## 1 Introduction  
Laser processing, with advantages such as its being contactless, fast, and noiseless, has become a key technique in industrial production and automation [1]. Galvanometer scanners (GSs) are core components of laser processing equipment, in which the laser path changes through the swinging of the motor. At present, the ScanLab in Germany, and GSI Group (GSI) and Cambridge Technology Inc. (CTI) in the United States have launched excellent two-dimensional (2-d) and three-dimensional (3-d) GSs. The driver of traditional GSs is generally controlled by an analogue driver board. Although this can also provide a favorable control effect, the performance and polyfunctionality of the system are limited due to the relatively fixed control structure and a finite number of fixed parameters. For example, vectors or steps fail to adapt to different dynamic requirements. When system parameters change due to variations of the scanning angle and temperature as well as the aging of parts, the fixed parameters cannot be adjusted accordingly. These limitations, however, can be overcome by the digital control of the driver [2]. The motor of a GS generally runs at a high speed when in service: this inevitably produces unwanted mechanical heat. The heat causes the system parameters (e.g., torque, rotational inertia, and damping) of the motor to change during the operation, which further degrades the tracking performance of the system [3]. Therefore, the following, as matters of particular concern, should be achieved in the control system of a GS: quick responses, a high position accuracy, and a low temperature-drift. To improve the performance of the control system, many methods have been proposed in recent years. [PERSON] et al. proposed the use of a transformation-based iterative learning control for the GS of a non-collocated sensor, which reduced output error during the actual control, and the resulting tracking accuracy was higher than that of the traditional integrated circuit design [4]. To improve the performance of laser scanning microscopes, they also proposed a GS based on an iterative learning control. The root mean square error (RMSE) of the controller was decreased by a factor of 73 compared with that of the GS with a feedback control in commercial systems [5]. By reducing the noise of angle sensors to realize a motion of a higher quality, [PERSON] et al. developed an iterative control technique based on inversion, which decreased the tracking error to 2.83 \(\times\) 10\({}^{-40}\)[6]. [PERSON] et al. proposed a PID-L1 controller based on the extension of the existing control structure that was characterized by a better resistance to constant disturbance than the traditional PD controller. On this basis, [PERSON] et al. studied the MPC control algorithm, further improving the disturbance resistance of the GS [7,8]. [PERSON] et al. developed an adaptive sliding-mode variable-structure control algorithm based on the Smith predictor and internal model principle to solve parameter perturbation and time delay problems in GSs. Compared with a traditional PD controller, the controller had a lower tracking error for sinusoidal signals [3]. To enhance the dynamic performance of the scanning system, [PERSON] et al. proposed an improved adaptive inverse control method that enabled a decrease of 90% in the error compared with the state-of-the-art scanning system [9]. To reduce the tracking error for galvanometer trajectories, [PERSON] et al. combined an adaptive inverse control with dynamic behaviors of structures, thus proposing a photo adaption inverse control method, whose tracking error was 90% lower than that of the reference controller [10]. [PERSON] et al. developed a scheme based on a multi-rate control to resist external disturbance caused by laser-material interaction, thereby inhibiting narrow-band non-Nyquist disturbance [11]. Considering the requirements for high accuracy and high dynamics of GSs, [PERSON] et al.