## 4 Simulations
### Numerical Example
Consequently, TSMC signals are suppressed in the following figures for better visualization of other controllers. It can be seen in Figure 1 that, although EOSMC can only drive the system state to a bounded region of about 0.08 around the equilibrium, DOBTSMC offers much higher accuracy, with a bound of \([-2.5\times 10^{-8},\ 1.5\times 10^{-8}]\). Figure 2 depicts the mismatched disturbance existing in the target system and the second state \(x_{2}(t)\) response of EOSMC and DOBTSMC. The figure illustrates that the mismatched disturbance is nonlinear and time varying in a bounded region. It can also be interpreted from the figure that, overall, the trajectory of the mismatched disturbance and that of the second state are symmetric about the horizontal axis 0. This phenomenon can be explained as follows. The system (38) formulation implies that, to stabilize the system state \(x_{1}(t)\) to the equilibrium point, the following inequality must be fulfilled:  
\[\left|\dot{x}_{1}\right|=\left|x_{2}+d_{1}\right|\leq\delta \tag{41}\]  
where \(\delta\) is a sufficiently small positive number and can represent the performance indicator for evaluation. Inequality (41) indicates that for the sake of stabilization, \(x_{2}(t)\) must counteract the time-varying effect caused by the mismatched disturbance \(d_{1}(t)\). Consequently, \(x_{2}(t)\) and \(d_{1}(t)\) should be opposite. This also explains why the system state \(x_{1}(t)\) converges to the equilibrium, but not \(x_{2}(t)\). Figure 3a records the control signals generated by EOSMC and the proposed algorithm. It clearly shows that both control signals are smooth without chattering and exhibit the same magnitude of peaking phenomenon at \(t=5\) s as the mismatched disturbance is suddenly initiated. The satisfactorily chattering-attenuated control signal results from an attenuation step depicted in Figure 3b, in which the chattering signal \(v\) is filtered to create the smooth switching control signal \(u_{\pi}\), as in (22). Therefore, it demonstrates the effectiveness of the proposed algorithm in attenuating chattering effects. Figure 1: Response trajectory of the state \(x_{1}\)  
Figure 4 compares EOSMC and the proposed control algorithm in mismatched disturbance estimation performance. Under the condition of time-varying mismatched disturbances, from \(t=5\) s onwards, the estimation error of EOSMC varies inside a bounded region of 0.1 around the equilibrium. In comparison, DOBTSMC can deliver significantly higher precision, with an estimation error of approximately \(\pm 5\times 10^{-7}\). This subsequently contributes to the higher accuracy in the system state response discussed earlier and proves the effectiveness of the proposed controller in mismatched disturbance estimation. Figure 3: Total control signal (**a**) and switching control signal (**b**). Figure 2: Mismatched disturbance and second state response trajectories. Figure 5 illustrates the sliding surface trajectory of the proposed control algorithm, which oscillates inside the interval \([-0.005;\ 0.005]\). Because the sliding surface \(\sigma\), defined in (19), is a function of the second state \(x_{2}\), which varies with time to suppress the influence of the mismatched disturbance \(d_{1}\), it is reasonable that the sliding surface \(\sigma\) is also time varying. Nonetheless, the proposed control algorithm only utilizes the sign, but not the magnitude, of the sliding surface. Thus, the oscillating trajectory of the sliding surface is believed to have little or no effect on controller performance. Figure 4: Mismatched disturbance estimation error of ESOSMC and DOBTSMC. Figure 5: Sliding surface response of the proposed control algorithm. To illustrate the influence of different design parameter values on the performance of the disturbance observer, three simulations were conducted with different values of the Lipschitz constant, \(L=1,3,5\).