## 3 Results
### Disturbance Observer-Based Terminal Sliding Mode Control
\right. \tag{29}\]  
where \(u=-c_{2}|w_{2}|^{\kappa_{2}}\text{sign}(w_{2})-c_{1}|w_{1}|^{\kappa_{1}}\text {sign}(w_{1})\), and \(c_{i},\alpha_{i}\) (\(i=1,2\)) are designed in the same manner as below equation (19) and equation (8). Then, according to Proposition 8.1 and its proof in [47], it can be concluded that the origin is a globally finite-time-stable equilibrium for the system (29), indicating that the state \(x_{1}(t)\) of the system (1) will eventually converge to zero in finite time. This completes the proof for Theorem 2. \(\square\)  
**Remark 2**. : _As stated in [30], the control signal (22) acts in the same manner as a low-pass filter, wherein \(\omega=T\) can be interpreted as the bandwidth of the filter. The non-smooth switching function \(u_{n}\) is therefore smoothed to prevent the chattering phenomena from occurring, given that the design parameters have been properly chosen. Moreover, because it is a full-order terminal sliding mode surface, it allows the corresponding control law to be derived without differentiating the term \(c_{i}|*|^{\kappa_{i}}\text{sign}(*)\), thus avoiding the singularity phenomena._  
**Remark 3**. : _It should be mentioned that the sliding surface (19) includes the \"acceleration\" signal \(\dot{x}_{2}(t)\), which can be obtained by differentiating the \"velocity\" signal \(x_{2}(t)\). Nonetheless, this action is sensitive to noise and thus can degrade system performance. Note that the proposed control law only utilizes the sign of the sliding surface \(\sigma\) rather than its value. Based on this observation, the same idea suggested in [30] is re-depicted here to avoid the use of the \"acceleration\" signal. In this approach, \(\text{sign}(\sigma)\) can be alternatively determined as follows:_  
\[\text{sign}(\sigma)=\text{sign}(h(t)-h(t-\tau)) \tag{30}\]  
_where \(h(t)\) is defined as:_  
\[ \begin{array}{ll}h(t)&=\int_{0}^{t}\sigma(t)dt\\ &=x_{2}+\int_{0}^{t}(c_{2}|x_{2}+z_{2}|^{\kappa_{2}}\text{sign}(x_{2}+z_{2})+ c_{1}|x_{1}|^{\kappa_{1}}\text{sign}(x_{1})+z_{3})dt\end{array}  \tag{31}\]  
_and \(\tau\) is chosen to be the fundamental sampling time._  
_Alternatively, the nonlinear disturbance observer structure can be applied again to meet this need and is presented below:_  
\[\left\{ \begin{array}{l}\dot{z}_{4}=v_{4}+x_{3}\\ v_{4}=-\lambda_{4}{L_{2}}^{1/3}|z_{4}-x_{2}|^{2/3}\text{sign}(z_{4}-x_{2})-\mu_ {4}(z_{4}-x_{2})+z_{5}\\ \dot{z}_{5}=v_{5}\\ v_{5}=-\lambda_{5}{L_{2}}^{1/2}|z_{5}-v_{4}|^{1/2}\text{sign}(z_{5}-v_{4})-\mu_ {5}(z_{5}-v_{4})+z_{6}\\ \dot{z}_{6}=-\lambda_{6}{L_{2}}\text{sign}(z_{6}-v_{5})-\mu_{6}(z_{6}-v_{5}) \end{array} \right. \tag{32}\]_where \(x_{3}=f(x,t)+g(x,t)u\), \(z_{4}=\hat{x}_{2}\), \(z_{5}=\hat{d}_{2}\), \(z_{6}=\hat{\hat{d}}_{2}\). Because the disturbance observer is proven to be finite-time convergent, after a transient period, \(\dot{z}_{4}=\hat{\dot{x}}_{2}\) can be regarded as the \"acceleration\" signal. In addition, the matched disturbance \(d_{2}(t)\) can be accurately estimated for further potential, such as reducing the switching gain in the control signal (23), thus decreasing the chance that chattering phenomena will occur. However, the use of another disturbance observer will increase the computational burden, and consequently, a longer response time is expected, which might represent a disadvantage in practice._  
**Remark 4**. : _Because the mismatched disturbance is absent from the control system, if the initial values of the observer states are selected as \(z_{1}(t_{0})=x_{1}(t_{0})\), \(z_{2}(t_{0})=z_{3}(t_{0})=0\), then it is derived from the observer formulation and observer error dynamics that \(\varepsilon_{1}(t)=\varepsilon_{2}(t)=\varepsilon_{3}(t)=0\) and \(v_{1}(t)=v_{2}(t)=0\) during the absence of mismatched disturbances. In this case, the sliding surface (19) and the control law (20) reduce to those of the conventional TSMC. This indicates the nominal control performance preservation capability of the proposed controller._  
**Remark 5**.