## 1 Introduction  
Lithography is the most important and complicated aspect in integrated circuit manufacturing; the manufacturing accuracy determines the integration degree and performance of the manufactured chip. The highly-accurate two-dimensional stepper is the core element of lithography. The two-dimensional stepper driven directly by the maglev permanent magnet planar motor has many advantages, such as quick response, high accuracy, and simple structure [1; 2; 3; 4]. It can also work in a vacuum environment, which can meet the requirement of next-generation ultraviolet lithography technology [5; 6]. Thus, it has caught the attention of many researchers in the field of high-accuracy planar directly-driven lithography [7; 8; 9; 10; 11; 12]. High thrust density is a very important factor in the design of the planar motor to achieve the goal of high acceleration. Meanwhile, the simple winding structure is needed for easy manufacturing. Therefore, a maglev core-less planar motor adopting the concentric winding structure is proposed in this paper. The most common structure of the primary of the planar motor consists of multiple sets of rectangular windings and the windings are deviated 90 degrees from each other [13]. The proposed maglev planar motor adopts concentric windings with a square structure. The primary section of the motor uses the same windings and they are arranged as a matrix. This motor has many advantages, such as high winding utilization rate, good dynamic performance, and no cogging force [14]. thus a motor of this structure is very suitable for the application in high-accuracy two-dimensional movement. Meanwhile, the effective winding length is increased in the concentric winding structure. Thus, on the condition that the whole system is controllable, fewer phases are needed for the same mover area, reducing the required number of drive units accordingly. This is another significant advantage of the proposed concentric winding structure. The electromagnetic model of the traditional rectangle winding is simple, but its short edge will not generate a horizontal force. As a result, to make this planar motor possible, other sets of rectangular windings rotated 90 degrees are needed [15; 16], while the concentric winding in the proposed maglev planar motor adopts the square structure. The square structure leads to a more complex electromagnetic model and, therefore, the decoupling control of the current in the concentric winding is important for realizing the high-precision control of the whole drive system. It has to be controlled in six degrees of freedom because of the active magnetic bearing, even though the planar motor mover can only move relatively small distances in the \(x\)-\(y\) plane. Therefore, the maglev permanent magnet synchronous planar motor control system is a strong coupling, nonlinear, and multivariable system because the motion of the mover has multiple degrees of freedom and the magnetic field produced by the permanent magnet decreases nonlinearly in the vertical direction. The classical PID controller is the most commonly used controller in industrial automation for its simple structure, high stability, high reliability, simple adjustment, etc. The extra external cable disturbance can impact the performance of the planar motor directly; therefore, the external disturbance force needs to be considered when designing the control system [17; 18]. To solve this problem, active disturbance rejection control (ADRC) is proposed for designing the controller [19; 20; 21]. ADRC has low requirements for the modeling accuracy and it can observe the disturbance in multiple degrees of freedom through the extended state observer and implement compensation for it. Thus, it has been widely used in the control field. The maglev planar motor system is a complex, multi-input, multi-output coupling system.