## 2 Problem Statement  
Consider a class of second-order nonlinear systems that are susceptible to both matched and mismatched disturbances as follows [41]:  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2}(t)+d_{1}(t)\\ \dot{x}_{2}(t)=f(x,t)+g(x,t)u(t)+d_{2}(t)\\ y(t)=x_{1}(t)\end{array} \right. \tag{1}\]  
where \(x=[x_{1},x_{2}]^{T}\) is the system state and assumed to be available, \(f(x,t)\) and \(g(x,t)\
eq 0\) are two known smooth nonlinear functions, \(u(t)\) is the control input, \(y(t)\) is the system output, and \(d_{i}(t),\ i=1,2\) are the time-varying mismatched and matched disturbances of the system. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Control** & **Matched** & **Mismatched** & **Finite-Time** \\
**Algorithm** & **Disturbance** & **Disturbance** & **Convergence** \\  & **Compensation** & **Compensation** & **Convergence** \\ \hline TSMC [30] & Yes & No & Yes \\ EOSMC [41] & Yes & Yes & No \\ The proposed & Yes & Yes & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the proposed controller and other works. **Assumption 1**. : _Both mismatched and matched disturbances \(d_{i}(t),\ i=1,2\), and their derivatives, are bounded and satisfy:_  
\[|d_{i}(t)|\leq\phi_{i},\ \ \ \left|\dot{d}_{i}(t)\right|\leq\rho_{i},\ \ \ i=1,2 \tag{2}\]  
_where \(\phi_{i}>0\) and \(\rho_{i}>0,\ (i=1,2)\) are positive constants. This assumption is reasonably realistic due to the physical limitation of machinery in practical applications._  
This work aims to design a disturbance observer-based terminal sliding mode controller to drive the system state \(x_{1}(t)\) to the origin in finite time in the presence of matched and mismatched disturbances. For this purpose, where the context is sufficiently explicit, the arguments of some functions might be omitted for brevity. For the system (1), the chattering-free full-order TSMC developed in [30] is derived as below. First, a new terminal sliding mode surface is introduced with the form:  
\[s=\dot{x}_{2}+c_{2}|x_{2}|^{\alpha_{2}}\text{sign}(x_{2})+c_{1}|x_{1}|^{\alpha_ {1}}\text{sign}(x_{1}) \tag{3}\]  
Then, a control law is designed accordingly as:  
\[u=g^{-1}(x,t)\big{(}u_{eq}+u_{n}\big{)} \tag{4}\]  
where \(u_{eq}\) denotes the equivalent control and \(u_{n}\) denotes the switching term. The equivalent control \(u_{eq}\) is designed as:  
\[u_{eq}=-f(x,t)-c_{2}|x_{2}|^{\alpha_{2}}\text{sign}(x_{2})-c_{1}|x_{1}|^{ \alpha_{1}}\text{sign}(x_{1}) \tag{5}\]  
the switching term \(u_{n}\) is defined and filtered by the following method:  
\[\dot{u}_{n}+Tu_{n}=v \tag{6}\]  
\[v=-(\rho_{2}+k_{T}+\eta)\text{sign}(s) \tag{7}\]  
where \(\eta\) is a positive constant, \(T\geq 0\), \(k_{T}\) is chosen satisfying \(k_{T}\geq T\phi_{2}\), \(c_{i}\) and \(\alpha_{i}\ (i=1,2)\) are positive constants. \(c_{i}\) can be selected to ensure that the polynomial \(p^{2}+c_{2}p+c_{1}\) is Hurwitz, and \(\alpha_{i}\) can be determined to satisfy [30; 47]:  
\[\left\{ \begin{array}{ll}\alpha_{1}=\frac{\alpha_{2}}{2-\alpha_{2}}\\ \alpha_{2}=\alpha,\ \alpha\in(1-\varepsilon,1),\ \varepsilon\in(0,1)\end{array} \right. \tag{8}\]  
By substituting the system (1) and the control law (4) and (5) into the sliding surface (3), one can obtain:  
\[ \begin{array}{ll}s&=\dot{x}_{2}+c_{2}|x_{2}|^{\alpha_{2}}\text{sign}(x_{2} )+c_{1}|x_{1}|^{\alpha_{1}}\text{sign}(x_{1})\\ &=f(x,t)+g(x,t)u+d_{2}(t)+c_{2}|x_{2}|^{\alpha_{2}}\text{sign}(x_{2})+c_{1}|x_{ 1}|^{\alpha_{1}}\text{sign}(x_{1})\\ &=d_{2}(t)+u_{n}\end{array}  \tag{9}\]  
Consider a candidate Lyapunov function of the form: \(V(s)=s^{2}/2\). It was proven in [48] that the switching term satisfies \(|u_{n}(t)|\leq\phi_{2}\). This statement is found to be crucial in proving the finite-time convergence of the sliding surface \(s\). Thus, it is re-depicted here for clarity. Initially, \(u_{n}(t=0)=0\) is taken, i.e., \(u_{n}(0)\in[-\phi_{2},\phi_{2}]\). Then, \(u_{n}\) indicates that it cannot leave the interval \([-\phi_{2},\phi_{2}]\).