}^{2}\, \tag{42}\]  
where \(\tilde{\xi}_{i}=\zeta_{i}-\dot{\zeta}_{i}\). By referencing Formula (41), the derivative of the Lyapunov function with respect to time can be obtained as follows:  
\[\begin{split}\dot{V}&=s\dot{s}-\chi_{1}^{-1}\tilde{ \sigma}_{1}\zeta_{1}-(1-\tau)\chi_{2}^{-1}\tilde{\xi}_{2}^{2}\tilde{\sigma}_{2 }-\varphi_{1}^{-1}\zeta_{1}\dot{\xi}_{1}-\varphi_{2}^{-1}\zeta_{2}\dot{\xi}_{2 }\\ &=s\dot{s}-\dot{\xi}_{1}|s|-(1-\tau)\xi_{2}\varpi|s|-\dot{\xi}_{1} s-\dot{\xi}_{2}\tau|s|\end{split} \tag{43}\]  
The derivative with respect to the sliding mode surface can be expressed as  
\[\begin{split}\dot{s}&=\dot{e}_{2}-c\dot{e}_{1}=(x_ {2}-\ddot{x}_{d})-ce_{1}\\ &=f_{0}(x)+\zeta_{1}(x)+g_{0}(x)u+\zeta_{2}(x)u+d_{w}-\ddot{x}_{d }-ce_{1}\.\end{split} \tag{44}\]Substituting the control law from Equation (41) into Equation (44) and replacing the control law concerning the known parameter \(g_{0}(x)\) yield  
\[\begin{array}{l}\dot{s}=f_{0}(x)+\xi_{1}(x)+\xi_{2}(x)u+d_{w}-\bar{x}_{d}-ce_{1 }\\ +g_{0}(x)\frac{1}{g_{0}(x)}\big{(}\bar{x}_{d}+ce_{1}-f_{0}(x)-\hat{\xi}_{1} \text{sgn}(s)\\ -\xi_{1}-\lambda\theta\text{sgn}(s)-\iota\text{sgn}(s))\\ =\xi_{1}(x)+\xi_{2}(x)u+d_{w}-\hat{\xi}_{1}\text{sgn}(s)\\ -\lambda\theta\text{sgn}(s)-\iota\text{sgn}(s)\end{array}. \tag{45}\]  
Furthermore, the derivative of the Lyapunov function can be extended to  
\[\begin{array}{l}\dot{V}=s\xi_{2}(x)u+sd_{w}-s\hat{\xi}_{1}\text{sgn}(s)\\ -s\lambda\theta\text{sgn}(s)-s\iota\text{sgn}(s)\\ -\xi_{1}|s|-(1-\tau)\xi_{2}\alpha|s|-\xi_{2}\tau|s|\\ =s\xi_{2}(x)u+sd_{w}-\hat{\xi}_{1}|s|-\lambda\theta|s|-\iota|s|\\ -\hat{\xi}_{1}|s|-(1-\tau)\xi_{2}\alpha|s|-\xi_{2}\tau|s|\end{array}. \tag{46}\]  
From the formula, \(\tau<1\) can be obtained. Therefore, according to the parameter \(\tau=\big{|}\xi_{2}/g_{0}\big{|}\), for part \(s\zeta_{2}(x)u\) of Formula (46), the following relation can be derived:  
\[\begin{array}{l}s\xi_{2}(x)u=s\xi_{2}\frac{1}{g_{0}}\big{(}\dot{x}_{d}+ce_{1 }-f_{0}(x)-\hat{\xi}_{1}\text{sgn}(s)\\ -\xi_{1}-\lambda\theta\text{sgn}(s)-\iota\text{sgn}(s)\\ \leq s\big{|}\frac{\xi_{2}}{g_{0}}\big{|}(\bar{x}_{d}+ce_{1}-f_{0}(x)-\hat{\xi }_{1}\text{sgn}(s)\\ -\xi_{1}-\lambda\theta\text{sgn}(s)-s\iota\text{sgn}(s))\\ \leq\tau|s|(|\bar{x}_{d}+ce_{1}-f_{0}(x)-\hat{\xi}_{1}\text{sgn}(s)\\ -\xi_{1}|-\xi_{2}+\xi_{2}+\lambda\theta+\iota)\\ \leq\tau|s|(\alpha+\xi_{2}+\lambda\theta+\iota)\end{array}. \tag{47}\]  
So, for Formula (46), we obtain  
\[\begin{array}{l}\dot{V}\leq\tau|s|(\alpha+\xi_{2}+\lambda\theta+\iota)+sd_{ w}-\hat{\xi}_{1}|s|\\ -\lambda\theta|s|-\iota|s|-\xi_{1}|s|\\ -(1-\tau)\xi_{2}\alpha|s|-\xi_{2}\tau|s|\end{array}. \tag{48}\]  
Because \(\zeta_{2}=1/1-\tau\), \(|d_{w}||s|-\zeta_{1}|s|\leq 0\), \(\theta=\hat{\zeta}_{2}\alpha\), it follows that  
\[\begin{array}{l}\dot{V}\leq\tau\alpha|s|+\tau\lambda\theta|s|+\tau_{1}|s|+sd_ {w}-\hat{\xi}_{1}|s|\\ -\lambda\theta|s|-\iota|s|-\hat{\xi}_{1}|s|-(1-\tau)\xi_{2}\alpha|s|\\ \leq\tau\alpha|s|-(1-\tau)(\lambda-1)\xi_{2}\alpha|s|\\ -(1-\tau)\xi_{2}\alpha|s|+\tau_{1}|s|+sd_{w}\\ -\zeta_{1}|s|-\iota|s|-(1-\tau)\xi_{2}\alpha|s|\\ \leq-(1-\tau)(\lambda-1)\xi_{2}\alpha|s|-(1-\tau)\alpha|s|\\ -(1-\tau)\iota|s|\end{array}. \tag{49}\]  
Since \(\tau<1,1-\tau>0\), according to the inequality relation in Equation (39), it is true that when \(\hat{\xi}_{2}>0\), there is \(\dot{V}<0\); and when \(\hat{\xi}_{2}\leq 0\), \(\lambda\leq-1\), the derivative of Lyapunov function \(\dot{V}<0\) still holds.Therefore, the system is asymptotically stable.