## 4 Simulation and Results
### The High-Frequency External Disturbances Within 1  
In this section, disturbances are added to the state functions, which have a high frequency but an amplitude of less than 1. Specifically, add \(cos(10t)\) to \(d_{1}(t)\), add a random number with frequency 0.1 to \(d_{2}(x,t)\), and add \(sin(10t)\) to \(d_{3}(x,t)\).  
The simulation results are basically consistent with Section 4.1. From Figure 7d, all three states can follow the desired trajectory \(sin(t)\). Firstly, compared with Figure 7a-c, the chattering phenomenon is the most obvious in Figure 7b, especially the error based on the SMC with exponential switching function, which starts to display obvious chattering. But the error based on the SMC with fuzzy switching function always displays almost no chattering. Secondly, when we compare Figure 7 to Figure 6, the value of errors is almost uniform, while the chattering becomes worse, except for the error based on the SMC with fuzzy switching function. The comparative analysis of tracking performance under small-amplitude reference signals reveals that both the conventional SMC employing exponential switching function and the proposed fuzzy-adaptive SMC satisfy the prescribed accuracy thresholds, with errors maintained below 0.005. However, the experimental  
Figure 5: Control inputs of the given system based on SMC with different switching functions. (**a**) Description of control input \(u1\) of the first state function. (**b**) Description of control input \(u2\) of the second state function. (**c**) Description of control input \(u3\) of the third state function.  
validation demonstrates that the fuzzy switching function exhibits superior chattering suppression characteristics.  
Compared with Figure 6d, Figure 7d shows different adjust time. From Table 1, we can obtain state tracking errors when the system reaches steady state. The system errors are less than 0.5%, which satisfies the tracking accuracy, as the amplitude of the desired state trajectory is 1. It is intuitive to see that the tracking accuracy of the SMC with the fuzzy switching function is the highest. However, the adjustment time is also the longest. In order to discuss the reason for this result, the attenuation oscillation curve \(sin(t)/t\) is selected  
Figure 6: (**a**) Tracking error \(e1\) of the first state function. (**b**) Tracking error \(e2\) of the second state function. (**c**) Tracking error \(e3\) of the third state function. (**d**) State tracking of each state.  
Figure 7: (**a**) Tracking error \(e1\) of the first state function. (**b**) Tracking error \(e2\) of the second state function. (**c**) Tracking error \(e3\) of the third state function. (**d**) State tracking of each state.  
as the input to simulate the function of each switching function in the SMC. As shown in Figure 8, as the input decreases, the amplitude of each switching function decreases. The amplitude of the fuzzy switching function decreases significantly with the decrease in the input, and the switching is smoothest when the positive and negative signs of the input change. Therefore, chattering can be well weakened in the SMC with fuzzy switching function. Similarly, the amplitude of each switching function is set to be the same at \(t_{1}\), and then the amplitude of the fuzzy switching function is the largest at \(t_{0}\), and \(t_{0}<t_{1}\). This is the reason why the SMC with fuzzy switching function has a large adjust time and overshoot.