## 3 Design of Global Fast Terminal Sliding Mode Control  
The conventional fast terminal sliding surface is designed as follows:  
\[s(t)=\dot{x}_{1}+nx_{1}^{v}+lx_{1} \tag{9}\]  
To enhance the global robustness of the system and accelerate its convergence to the steady state, a nonlinear time-varying function \(g(t)\) is incorporated into Equation (9). Consequently, the improved global fast terminal sliding mode surface can be designed as follows:  
\[s(t)=\dot{x}_{1}+m\int_{0}^{t}x_{1}(\tau)d\tau+nx_{1}^{v}+lx_{1}+g(t) \tag{10}\]  
where \(m\), \(n\), and \(l\) are the controller parameters to be set; \(m>0\), \(n>0\), \(l>0\), \(\
u=\frac{p}{q}\), \(p\), and \(q\) are positive odd numbers. The function \(g(t)\) meets the following conditions: (1) it possesses a first-order derivative; (2) when \(t=0\), \(s(t)=0\); (3) when \(t\rightarrow\infty\), \(g(t)\to 0\).  
Given the above conditions, \(g(t)\) is formulated as a monotonically decreasing exponential function, as shown in Equation (11):  
\[g(t)=ke^{-\lambda t} \tag{11}\]  
where \(k\) and \(\lambda\) are the controller parameters to be designed according to the sliding mode surface, \(\lambda>0\).  
The global fast terminal sliding mode control (GFTSMC) controller is designed as follows:  
\[\begin{array}{ll}u(t)&=-\alpha_{1}s(t)-\alpha_{2}\text{sgn}(s(t))-\alpha_{ 3}|s(t)\,^{\dagger}\text{sgn}(s(t))+Lx(t)\\ &+\bigg{[}\Big{(}\frac{D_{1}}{M_{1}}+\frac{1}{M_{1}}\Big{)}Q_{1}+\Big{(}\frac {\alpha}{M_{1}^{2}}+\frac{1}{M_{1}}\Big{)}Q_{2}\bigg{]}\text{sgn}(s(t))\\ &+k\lambda\text{e}^{-kt}\end{array} \tag{12}\]  
where \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are positive numbers; \(L\) is a constant matrix; \(\alpha>0\), \(Q_{1}\), and \(Q_{2}\) are constants; and \(\|\omega(t)\|\leq Q_{1}\), \(\|\omega(t)\|\leq Q_{2}\).  
The Lyapunov function can be expressed as follows:  
\[V(t)=\frac{1}{2}s^{2}(t) \tag{13}\]  
Taking the derivative of \(V(t)\), the result is given below:  
\[\dot{V}(t)=s(t)\dot{s}(t) \tag{14}\]The derivative of \(s(t)\) can be written as follows:  
\[\begin{array}{ll}\dot{s}(t)&=\ddot{x}_{1}+l\dot{x}_{1}+mx_{1}+nv\chi_{1}^{v-1} \dot{x}_{1}+k\lambda\mathrm{e}^{-kt}\\ &=-\frac{D_{i}}{M_{i}}\dot{x}_{1}+\frac{1}{M_{i}}\dot{x}_{2}-\frac{1}{M_{i}} \dot{x}_{4}\\ &+\frac{1}{M_{i}}\dot{x}_{6}+\frac{1}{M_{i}}\dot{x}_{7}-\frac{1}{M_{i}}\dot{ \omega}(t)\\ &+l[-\frac{D_{i}}{M_{i}}x_{1}+\frac{1}{M_{i}}x_{2}-\frac{1}{M_{i}}\dot{\omega} (t)]\\ &+\frac{1}{M_{i}}x_{6}+\frac{1}{M_{i}}x_{7}-\frac{1}{M_{i}}\dot{\omega}(t)]\\ &+mx_{1}+nv\chi_{1}^{v-1}\dot{x}_{1}+k\lambda\mathrm{e}^{-kt}\end{array} \tag{15}\]  
By integrating Equations (1), (12), (14) and (15), the following is obtained:  
\[\begin{array}{ll}\dot{V}(t)&=s(t)\dot{s}(t)\\ &=s(t)[-\alpha_{1}s(t)-\alpha_{2}\mathrm{sgn}(s(t))-\alpha_{3}|s(t)\,\mathrm{ l}^{v}\mathrm{sgn}(s(t))\\ &\quad+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\omega(t)+k\lambda \mathrm{e}^{-kt}-\Big{(}\frac{\alpha}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\dot{ \omega}(t)\\ &\quad+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}Q_{1}\mathrm{sgn}(s(t ))-\Big{(}\frac{\alpha}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}Q_{2}\mathrm{sgn}(s( t))\\ &\quad-k\lambda\mathrm{e}^{-kt}]\\ &\leq-\alpha_{1}s^{2}(t)-\alpha_{2}s^{2}(t)-\alpha_{3}\Big{|}s(t)\,\mathrm{ l}^{1+v}\\ &\quad+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}Q_{1}|s(t)|-\Big{(} \frac{\alpha}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}Q_{2}|s(t)|\\ &\quad-\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}Q_{1}|s(t)|+\Big{(} \frac{\alpha}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}Q_{2}|s(t)|\\ &\leq-\alpha_{1}s^{2}(t)-\alpha_{2}s^{2}(t)-\alpha_{3}\Big{|}s(t)\,\mathrm{ l}^{1+v}\\ &\quad\leq-(\alpha_{1}+\alpha_{2})s^{2}(t)-\alpha_{3}\Big{|}s(t)\,\mathrm{l}^{1+v}<0 \end{array} \tag{16}\]  
Based on the above analysis, it can be concluded that the system is stable.