## 1 Introduction  
With the development of power electronics technology and high-performance permanent magnet materials, the permanent magnet linear synchronous motor (PMLSM) has gained significant attention. Because its structure has no mechanical friction problem, a large thrust force, and small volume, the PMLSM is widely used in transportation systems, machine tools, precision positioning platforms, and other fields [1, 2]. In addition, the market demand for chip semiconductors has increased in recent years, leading to the rapid development of the semiconductor manufacturing industry; therefore, the high-speed and high-acceleration motion platform built by the PMLSM has gradually become an essential part of the equipment [3]. However, the performance requirements are becoming increasingly stringent, and the single-motor driving method can no longer meet the high-acceleration requirement. To overcome the thrust force limit of the PMLSM and further improve the efficiency of semiconductor manufacturing equipment, a dual-motor driving method has been designed and widely used [4, 5]. A redundant direct-drive gantry platform is a common application of the dual-motor driving method. The Y-degree-of-motion freedom is driven by two PMLSMs simultaneously, and the rigid connection between the PMLSMs ensures that the thrust force can act on the beam together, realizing high speed, high acceleration, and a large stroke motion [6]. A dual-actuator system can usually achieve a higher acceleration motion, but it also introduces the asynchronous motion problem, which will cause tension in the connection structure and decrease the service life or even seriously damage the mechanical structure [7]. Scholars have proposed a variety of control methods for the synchronous motion of a dual-motor system, including the series synchronous control method [8], parallel synchro-nous control method [9], and cross-coupling synchronous control method [10]. Because the cross-coupling synchronous control method can adjust the control voltage of each motor in real time according to the synchronization error, it has certain advantages in ensuring the synchronous motion of the dual-motor system [11; 12]. The control structure of the cross-coupling synchronous control method is simple; therefore, the accuracy of the control parameter selection is the key factor that determines the synchronous motion performance of the dual-motor system. Some scholars improved the synchronous control method by considering cross-coupling, such as the variable-gain cross-coupling synchronous control method [13], fuzzy neural network cross-coupling synchronous control method [14], and self-tuning cross-coupling synchro-nous control method [15]. These methods can improve the real-time adjustment ability of the parameters of the cross-coupling controller, thereby ensuring the adaptability of the control parameters under different synchronization errors. However, accurately establishing the parameter adjustment law of the cross-coupling synchronous control method requires long-term pre-learning, and the control parameter adjustment algorithm has a high degree of complexity, making it difficult for practical applications. Considering that the characteristics of the PMLSM system can be obtained through system identification, several methods were studied, for example, a model-based feed-back-feedforward decoupling control method [16], model-based adaptive synchronous control method [17], robust immersion and invariance adaptive coordinated control method [18], adaptive thrust-allocation based synchronous control method [19], and other related methods [20; 21; 22]. The introduction of model information can reduce the design complexity of the parameter adjustment method and provide a basis for control parameter selection, thus effectively suppressing the dual-motor system's synchronization error. However, these synchronous control methods adjust the control voltage of the PMLSM through position compensation or force compensation when synchronization error is observed.