# High-Performance Flux Tracking Controller for Reluctance Actuator
## 4 Simulation Analysis
### Model Identification  
According to the Hammerstein structure in Figure 3, which was determined in Chapter 2. Owing to the intermediate variable w(t) not being measurable, the model identification was difficult. In application, rate-dependent characteristics of the linear reluctance actuator, which is caused by the eddy current effect, are in Figure 7.  
**Remark 2**.: _It is worth noting that the hysteresis between the input current and output flux in the reluctance actuator is not only the B-H hysteresis loop of ferromagnetic materials but also closely related to the structure of a linear reluctance motor (such as the flux length). This gives rise to the hysteresis loops shown in Figure 7, which seem larger than is normal for actuators made with soft magnetic steels._  
Theoretically, when the input frequency is low enough, there is no eddy current effect, and it can be considered that it is rate-independent. Thus, at low frequencies, the static nonlinear and the dynamic linear of the Hammerstein model can be separated. Therefore, in this paper, the separation identification method is used to identify the parameters of the Hammerstein structure.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sign** & **Physical Meaning** & **Value** \\ \hline A & The cross-sectional area of teeth & 960 (mm\({}^{2}\)) \\ N & Coil turns & 600 \\ i & current & See paper \\ g & Air gap & 2 mm \\ Material & Material & CoFe \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of reluctance actuator.  
Figure 6: Structure diagram of linear reluctance actuator.  
Figure 7: Rate-dependent hysteresis.  
To realize the identification of Hammerstein model parameters, the frequency turning point, which is the frequency inflection point between rate-independent and rate-dependent in the reluctance motor model, was determined by COMSOL first. The result is shown in Figure 8.  
From Figure 8, we can draw a conclusion that the hysteretic is rate-independent when the reference input signal is lower than 0.1 Hz. Thus, the parameter identification step of the linear reluctance actuator model is shown as follows:  
Step 1: A sinusoidal signal i(t) with an excitation frequency not exceeding 0.1 Hz is used to excite the reluctance actuator. Next, the output flux B(t) of the reluctance motor is obtained by the sense coil.  
Step 2: From the measurement data (i(t), B(t)), which is obtained by Step 1, the Bouc-Wen model was identified.  
Step 3: A sinusoidal scanning signal, with a frequency range of [0 100] Hz, was used to excite the Bouc-Wen model and reluctance motor, respectively. Next, the output data of the Bouc-Wen model was acquired, that is, the internal state variable x(t), as well as the reluctance motor output B(t) were acquired. Eventually, by dataset (x(t), B(t)), the linear part G(z) was identified using the least squares method. The process is shown in Figure 9.  
The accuracy of modeling is expressed by root mean square error (RMSE) and relative error (RE), which are defined as follows:  
\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{N}{(\hat{y}(i)-y(i))^{2}}}{N}} \tag{13}\]  
Figure 8: Hysteresis at low frequency.  
Figure 9: ARX model identification principle.  
\[RE=\sqrt{\frac{\sum\limits_{i=1}^{N}{{{({\hat{y}}(i)-y(i))}^{2}}}}}{\sum\limits_{i= 1}^{N}{{{(y(i))}^{2}}}} \tag{14}\]  
Here, \(\hat{y}(i)\) denotes the output of the model and \(y(i)\) is the output of the reluctance actuator.