## 2 Establishment of LIM Dynamic Model and Preliminaries  
### Establishment of LIM Dynamic Model  
The sketch map of LIM can be seen in Figure 1. LIM is like a rotary induction motor (RIM) that is tiled. The difference between LIM and RIM lies in the existence of end effect in LIM, and the dimensionless coefficient Q represent the magnitude of end effect, it can be derived from the speed of LIM calculated as [5; 7; 8]:  
\[Q=\frac{l\cdot R_{r}}{L_{r}\cdot v} \tag{1}\]  
where \(l\) means the primary length; \(R_{r}\), \(L_{r}\) represent secondary resistance and inductance, respectively; \(v\) denotes the speed.  
In this paper, the main goal is to control LIMs like DC motors, while ensuring superior performance. According to [7; 10], the equivalent circuit and flux linkage in \(dq-axis\) are expressed as follows.  
\[\begin{split} U_{ds}&=R_{s}i_{ds}+p\psi_{ds}- \omega_{e}\psi_{sq}\\ U_{qs}&=R_{s}i_{qs}+p\psi_{qs}-\omega_{e}\psi_{sd} \\ U_{dr}&=R_{r}i_{dr}+p\psi_{dr}-\left(\omega_{e}- \omega_{r}\right)\psi_{rq}\\ U_{qr}&=R_{r}i_{qr}+p\psi_{qr}+\left(\omega_{e}- \omega_{r}\right)\psi_{rd}\\ \psi_{ds}&=L_{sl}i_{ds}+L_{m}(1-f(Q))(i_{ds}+i_{dr} )\\ \psi_{qs}&=L_{sl}i_{qs}+L_{m}(1-f(Q))(i_{qs}+i_{qr} )\\ \psi_{dr}&=L_{rl}i_{dr}+L_{m}(1-f(Q))(i_{ds}+i_{dr} )\\ \psi_{qr}&=L_{rl}i_{qr}+L_{m}(1-f(Q))(i_{qs}+i_{qr} )\\ \end{split} \tag{2}\]  
where \(f(Q)=(1-e^{-Q})/Q\); \(U_{ds}\), \(U_{qs}\), \(i_{ds}\), \(i_{qs}\), \(\psi_{ds}\), \(\psi_{qs}\) denote the primary voltage, current and flux linkage in \(d-q\) axis, and \(U_{dr}\), \(U_{qr}\), \(i_{dr}\), \(i_{qr}\), \(\psi_{dr}\), \(\psi_{qr}\) denote the secondary voltage, current and flux linkage in \(d-q\) axis, respectively. Notice that the d-q axis is relative to the rotor. For the motor, the d-axis is the axis where the magnetic pole of the rotor is located while the q-axis is perpendicular to the d-axis and rotates 90 degrees along the d-axis anticlockwise. Besides, \(R_{s}\) is the primary resistance; \(\omega_{e}\) and \(\omega_{r}\) are the synchronous speed and secondary speed; \(p\) is the differential operator; \(L_{sl}\) and \(L_{rl}\) are the primary and secondary leakage inductance.  
By virtue of the indirect vector control (IVC) technique, the linear induction motor model is transformed into a DC motor model, which is shown as follows [8; 24].  
\[\begin{split}\psi_{qr}&=\psi_{qr}=0\\ U_{dr}&=U_{qr}=0\\ \end{split} \tag{4}\]  
Figure 1: Structure of linear induction motor (LIM).  
Then the LIM mathematical model can be described as [7; 10]  
\[\dot{i}_{ds} =-\frac{R_{s}}{L(Q)}i_{ds}+\frac{U_{ds}}{L(Q)}+\omega_{e}i_{qs} \tag{5}\] \[\dot{i}_{qs} =-\omega_{e}\left[i_{ds}+\frac{L_{m}\left(1-f(Q)\right)}{L(Q)\left( L_{r}-L_{m}f(Q)\right)}\psi_{dr}\right]\] \[\quad-\frac{R_{s}}{L(Q)}i_{qs}+\frac{U_{qs}}{L(Q)}\] (6) \[\psi_{dr} =\frac{L_{m}\left[1-f\left(Q\right)\right]}{1+\left[T_{r}-L_{m}f \left(Q\right)/R_{r}\right]p}i_{ds}\] (7) \[\omega_{sl} =\omega_{e}-\omega_{r}=\frac{L_{m}\left[1-f(Q)\right)}{T_{r}-L_{ m}f(Q)/R_{r}}\frac{i_{qs}}{\psi_{dr}}\] (8) \[F_{e} =K_{r}i_{qs}=M\cdot\vartheta+D\cdot v+F_{L} \tag{9}\]  
where \(T_{r}=L_{r}/R_{r}\); \(K_{T}=\frac{3}{2}P\frac{\pi}{R}\frac{L_{m}\left(1-f(Q)\right)}{L_{r}-L_{m}f( Q)}\psi_{dr}\); \(L(Q)=L_{s}-L_{m}f(Q)-\frac{\left|L_{m}\left(1-f(Q)\right)\right|^{2}}{L_{r}-L_{ m}f(Q)}\); \(\omega_{sl}\) denotes the slip frequency; \(D\) denotes the viscous friction coefficient; \(M\) denotes the total mass of mover; \(P\) denotes the number of pole pairs and \(h\) denotes the number of pole pitch; \(F_{L}\) denotes the external force disturbance.  
According to (9), we can get the accelerated velocity of LIM as:  
\[\vartheta=\frac{K_{T}}{M}i_{qs}+\Phi\upsilon+F \tag{10}\]  
where \(\Phi=-D/M\), \(F=-F_{L}/M\).