## 6 Simulation Results and Analysis
### Comparison of Dual-Motor Speed Synchronization Performance Without Disturbance Observer  
In this experiment, to verify the performance of different control strategies, we designed synchronous PI controllers, SMC, and GFTSMC, and provided the corresponding parameter settings. First, in the synchronous PI controller design, the PI control parameters were set as follows: \(k_{ps}=0.06\),\(k_{is}\)=10. Next, in the synchronous 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: \(c=60,\epsilon=150,k=400\)Finally, in the synchronous GFTSMC design, the controller parameters were set as follows: \(\alpha=250\), \(\beta=250\), \(\gamma=250\), \(\varphi=200,p=7,q=5\). When the motor operates at a speed of 1000 r/min, the dual-motor speed waveforms under different control strategies are shown in Figure 9. At 0.1 s, after applying a friction disturbance to Motor 2, significant fluctuations in motor speed occur. In Figure 9a, where the improved GFTSMC cross-coupling control system is applied, the maximum motor speed decreases by about 25 r/min. In contrast, in Figure 9b, where the traditional SMC cross-coupling control strategy is applied, the maximum motor speed decreases by only about 33 r/min, and in Figure 9c, where the PID cross-coupling control strategy is used, the maximum motor speed decreases by about 56 r/min. Additionally, the comparison of the dual-motor speed synchronization errors under different control strategies is shown in Figure 10. When the PID cross-coupling control system is used, the maximum synchronization error of the system is 34 r/min, and it takes about 0.03 s to restore to the reference speed. When the traditional SMC cross-coupling control system is applied, the maximum synchronization error is 21 r/min, and the adjustment time is shortened to about 0.02 s. Under the improved GFTSMC cross-coupling control system, the maximum synchronization error is reduced to 17 r/min, with an adjustment time of about 0.01 s.  
When the motor operates at a speed of 300 r/min, the effects of different control strategies on the dual-motor speed waveform are shown in Figure 11. At 0.1 s, a frictional disturbance was applied to Motor 2, and the system exhibited different dynamic response characteristics. As shown in Figure 11a, when the improved GFTSMC cross-coupling control system was used, the maximum motor speed decreased by approximately 16 r/min. In contrast, in Figure 11b, when the traditional SMC cross-coupling control strategy was used, the maximum motor speed decreased by only about 25 r/min. Meanwhile, in Figure 11c, when the PID cross-coupling control system was applied, the maximum motor speed dropped significantly by about 40 r/min. In addition, Figure 12 compares the dual-motor speed synchronization errors under different control strategies. When the PID cross-coupling control system was used, the maximum synchronization error reached 24 r/min, and approximately 0.05 s was needed to recover to the reference speed. When the traditional SMC cross-coupling control system was used, the synchronization error was reduced to 15 r/min, with a regulation time of about 0.02 s. In the improved GFTSMC cross-coupling control system, the maximum synchronization error was further reduced to 12 r/min, and the regulation time was about 0.01 s.  
Figure 8: Comparison of single\(-\)motor speed\(-\)tracking waveforms under different control strategies: (**a**) speed\(-\)tracking waveform of the single motor under no\(-\)load conditions; (**b**) speed\(-\)tracking waveform of the single motor under load conditions. In summary, the improved GFFSMC shows significant advantages over traditional SMC and PID control strategies in terms of disturbance rejection and synchronization accuracy. It not only limits the speed drop caused by frictional disturbances to a minimum but also significantly reduces the dual-motor synchronization error, achieving a shorter adjustment time.