## 4 Simulations
### Numerical Example  
To demonstrate the effectiveness of the proposed control algorithm, the succeeding nonlinear second-order system, which is subject to time-varying mismatched and matched disturbances, is considered:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}+d_{1}(t)\\ \dot{x}_{2}=-2x_{1}-x_{2}+e^{x_{1}}+u+d_{2}(t)\end{array} \right. \tag{38}\]  
Here, the primary objective is to stabilize the above system, and the initial system state is selected away from the equilibrium as \(x=[1,-1]^{T}\). Mismatched and matched disturbances are deliberately defined as \(d_{1}(t)=1+0.3\cos(2t)-\sin(t)\) and \(d_{2}(t)=\sin(2t)\), respectively. The mismatched disturbance is selected such that as the running time increases and the system state \(x_{1}(t)\) reaches its steady-state value. The mismatched disturbance is still time varying in a relatively large, bounded region rather than gradually converging to a constant as in the reference [41]. This is a decisive factor in demonstrating the superiority of the proposed control algorithm compared to strategies considered, which will be illustrated by later simulation results. Moreover, to illustrate the nominal control performance preservation characteristic of the proposed control algorithm, the mismatched disturbance is only active after \(t=5\) s.  
For comparison purposes, the chattering-free full-order TSMC approach [30], denoted as TSMC, the extended-state observer-based chattering free SMC [41], denoted as ESOSMC, and the proposed control algorithm, denoted as DOBTSMC, are employed. It should be noted that both DOBTSMC and ESOSMC stem from TSMC, and that the proposed control algorithm, to a certain extent, can be interpreted as an extension and improvement of TSMC and ESOSMC in dealing with mismatched disturbances. Hence, the analogy is drawn here to demonstrate this argument. Briefly, the ESOSMC proposed in [41] is constructed of a nonlinear extended-state observer (ESO) and an ESO-based SMC. The ESO is designed based on a hyperbolic tangent function of the form:  
\[\left\{ \begin{array}{l}\dot{Z}_{1}=Z_{2}-a_{1}(Z_{1}-x_{1})+x_{2}\\ \dot{Z}_{2}=-a_{2}\text{tanh}(b(Z_{1}-x_{1}))\end{array} \right. \tag{39}\]  
where \(Z_{1}=\hat{x}_{1}\), \(Z_{2}=\hat{d}_{1}\), \(\text{tanh}(*)=(e^{*}-e^{-*})/(e^{*}+e^{-*})\), \(a_{i}>0\), \(i=1\), \(2\), \(b>0\). The sliding mode surface integrated with the ESO is designed as:  
\[s=\dot{x}_{2}+c_{2}(x_{2}+Z_{2})+c_{1}x_{1}+\dot{Z}_{2} \tag{40}\]  
where \(c_{i},\ i=1,2\) are chosen in the same manner as in previous sections. The design control parameters are specifically selected to be the same for all control algorithms as \(c_{1}=5\), \(c_{2}=5\), \(T=0.1\), \((\rho_{2}+k_{T}+\eta)=3\). In addition, for the TSMC and the DOBTSMC, following (8), \(\alpha_{i}(i=1,2)\) are designed as \(\alpha_{1}=3/7\), \(\alpha_{2}=3/5\). For the mismatched disturbance observers, ESO parameters are selected as \(a_{1}=20\), \(a_{2}=14\) and \(b=21\); and DOB parameters are chosen as \(L=3\), \(\lambda_{1}=6\), \(\lambda_{2}=11\), \(\lambda_{3}=6\), \(\mu_{1}=18\), \(\mu_{2}=6\), and \(\mu_{3}=2\). The sampling time is set to be 0.1 ms. The system state \(x_{1}(t)\) delivered by all control algorithms is shown in Figure 1. For the first 5 s, all controllers are able to stabilize the target system. However, the convergence rate of ESOSMC is lower than that of the other two approaches. It takes ESOSMC about 3.4 s to reach the 0.01 mark (-1% of the initial value), whereas it takes TSMC and DOBTSMC only around 1.8 s to reach this mark. Moreover, because the trajectory of TSMC and DOBTSMC are identical in this period, it proves that the proposed control algorithm possesses the nominal performance preservation characteristic in the absence of mismatched disturbances. After \(t=5\) s, mismatched disturbance starts to affect the control system. It is noticeable that TSMC fails to stabilize the target system, which is already anticipated in the Problem Statement section.