## 6 Simulation Results and Analysis
### Comparison of Dual-Motor Speed Synchronization Performance with Disturbance Observer  
When a given sinusoidal signal is applied, the tracking curve exhibits distortion near zero speed due to frictional disturbances. From the speed tracking error curve, it can be seen that the speed tracking error is minimal except at zero speed, indicating the need for effective compensation of the frictional force to reduce the tracking error. The comparison between the reference frictional force value and the disturbance observer's estimated value is shown in Figure 13. It can be observed that the disturbance  
Figure 11: Comparison of dual\(-\)motor speed waveform under different control strategies at low-speed operation without disturbance observer: (**a**) speed waveform under improved GFTSMC control; (**b**) speed waveform under SMC control; (**c**) speed waveform under PID control. Figure 12: Comparison of dual\(-\)motor speed synchronization errors under different control strategies at low-speed operation without disturbance observer. observer quickly tracks the changes in frictional force, with the error remaining within an extremely small range, indicating that the observer has excellent estimation accuracy. When the motor operates at a speed of 1000 r/min, the dual-motor speed waveforms under different control strategies are shown in Figure 14. At 0.1 s, after applying a friction disturbance to motor 2, the motor speed shows significant fluctuations. In Figure 14a, when the improved GFTSMC cross-coupling control system is used, the motor's maximum speed decreases by about 16 r/min. In contrast, in Figure 14b, with the traditional SMC cross-coupling control strategy, the motor's maximum speed decreases by only about 17 r/min. In Figure 14c, when the PID cross-coupling control strategy is used, the motor's maximum speed decreases by about 22 r/min. Furthermore, Figure 15 compares the dual-motor speed synchronization errors under different control strategies. With the PID cross-coupling control system, the system's maximum synchronization error is 14 r/min, and about 0.01 s of adjustment time is required to restore to the reference speed. When the traditional SMC cross-coupling control system is used, the system's maximum synchronization error is 13 r/min, and the adjustment time is shortened to about 0.008 s. Under the improved GFTSMC cross-coupling control system, the system's maximum synchronization error decreases to 11 r/min, with an adjustment time of approximately 0.001 s.  
When the motor operates at a speed of 300 r/min, the effects of different control strategies on the dual-motor speed waveforms are shown in Figure 16. At 0.1 s, after applying a friction disturbance to motor 2, the system exhibits different dynamic response characteristics. As shown in Figure 16a, when the improved GFTSMC cross-coupling control system is used, the motor's maximum speed decreases by about 7 r/min. In contrast, in Figure 16b, with the traditional SMC cross-coupling control strategy, the motor's maximum speed drop is controlled to about 10 r/min. In Figure 16c, when the PID cross-coupling control system is used, the motor's maximum speed decreases significantly, by about 12 r/min. Furthermore, Figure 17 compares the dual-motor speed synchronization errors under different control strategies. With the PID cross-coupling control system, the system's maximum synchronization error reaches 8 r/min, and about 0.01 s of adjustment time is required to restore to the reference speed. With the traditional SMC cross-coupling control system, the synchronization error is reduced to 6 r/min, with an adjustment time of about 0.005 s. Under the improved GFTSMC cross-coupling control system, the maximum synchronization error is further reduced to 5 r/min, with an adjustment time of approximately 0.002 s.  
Figure 13: Comparison of reference and estimated friction torque. To more clearly compare the performance of synchronization accuracy of dual motors under external disturbances, key metrics have been quantitatively assessed.