## 1 Introduction  
Automated mechanical transmission (AMT) has a mechanical structure similar to manual transmission (MT), and it is equipped with additional clutch and gear-shifting actuation systems to realize automatic control. AMT inherits the advantages of high transmission efficiency and the low cost of MT, and provides convenient operation similar to automatic transmission (AT). AMT has been widely applied in trucks, buses and small cars [1]. However, a poorly designed and controlled AMT may not achieve desired shifting quality. To improve the performance of AMT, many researchers have focused on control strategies in terms of clutch, engine, shift schedule or integrated driveline control [2; 3; 4], while the details of gear-shifting control are seldom studied. The shifting duration is one of the important indices that determine the torque interruption during the gear-shifting process. The key to minimizing the shifting duration is to achieve independent gear-shifting control. Nowadays, different varieties of gear-shifting actuation systems are applied in AMT, such as, electrohydraulic types [5; 6], electro pneumatic types [7; 8], and electromechanical types [9; 10]. However, these systems are limited by their complex structure and can hardly meet the need to implement independent gear-shifting actuation systems. As a result of advances in permanent magnet materials, power electronics and control technologies, the linear electromagnetic actuator (LEMA) is widely applied to high-speed and high-precision positioning systems, since it is of simple structure and responds quickly [11; 12]. An LEMA-based independent gear-shifting actuation system offers tremendous potential for shortening the duration of power interruption, because it provides feasible overlap control of gear-disengaging and gear-engaging phases, which can significantly reduce the shifting time [13]. Gear-shifting with the synchronizer is performed by changing the position of the shifting fork according to the appropriate time, i.e., the control of the actuator requires the position information of the actuator as feedback [14]. The most common approach is to install the position sensor on each actuator, but this introduces a number of position sensors, increases the system cost and takes up more installation space. In addition, the position sensors are susceptible to electromagnetic interference when packed in the same shell as the actuators. Once the position sensor fails, the actuation system cannot work normally [15]. In order to overcome these shortcomings, the sensorless control scheme is adopted in this paper. Compared with the separate actuator/sensor systems, sensorless control provides many advantages such as low-cost, simplicity and robustness [16; 17]. In this context, a great number of papers have been presented for the piezoelectric actuator [18], induction motor [19; 20], permanent magnet synchronous machines [21], etc. Compared to rotating electrical machines, studies of sensorless control for LEMAs are not frequently found in the literature. In [22], an adaptive and optimized switching velocity estimator is designed, but this method only applies to a system that satisfies the persistent excitation conditions. Furthermore, some cascade structures, which combine a position observer with a velocity estimator, are proposed to achieve sensorless control of the electromagnetic system. In particular, the Luenburger observer [23], extended Kalman filter [24] or sliding-mode observer [25] are widely used to estimate the position. However, these approaches require relatively accurate mechanical models that are often hard or difficult to obtain. Currently, sensorless control for LEMAs has been extensively used in industrial applications, such as, active vibration control [26], loudspeaker systems [27] or linear compressors [28; 29], but few papers pay attention to the gear-shifting control. Since the actuator is directly connected to the load, it is easily affected by various uncertainties or disturbances from the complex gear-shifting process.