## 1 Introduction  
The linear survey system of a coordinate measuring machine (CMM) is composed of a grating scale, a servo motor, and a linear motion mechanism, enabling the survey of free-form surface components [1, 2]. The maglev ruler has the potential to simplify this system, as its mover core (MRC) undertakes the functions of autonomous displacement and displacement survey [1, 2]. The MRC of the maglev ruler (MAR) moves in a step-by-step linear fashion with a fixed step size. The microcontroller determines its displacement by recording the number of steps, and the horizontal positioning precision of the MRC directly influences the survey accuracy of the MAR [1, 2]. This paper delves into the research on the horizontal displacement system of the MAR. A mathematical model is established based on the principles of magnetic circuits and dynamics, the improved active disturbance rejection control algorithm is utilized to solve the nonlinear and strongly coupled problems of the system, and the sensor-less control technology is adopted to simplify the system structure [1,2]. Active disturbance rejection control (ADRC), particularly effective in handling nonlinear and coupled systems [3, 4, 5, 6, 7], is adopted to address the horizontal motion control challenges. Its core mechanism--real-time estimation and compensation of total disturbances via an extended state observer (ESO)--aligns with the requirements of the maglev ruler's horizontal system. ADRC has demonstrated success in maglev systems, such as yaw angle control [8] and rotor stabilization [9]. However, its application to horizontal motion control of maglev rulers remains unexplored. The rotor of a maglev turbo machine is subject to various disturbances, and traditional PID control has limitations [10]. The linear active disturbance rejection controller (LADRC) improves upon the shortcomings of PID but still has issues such as overshoot. The improved LADRC suppresses uncertain factors through specific design, facilitating its application in engineering [10]. Conventional tracking differentiators (TDs) face challenges in balancing noise suppression and response speed [11, 12, 13, 14]. To overcome these limitations, a novel TD (SYSTD) is designed, as detailed in Section 5. The fhan-TD with double \(r\) improves its performance through innovative design, and its advantages have been verified by simulations and experiments [14]. In the field of electric vehicles, to address the speed-regulation contradiction of permanent-magnet synchronous motors, a study proposed MTD-DADRC [15]. This controller is applied in the speed loop control, which improves the system stability and anti-noise ability and enhances the speed tracking performance of the motor. Compared with the traditional PI controller, it can shorten the shift-speed-regulation time and reduce the shift shock [15]. There is also a study that proposed an active disturbance rejection fractional-order controller (ADRFOC) for time-delay systems to achieve independent control of servo and regulation [16]. Comprising FOC and MESO, its effectiveness was verified through mathematical derivation. The air-floating motion platform experiment shows that its performance outperforms that of typical time-delay ADRC and SIMC-PID controllers [16]. When the MAR system conducts surveys, the MRC moves in a step-by-step manner, resulting in a high-frequency fluctuation of its position feedback signal. Classical TD and MTD cannot meet the response speed requirements when dealing with such fast-changing signals, leading to tracking delays. This will affect the real-time tracking precision of the MRC position, thereby reducing the positioning accuracy of the horizontal control system (HCS) of the MAR system and making it impossible to quickly and accurately obtain the position information of the MRC. Furthermore, in the MAR system, obtaining accurate position and speed information is of utmost importance. In the field of motor control, relevant technologies can offer valuable references [17, 18, 19, 20, 21].