## 4 Simulation
### Mixed Control Input Simulation Result  
The simulation proceeds using Figure 6. The mixed control input is not an actual control input, assuming the worst case of a controller using neural network. Therefore, we verify through simulation whether the proposed controller shows satisfactory control performance even in the worst case.  
Figures 16-19 are the result of controller performance for mixed input. As shown in Figure 16, when only NTSMC is used, the sinusoidal wave reference signal part shows the same characteristics as Figure 8. Nevertheless, in a constant reference signal with a large bias, it follows the steady state well. However, it is slow to reach a steady state. In Figure 17 using RBF NN, the sinusoidal wave reference signal part is improved as shown Figure 9. The effect of sudden signal change, which is a weak part of the neural network, seems to be suppressed by NTSMC, so overshoot does not appear. Whereas, as with the previous results, there is a delay. Figure 18 shows the previous robust term, and Figure 19 shows the adaptive robust term. Both controllers improved control performance for suddenly changing signals. Based on 15 s when the signal changes suddenly, the previous robust term is 0.73 s faster than NTSMC-NN, and the adaptive robust term is 0.93 s faster than NTSMC-NN. As a result, the proposed controller was able to increase the convergence speed compared with the previous robust term.  
Next, we proceed to compare the control inputs of the controllers to the mixed inputs and the error comparisons.  
Figures 20 and 21 are compared for control input performance result. Results from the mixed input are similar to those from continuous input. As shown is Figures 20 and 21b, the controller using the neural network generates a large control force, and the chattering of the control input is also increased by the switching term because the effect of the neural network is included in the system output. Figures 20 and 21c are results for the controller using neural network and robust term. As shown in Figure 18, the control performance is much better than NTSMC-NN; however, input chattering is larger than that of NTSMC-NN. The proposed controller, as shown in Figures 20 and 21d, significantly reduced chattering compared with the other controller. Due to the reverse saturation function and the adaptive coefficient, the control input decreases as it approaches the steady state, and it can be confirmed that chattering occurs less by removing the robust term, which can cause chattering in the vicinity of the steady state. Additionally, as shown in 15 s and 22.5 s, the proposed controller can generate control input according to errors; therefore, the proposed controller is faster than the previous controller.  
Figures 22 and 23 compare error performance for mixed input. A large error change occurs at 7.5 s, 15 s, and 22.5 s, when the signal changes suddenly. On average, the proposed controller showed a faster convergence speed of about 0.2 s than the previous robust term. Moreover, it can be seen that the steady state error converges to 0 for a constant signal, unlike the sinusoidal signal.  
Figure 19: Proposed controller performance for mixed input.  
Figure 18: RT-NTSMC-NN controller performance for mixed input.  
Figure 21: Mixed input control input \(\theta_{2}\) control performance. (**a**) NTSMC input result; (**b**) NTSMC-NN input result; (**c**) NTSMC-NN-RT input result; (**d**) proposed controller input result.  
Figure 22: Comparing \(\theta_{1}\) error for mixed input.  
Figure 20: Mixed input \(\theta_{1}\) control input performance. (**a**) NTSMC input result; (**b**) NTSMC-NN input result; (**c**) NTSMC-NN-RT input result; (**d**) proposed controller input result.