## 2 System Overview and Modeling  
Figure 1a shows the prototype of the novel independent gear-shifting actuation system. Figure 1b shows the schematic diagram of the independent gear-shifting actuation system. Each shifting fork is driven by one LEMA, which can move axially with bi-directional movement to actuate the synchronizer to change gears between three positions (two gear-engaging positions and one neutral position).  
Since the axial inertia of the moving parts is minimized in the shifting mechanism, high response performance could be obtained via the high driving capability of the LEMA. The common approach for the control of the actuator uses the position sensor linked to the actuator as feedback, but this will introduce a number of position sensors, increase the system cost and take up more installation space. Therefore, this paper applies the sensorless control scheme to the position control of the LEMA. A velocity estimator is designed by using only the electrical subsystem, thus avoiding the complete system model that contains mechanical uncertainties.  
The structure and composition of the LEMA is represented schematically in Figure 2. The LEMA basically consists of permanent magnets, a moving coil, outer core, inner core and connecting plate [39]. A tubular structure is applied since it has no end effect and cogging effect. The high energy product sintered NdFeB magnet (45 SH) is adopted to achieve a high driving force, and a maximum operating temperature of 150 \({}^{\circ}\)C is selected for the automotive application environment. Permanent magnets are mounted on the outer core parts, and the Halbach magnetized topology is used to maximize the driving force of the LEMA. When current is applied to the moving coil, it interacts with the radial  
Figure 1: The independent gear-shifting actuation system: (**a**) prototype; and (**b**) schematic diagram. LEMA: linear electromagnetic actuator; TCU: transmission control unit.  
air-gap magnetic field via Lorentz force principle to create a proportional axial force, whose direction is determined by the polarity of the current-producing voltage, according to [PERSON]'s left-hand rule.  
As shown in Figure 3, the LEMA can be modeled mathematically as follows:  
\[I(t)=-\frac{R}{L}I(t)-\frac{k_{e}}{L}v(t)+\frac{1}{L}U(t) \tag{1}\]  
\[S(t)=v(t) \tag{2}\]  
\[v(t)=\frac{k_{e}}{m}I(t)-\frac{c}{m}v(t)-\frac{1}{m}F_{d} \tag{3}\]  
where, Equation (1) represents the electrical subsystem of the actuator. \(U\) is the applied voltage of the LEMA, \(R\) is the resistance of the moving coil, \(I\) is the current through the moving coil, \(L\) is the inductance of the moving coil. Equations (2) and (3) both describe the mechanical subsystem of the actuator. \(S\) is the position of the moving coil, \(v\) is the velocity of the moving coil, \(c\) is the viscous friction damping coefficient, \(F_{d}\) is the total bounded disturbance force applied on the moving coil, \(m\) is the mass of the moving part in the gear-shifting mechanism, \(k_{e}\) is the actuator coefficient.  
Define the state vector \(\boldsymbol{x}=\left[\begin{array}{cc}x_{1}&x_{2}&x_{3}\end{array}\right]^{ \mathrm{T}}=\left[S,v,I\right]^{\mathrm{T}}\), then the model of the LEMA can be rewritten in the following standard form:  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f_{1}(x_{1},x_{2})+r_{1}x_{3}+d_{1}\\ \dot{x}_{3}=f_{2}(x_{1},x_{2},x_{3})+r_{2}u+d_{2}\end{array}\right. \tag{4}\]  
where  
\[f_{1}(x_{1},x_{2})=-\frac{c}{m}x_{2}f_{2}(x_{1},x_{2},x_{3})=-\frac{k_{e}}{L} x_{2}-\frac{R}{L}x_{3},r_{1}=\frac{k_{e}}{m},r_{2}=\frac{1}{L}.\]  
Figure 3: System block diagram of the LEMA.  
Figure 2: The structure and composition of the LEMA.  
Since \(x_{1}\)-subsystem is an integral system, there exists no uncertainty. The disturbance \(d_{1}\) is determined by the parameter uncertainties and external load forces from the \(x_{2}\)-subsystem, and the disturbance \(d_{2}\) is determined by the parameter uncertainties from the \(x_{3}\)-subsystem.