## 3 Main Results
### The Designed FTSMM  
To enhance the response, fast convergence time, and accuracy of the tracking position, the novel FTSMM is developed as follows:  
\[s_{i}=x_{dei}+\frac{2\gamma_{1}}{1+e^{-\mu_{1}(|x_{di}|-\phi)}}x_{ei}+\frac{2 \gamma_{2}}{1+e^{\mu_{2}(|x_{di}|-\phi)}}|x_{ei}|^{\alpha}\text{sgn}(x_{ei}) \tag{8}\]  
where \(s_{i}\) is the proposed sliding mode manifold, \(\gamma_{1},\gamma_{2},\mu_{1},\mu_{2}\) are the positive constants, \(0<\alpha<1\), and \(\phi=\left(\frac{\gamma_{2}}{\gamma_{1}}\right)^{1/(1-\alpha)}\).  
Based on the SMC, when the control errors operate in the sliding mode, the following constrain is satisfied [1]:  
\[\begin{array}{l}s_{i}=0;\\ \dot{s}_{i}=0\end{array} \tag{9}\]From condition in Equation (9), it is pointed out that:  
\[x_{dei}=-\frac{2\gamma_{1}}{1+e^{-\mu_{1}(|x_{ei}|-\phi)}}x_{ei}-\frac{2\gamma_{2 }}{1+e^{\mu_{2}(|x_{ei}|-\phi)}|x_{ei}|^{\alpha}\text{sgn}(x_{ei})} \tag{10}\]  
**Remark 1**.: _When the control error of \(|x_{ei}|\) is much greater than \(\phi\), the first component of Equation (10) offers the role of providing a quick convergence rate and the second component has a smaller role. Contrariwise, when the control error of \(|x_{ei}|\) is much smaller than \(\phi\), the second component of Equation (10) offers a greater role than the first one._  
The following theorem is launched to guarantee that convergence takes place within the defined time.  
**Theorem 1**.: _Let us consider dynamic of Equation (10). \(x_{ei}=0\) is defined as the equilibrium point and the state variables of the dynamic of Equation (10), including \(x_{ei}\) and \(x_{dei}\) stabilize to zero in finite-time._  
Proof.: To validate the correctness of Theorem 1, the Lyapunov function candidate is proposed as follows:  
\[L_{1}=0.5x_{ei}^{2}, \tag{11}\]  
and its time derivative is  
\[\begin{array}{l}\dot{L}_{1}=x_{ei}x_{dei}\\ =-\frac{2\gamma_{1}}{1+e^{-\mu_{1}(|x_{ei}|-\phi)}}x_{ei}^{2}-\frac{2\gamma_{ 2}}{1+e^{\mu_{2}(|x_{ei}|-\phi)}}|x_{ei}|^{\alpha+1}\text{sgn}(x_{ei})\\ <0\end{array} \tag{12}\]  
It is shown that \(\dot{L}_{1}<0\), hence, \(x_{ei}\) and \(x_{dei}\) concentrate on the equilibrium state in finite time.  
When \(\big{|}x_{ei}(0)\big{|}>\phi\), the sliding motion consists of two phases:  
The first phase: \(x_{ei}(0)\rightarrow|x_{ei}|=\phi\), the first component of Equation (10) offers the role of providing a quick convergence rate and the second component has a smaller role.  
\[\begin{array}{l}\int\limits_{0}^{t_{1}}dt=\int\limits_{\phi}^{x_{ei}(0)} \frac{2\gamma_{1}}{1+e^{-\mu_{1}(|x_{ei}|-\phi)}}\frac{1}{x_{ei}+\frac{2\gamma _{2}}{1+e^{\mu_{2}(|x_{ei}|-\phi)}}|x_{ei}|^{\alpha}}d(|x_{ei}|)\\ <\int\limits_{\phi}^{x_{ei}(0)}\frac{1}{\gamma_{1}|x_{ei}|}d(|x_{ei}|)=\frac{ \ln(|x_{ei}(0)|)-\ln(\phi)}{\gamma_{1}}\end{array} \tag{13}\]  
The second phase: \(|x_{ei}|=\phi\to x_{ei}=0\), the second component of Equation (10) offers a role greater than the first one.  
\[\begin{array}{l}\int\limits_{0}^{t_{2}}dt=\int\limits_{0}^{\phi}\frac{\frac{ 2\gamma_{1}}{2\gamma_{1}}}{1+e^{-\mu_{1}(|x_{ei}|-\phi)}x_{ei}+\frac{2\gamma _{2}}{1+e^{\mu_{2}(|x_{ei}|-\phi)}}|x_{ei}|^{\alpha}}d(|x_{ei}|)\\ <\int\limits_{\phi}^{\phi}\frac{1}{\gamma_{2}|x_{ei}|^{\alpha}}d(|x_{ei}|)= \frac{1}{\gamma_{2}(1-\alpha)}\big{|}\phi\big{|}^{1-\alpha}\end{array} \tag{14}\]  
The total time of the sliding motion phase is defined as:  
\[T_{s}=t_{1}+t_{2}<\frac{\ln\Big{(}|x_{ei}(0)|\Big{)}-\ln(\phi)}{\gamma_{1}}+ \frac{1}{\gamma_{2}(1-\alpha)}\big{|}\phi\big{|}^{1-\alpha} \tag{15}\]The state variable of the dynamic in Equation (10) converges to sliding manifold (\(s(0)\to 0\)) within the defined time \(T_{r}\), which was pointed out in [8]. Therefore, the total time for stability on the sliding manifold is computed as:  
\[T\leq T_{r}+T_{s} \tag{16}\]