## 2 Modeling of a Mobile Satcom Antenna
### Non-Singularity Fast Terminal Sliding Mode  
The nonlinear system is defined by Equation (7). In order to improve the control performance of terminal sliding mode, the paper proposed an improved NFTSM as shown in Equation (8), which is similar to reference [14].  
\[s=\dot{e}+\frac{1}{\alpha}\|\dot{e}\|^{r+1}sign(\dot{e})+\frac{1}{\beta}\| \dot{\dot{e}}\|^{p+1}sign\Big{(}\dot{\dot{e}}\Big{)}=0 \tag{8}\]  
in which, \(\alpha\), \(\beta>0\), \(0<p<1\) and \(r>p\), \(\dot{e}=\theta_{a}-\theta_{a}^{c}\). The first derivative can be expressed as  
\[\dot{s}=\dot{\dot{e}}+\frac{r+1}{\alpha}\|\dot{e}\|^{r}\dot{e}+\frac{p+1}{\beta }\|\dot{\dot{e}}\|^{p}\dot{\dot{e}} \tag{9}\]  
**Theorem 1**.: _For System (7) with the adopted NFTSM, if the control law is designed as:_  
\[u_{n}=-{B_{a}}^{-1}\Bigg{[}\begin{array}{cc}\frac{\beta}{p+1}\|\dot{\dot{e}} \|^{1-p}sign\Big{(}\dot{\dot{e}}\Big{)}\Big{(}1+\frac{r+1}{\alpha}\|\dot{e}\|^ {r}\Big{)}\\ +A_{a}\dot{e}+(l_{d}+\delta)sign(s)+\Big{(}A_{a}\dot{\theta}_{a}^{c}-\dot{ \theta}_{a}^{c}\Big{)}\end{array}\Bigg{]} \tag{10}\]  
_the NFTSM manifold will be reached in a finite time. In addition, the tracking error on the sliding mode surface will also converge to zero in a finite time._  
Figure 2: Control block diagram of azimuth gimbal.  
**Proof.** Consider the Lyapunov candidate function as:  
\[V=\frac{1}{2}s^{2} \tag{11}\]  
The derivative of \(V\) is given by the following function,  
\[\begin{array}{ll}\dot{V}&=\dot{s}\dot{s}=s\left(\begin{array}{c}\dot{\dot{ \epsilon}}+\frac{r+1}{\kappa}\|\dot{\epsilon}\|^{p}\dot{\epsilon}+\frac{p+1}{ \beta}\|\dot{\epsilon}\|^{p}A_{a}\dot{\epsilon}+\frac{p+1}{\beta}\|\dot{ \epsilon}\|^{p}B_{a}u_{n}(t)+\\ \frac{p+1}{\beta}\|\dot{\epsilon}\|^{p}C_{a}d_{n}(t)+\frac{p+1}{\beta}\|\dot{ \epsilon}\|^{p}D_{a}w_{n}(t)+\frac{p+1}{\beta}\|\dot{\epsilon}\|^{p}\left(A_{ a}\dot{\theta}_{a}^{c}-\ddot{\theta}_{a}^{c}\right)\\ &=s\left(-\frac{p+1}{\beta}\|\dot{\epsilon}\|^{p}\left((I_{d}+\delta)sign(s) \right)+\frac{p+1}{\beta}\|\dot{\epsilon}\|^{p}C_{a}d_{n}(t)+\frac{p+1}{\beta }\|\dot{\epsilon}\|^{p}D_{a}w_{n}(t)\right)\\ &\leq-s(\delta sign(s))=-|s|\delta\end{array} \tag{12}\]  
It is obvious that \(\dot{V}<0\) if \(\dot{\sigma}=0\). Thus, the condition for Lyapunov stability is satisfied. Then the system state will converge to zero within finite time. As shown in Equation (10), the value of \(1-p<0\) is always greater than zero, therefore, the controller is non-singular. \(\square\)  
We can also obtain the reaching time \(t_{r}\) satisfies (see Appendix A)  
\[t_{r}\leq\frac{p+1}{p\beta^{\frac{1}{p+1}}}|e(t_{0})|^{\frac{p}{p+1}} \tag{13}\]  
That completes the proof. Therefore, the system can reach the terminal sliding mode surface within a finite time.  
For terminal sliding mode control methods, the upper disturbance bound is generally required, but in practical applications, a large upper bound can be used to eliminate the external disturbance item and guarantee the robustness. However, the chattering phenomena is enhanced correspondingly. The paper attempts to approximate nonlinear disturbance by using BSNN. A novel BSNN is proposed in this paper. As its name implies, the B-spline curve is adopted as activation function.