# Two-Phase Non-Singular Terminal Sliding Mode Control of Nonlinear Systems
## 3 Problem Formulation
### NTSM with FD  
Considering the aforementioned analysis of the TSM surface, the amendment should be carried out to upgrade the controller. On the basis of the two-phase sliding mode surface (12), the non-singular terminal sliding mode controller is proposed:  
\[u = -\frac{1}{g}\bigg{(}f+\gamma\text{sign}(s)+k_{1}\alpha_{1}\text{ sat}(x_{1}^{\alpha_{1}-1}x_{2},\zeta)+ \tag{20}\] \[((\frac{k_{1}}{2}+\frac{k_{2}}{2})+(\frac{k_{2}}{2}-\frac{k_{1}} {2})\text{sign}(|x_{1}|-x_{t}))\times\] \[(\frac{3-\alpha_{1}}{2}+\frac{1-\alpha_{1}}{2}\text{sign}(|x_{1}| -x_{t}))\times\] \[x_{1}^{\frac{1-\alpha_{1}}{2}+\frac{1-\alpha_{1}}{2}\text{sign}(| x_{1}|-x_{t})}x_{2}+o_{1}s^{\alpha_{2}}\] \[+((\frac{\alpha_{1}}{2}+\frac{\alpha_{2}}{2})+(\frac{\alpha_{2}} {2}-\frac{\alpha_{1}}{2})\text{sign}(|s|-s_{t}))\times\] \[s^{\frac{3-\alpha_{2}}{2}+(\frac{1-\alpha_{2}}{2})\text{sign}(|s |-s_{t})}\bigg{)}\]  
where \(0<\alpha_{1},\alpha_{2}<1,o_{1},o_{2},k_{1},k_{2}>0\), \(\gamma\geq\delta\), and \(\text{sat}(a,b)=\min\{|a|,b\}\text{sign}(a)\) is the saturation function to restrain the singularity term, and \(\zeta\) is the given bound of \(|x_{1}^{\alpha_{1}-1}x_{2}|\) to limit the infinity term. The conclusions about NTSM controller (20) is drawn in the following theorem. **Theorem 1**. : _Considering the second-order nonlinear system (11), both NTSM surface (12) and NTSM controller (20) are proposed to guarantee the bounded control input and fast convergence, including in regions both close to and far away from the control objective. Moreover, the upper bound of the settling time during the whole process is given in (23)._  
Proof. : Case 1: \(|x_{1}^{\alpha_{1}-1}x_{2}|<\zeta\). In this area, the analysis of the terminal sliding mode surface and the settling time is consistent with before (section A), as the singularity does not appear. The convergence time from \(x(0)\) to the origin \(0\) is identical to equation (19), written as  
\[t_{s,n1} = \frac{\arctan(\sqrt{\frac{\alpha_{2}}{\alpha_{1}}}V(0)^{\frac{1- \alpha_{2}}{2}})-\arctan(\sqrt{\frac{\alpha_{2}}{\alpha_{1}}}V_{t}^{\frac{1- \alpha_{2}}{2}})}{\sqrt{\alpha_{1}\alpha_{2}}(1-\alpha_{2})}+ \tag{21}\] \[\frac{\ln(1+V_{t}^{\frac{1-\alpha_{2}}{2}})}{o_{1}(1-\alpha_{2}) }+\frac{\ln(1+x_{t}^{1-\alpha_{1}})}{k_{1}(1-\alpha_{1})}+\] \[\frac{\arctan(\sqrt{\frac{k_{2}}{k_{1}}}x_{1}(0)^{1-\alpha_{1}}) -\arctan(\sqrt{\frac{k_{2}}{k_{1}}}x_{t}^{1-\alpha_{1}})}{\sqrt{k_{1}k_{2}}( 1-\alpha_{1})}\]  
Case 2: \(|x_{1}^{\alpha_{1}-1}x_{2}|>\zeta\). The singularity phenomenon happens in this situation, and we define \(|x_{1}^{\alpha_{1}-1}x_{2}|>\zeta\) as the singularity area. It is obtained from (11) that  
\[x_{1}(t) = x_{1}(0)+\int_{0}^{t}\hat{x}_{2}(\tau)d\tau \tag{22}\]  
which indicates that \(x_{1}(t)\) will increase monotonically in the presence of \(x_{1}(0)>0\), and \(x_{1}(t)\) will decrease monotonically if \(x_{2}(t)<0\). Therefore, the system state will pass the singularity area in finite time, and the crossing time is denoted as \(t(\zeta)\), and it has noimpact on the finite convergence analysis. In the sequel, the settling time of system (11) is presented as  
\[((\frac{k_{1}}{2}+\frac{k_{2}}{2})+(\frac{k_{2}}{2}-\frac{k_{1}} {2})\text{sign}(|x_{1}|-x_{t}))\times\]0 \[((\frac{k_{1}}{2}+\frac{k_{2}}{2})+(\frac{k_{2}}{2}-\frac{k_{1}} {2})\text{sign}(|x_{1}|-x_{t}))\times\]1 \[((\frac{k_{1}}{2}+\frac{k_{2}}{2})+(\frac{k_{2}}{2}-\frac{k_{1}} {2})\text{sign}(|x_{1}|-x_{t}))\times\]2  
**Remark 4**. : _Due to the lack of a specific description of the boundary of the singularity area, \(t(\zeta)\) can not be given exactly, but the crossing time \(t(\zeta)\) has no impact on the finite-duration convergence process. Moreover, it is seen that (23) is dependent on the initial state \(x_{1}(0)\), and the width of settling time is finite for every given \(x_{1}(0)\).