## 1 Introduction
At the same time, the combination of fuzzy control and cross-coupling control further improves the system's synchronization and accuracy under load variations and external disturbances [18; 19; 20]. In servo systems, the presence of friction often leads to instability during low-speed operation, causing issues such as waveform distortion when the speed passes through zero, which significantly affects the accuracy of dual-motor control systems. To improve the system's low-speed stability and control accuracy, the study of friction torque compensation has become a key topic. After years of research, scholars both domestically and internationally have proposed various friction compensation methods, mainly including compensation methods based on friction models, disturbance suppression methods using robust controllers or disturbance observers, and compensation methods based on intelligent control strategies [21; 22]. Among these, friction disturbance suppression methods based on disturbance observers, such as disturbance observer (DOB), extended state observer (ESO) [23], and generalized proportional integral observer (GPIO), have made significant progress. Common strategies for addressing the friction issue include friction compensation and controller improvement. Friction compensation methods based on friction models can be divided into two categories: one is model compensation based on fixed friction parameters, and the other is adaptive friction compensation based on friction characteristics. The former obtains the friction model parameters through offline identification, while the latter adjusts the model parameters through online estimation. Since adaptive friction compensation based on friction characteristics offers advantages in real-time performance and has a relatively simple structure, it remains the focus of current friction compensation research [24]. In existing dual-motor synchronization control methods, many studies have adopted traditional control strategies, such as the method proposed in reference [14]. Although these methods can effectively adjust motor speeds, they often struggle to maintain high-precision speed synchronization in dual-motor control systems when faced with nonlinear disturbances, such as friction. In contrast, this paper proposes a cross-coupling control strategy based on an improved GFTSMC and DOB. This method significantly enhances the system's dynamic performance and speed synchronization accuracy by combining the cross-coupling controller with the improved GFTSMC. Additionally, the DOB, with its simple structure, effectively suppresses external disturbances, further enhancing the system's robustness against external perturbations. The structure of this paper is arranged as follows: In Section 1, the mathematical model of the PMSM is established. Section 2 designs a speed regulator for the single motor and proves the system's stability. Section 3 designs a cross-coupling control strategy based on GFTSMC, with stability analysis conducted using a Lyapunov function. Section 4 designs a disturbance observer to compensate for the friction torque disturbances in the system. Finally, MATLAB simulations are conducted, and the experimental results are analyzed in detail. The paper concludes with a summary of the main findings.