## 6 Simulation Results and Analysis
### Comparison of Single-Motor Control Performance Under Different Control Methods  
The input signal of the servo system is set as a sine wave \(n(t)=300\sin(\frac{\pi}{2}t)\mathrm{r}/\mathrm{min}\), where \(n(t)\) represents the input speed of the motor. To simulate the impact of external loads on the system, a 1 N-m load torque is applied to the motor at 0.2 s to study its effect on the dynamic response of the servo system.  
The parameters of the PI controller need to be properly tuned before conducting comparative experiments. In this paper, the bandwidth method is used to calculate the PI controller parameters. First, an appropriate bandwidth is selected based on the system's dynamic requirements for response speed. Then, the initial values of the proportional gain and integral gain are derived based on the selected bandwidth. Finally, the initial parameters are fine-tuned through the actual system response to ensure that the system's dynamic performance meets the design requirements. This ensures the rationality and comparability of the PI controller parameters in the comparative experiments.  
In this experiment, to verify the performance of different control strategies, we designed a PI controller, an SMC, and a GFTSMC, and provided the corresponding parameter settings. First, in the PI controller design, the PI control parameters were set as follows: \(k_{ps}=0.02\),\(k_{is}=2.5\). Next, in the SMC design, the sliding surface was defined as \(s=ce+\dot{e}\), and the control law adopted the exponential reaching law: \(s=-\epsilon\dot{g}n(s)-ks\). The controller parameters were set as follows: \(c=350,\epsilon=600,k=1000\). Finally, in the GFTSMC design, the controller parameters were set as follows:\(\alpha=250\), \(\beta=140\), \(\gamma=1800\), \(\varphi=300\), \(p=7,q=5\).  
The speed-tracking waveform of the single motor under no-load conditions is shown in Figure 8a. GFTSMC can quickly reduce the tracking error during the transient response stage, and its fast convergence property allows the error to approach zero in a short period of time. Traditional SMC also has good error convergence capabilities, but compared to GFTSMC, the error convergence speed is slower, especially in the initial stage, where the reduction in tracking error is relatively slow. The dynamic response of PID is smoother, and the error reduction is slower, especially under rapidly changing reference signals, where the error is significantly larger than that of GFTSMC and traditional SMC. The speed-tracking waveform of the single motor under load conditions is shown in Figure 8b. After applying the load disturbance at 0.2 s, GFTSMC demonstrates superior disturbance rejection performance, quickly stabilizing the speed fluctuations, with the maximum motor speed decreasing by only about 16 r/min, effectively keeping the impact of the disturbance at a minimal range. In contrast, traditional SMC also shows some disturbance rejection capability, with the maximum motor speed dropping by about 32 r/min, but the fluctuation amplitude is significantly larger than that of GFTSMC, showing weaker dynamic response capability. In comparison, PID performs the weakest, generating significant speed fluctuations after the disturbance is applied, with the maximum speed decreasing by about 40 r/min, and the fluctuation amplitude is much larger than that of GFTSMC and traditional SMC.