## 3 Results
### Disturbance Observer-Based Terminal Sliding Mode Control  
In this section, based on the disturbance estimation, a new sliding surface and, accordingly, a new control law are proposed to drive the system state to the equilibrium point in finite time. This work can be considered as an extension to the control strategy developed in [30] and an improvement of the work [41]. A new disturbance observer (DOB)-based terminal sliding mode surface for the system (1) is proposed as:  
\[\sigma=\dot{x}_{2}+c_{2}|x_{2}+z_{2}|^{\alpha_{2}}\text{sign}(x_{2}+z_{2})+c_{1 }|x_{1}|^{\alpha_{1}}\text{sign}(x_{1})+z_{3} \tag{19}\]  
where \(z_{2}=\hat{d}_{1}\), \(z_{3}=\hat{\hat{d}}_{1}\) are obtained from the disturbance observer, and \(c_{i}\) and \(\alpha_{i}\)\((i=1,2)\) are positive constants. \(c_{i}\) are chosen to guarantee that the polynomial \(p^{2}+c_{2}p+c_{1}\) is Hurwitz, and \(\alpha_{i}\) are determined in the same manner as in (8). **Theorem 2**. : _Given the sliding surface (19), if the control law is developed as:_  
\[u=g^{-1}(x,t)\big{(}u_{eq}+u_{n}\big{)} \tag{20}\]  
_with the equivalent control:_  
\[u_{eq}=-f(x,t)-c_{2}|x_{2}+z_{2}|^{\alpha_{2}}\text{sign}(x_{2}+z_{2})-c_{1}|x _{1}|^{\alpha_{1}}\text{sign}(x_{1})-z_{3} \tag{21}\]  
_and the switching term filtered for chattering attenuation is:_  
\[\dot{u}_{n}+Tu_{n}=v \tag{22}\]  
\[v=-(\rho_{2}+k_{T}+\eta)\text{sign}(\sigma) \tag{23}\]  
_where \(T,\ k_{T},\ \eta\) are defined as same as below (7), then the system (1) will reach the sliding mode surface \(\sigma=0\) in finite time, and the system state \(x_{1}(t)\) will also converge to the origin in finite time._  
**Proof of Theorem 2. ** By substituting the expression of \(\dot{x}_{2}(t)\) in (1) and the control law (20), (21) into the sliding surface (19), one can derive the following equality:  
\[ \begin{array}{ll}\sigma&=\dot{x}_{2}+c_{2}|x_{2}+z_{2}|^{\alpha_{2}}\text{ sign}(x_{2}+z_{2})+c_{1}|x_{1}|^{\alpha_{1}}\text{sign}(x_{1})+z_{3}\\ &=[f(x,t)+g(x,t)u+d_{2}(t)]+c_{2}|x_{2}+z_{2}|^{\alpha_{2}}\text{sign}(x_{2}+z_ {2})+c_{1}|x_{1}|^{\alpha_{1}}\text{sign}(x_{1})+z_{3}\\ &=d_{2}(t)+u_{n}\end{array}  \tag{24}\]  
Consider a candidate Lyapunov function as \(V(\sigma)=\sigma^{2}/2\). Differentiating the Lyapunov function \(V(\sigma)\) and following the same procedure in (11) yields:  
\[ \begin{array}{ll}\dot{V}(\sigma)&=\sigma\dot{\sigma}\\ &=\sigma\Big{[}\dot{d}_{2}(t)+\dot{u}_{n}\Big{]}\\ &\leq-\eta|\sigma|\end{array}  \tag{25}\]  
which implies that the system will arrive at the sliding surface \(\sigma=0\) in finite time. Once the desired sliding mode occurs, combining system (1) and sliding surface (19) yields:  
\[\left\{ \begin{array}{ll}\dot{x}_{1}=x_{2}+d_{1}\\ \dot{x}_{2}=-c_{2}|x_{2}+z_{2}|^{\alpha_{2}}\text{sign}(x_{2}+z_{2})-c_{1}|x_ {1}|^{\alpha_{1}}\text{sign}(x_{1})-z_{3}\end{array} \right. \tag{26}\]  
From the equation set (26), the following relationship can be established:\[ \begin{array}{ll}\ddot{x}_{1}&=\dot{x}_{2}+\dot{d}_{1}\\ &=\left[-c_{2}|x_{2}+z_{2}|^{\kappa_{2}}\text{sign}(x_{2}+z_{2})-c_{1}|x_{1}|^{ \kappa_{1}}\text{sign}(x_{1})-z_{3}\right]+\dot{d}_{1}\\ &=-c_{2}|\big{(}\dot{x}_{1}-d_{1}\big{)}+2\big{|}^{\kappa_{2}}\text{sign}\big{(} \big{(}\dot{x}_{1}-d_{1}\big{)}+z_{2}\big{)}-c_{1}|x_{1}|^{\kappa_{1}}\text{ sign}(x_{1})-z_{3}+\dot{d}_{1}\\ &=-c_{2}\big{|}\dot{x}_{1}+(z_{2}-d_{1})\big{|}^{\kappa_{2}}\text{sign}\big{(} \dot{x}_{1}+(z_{2}-d_{1})\big{)}-c_{1}|x_{1}|^{\kappa_{1}}\text{sign}(x_{1})- \big{(}z_{3}-\dot{d}_{1}\big{)}\end{array}  \tag{27}\]  
According to Theorem 1, \(z_{2}=d_{1}\) and \(z_{3}=\dot{d}_{1}\) are achieved in finite time by the disturbance observer. Thus, after the convergence of the disturbance observer, the dynamics (27) become:  
\[\ddot{x}_{1}=-c_{2}|\dot{x}_{1}|^{\kappa_{2}}\text{sign}\big{(}\dot{x}_{1} \big{)}-c_{1}|x_{1}|^{\kappa_{1}}\text{sign}(x_{1}) \tag{28}\]  
Let \(w_{1}=x_{1}\), \(w_{2}=\dot{x}_{1}\); then, the dynamic Equation (28) can be transformed to:  
\[\left\{ \begin{array}{l}\ddot{w}_{1}=w_{2}\\ \ddot{w}_{2}=u\end{array}