## 5 Simulations and Experimental Results  
### Comparative Simulations  
To validate the effectiveness of the proposed discrete control strategy in Section 4, comparative simulations were implemented on the LEMA with the prototype parameters given in Table 1 by MATLAB/Simulink. The sampling frequency is set to 10 kHz. First, in order to verify the control effect of the current loop, the target current was set as \(I_{v}=5\sin(100\pi t)\). Due to the influence of temperature variation, the coil resistance \(R\) may be different from the nominal values. Here, suppose that the actual resistance of the LEMA has a random error of \(\pm 20\%\) compared to the nominal one. To highlight the importance of the reduced-order ESO, the controllers with and without the reduced-order ESO are tested. As shown in Figure 6, the simulation results demonstrate that the control law of the current loop is effective, and the reduced-order ESO based controller can effectively restrain the influence of parameter variations in the current subsystem. Figure 6d shows the tracking error of three control cases. Under nominal model conditions, the maximum tracking error is about 5%. With the help of the reduced-order ESO, the maximum tracking error can still be controlled to around 5% in the case of parameter changes, but without the reduced-order ESO, the maximum tracking error can only be controlled to 15% in the case of parameter changes. Therefore, the proposed current loop controller that is based on reduced-order ESO, is insensitive to the parameter variation in the electrical subsystem. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Moving mass & \(m\) & 0.15 & kg \\ Active stroke & - & 18 & mm \\ Resistance & \(R\) & 0.68 & \(\Omega\) \\ Inductance & \(L\) & 0.89 & mH \\ Actuator coefficient & \(k_{e}\) & 15.8 & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the linear electromagnetic actuators (LEMA). Next, in order to verify the control effect of the position loop, the target position is set as 9 mm. Figure 7 shows the position simulation results of the proposed controller with the same parameter, \(\omega_{n}=300\) and the different parameter \(\omega_{c}\), and Figure 8 shows the results with the same parameter \(\omega_{c}=100\) and the different parameter \(\omega_{n}\). Note that the settling time of the position mainly depends on the parameter \(\omega_{n}\), and the margin of position error mainly depends on the parameter \(\omega_{c}\). Furthermore, the settling time and the margin of position error for the position control are almost unrelated, and can be designed independently. The controller with the model-assisted reduced-order ESO and that with the reduced-order ESO are compared, and the results of the position error are shown in Figure 9. Note that, with the help of the known model, the margin of position error is significantly reduced, and the convergence speed of position error is increased. Figure 6: Simulation results of the current loop: (**a**) \(R\) equals the nominal value, controller with reduced-order extended state observer (ESO); (**b**) \(R\) is 20% higher than the nominal value, controller with reduced-order ESO; (**c**) \(R\) is 20% higher than the nominal value, controller without reduced-order ESO; and (**d**) Current tracking error. To consider the effects of parameter uncertainties on the controller performance, suppose that the model parameters of the LEMA have the random error, which is described as follows:  
\[\left\{ \begin{array}{l}R=R_{0}+0.2R_{0}\lambda\\ L=L_{0}+0.02L_{0}\lambda\\ k_{e}=k_{e0}+0.1k_{e0}\lambda\\ m=m_{0}+0.02m_{0}\lambda\\ c=c_{0}+0.2c_{0}\lambda\end{array} \right. \tag{59}\]  
where \(R_{0},L_{0},k_{e0},m_{0},c_{0}\) are the nominal values of corresponding parameters, and \(\lambda\in[-1~{}1]\) is normally distributed random function. Figure 10 shows the robustness of the controller when the model parameters have random uncertainties.