# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 2 Materials and Methods
### Model of the Motor System  
Due to the nonlinear factors of the motor, the linear model in the form of the transfer function is not sufficient to accurately describe the characteristics of the system. Thus, we use the MATLAB/SIMULINK interface that provides various nonlinear functions for modeling. The MATLAB model of the permanent magnet synchronous motor (PMSM) is established. Figure 2 shows the structure diagram of the PMSM system.  
As shown in Figure 2, the voltage and the current of the PMSM can be transformed from the ABC coordinates to the d-q coordinates by using the Clarke transformation and the Park transformation. The encoder obtains the rotor position \(\theta_{e}\) for transformation calculation. The double PI controller is used to adjust the rotational speed. A PI speed controller is adopted in the speed loop, and a vector controller is used in the current loop. In the current loop, the Id = 0 method is used [35]. The voltage in the d-q coordinates can be transformed to \(\alpha-\beta\) coordinates through the Park inverse transformation, generating the control signal to control the SVPWM controller. The SVPWM controller generates the SVPWM signal by using a three-phase inverter (FSTPI). In the d-q coordinates, the relationships of the motor voltage, current, and speed are as follows [34]:  
\[u_{d}=R_{s}i_{d}+L_{d}\frac{di_{d}}{dt}-\omega_{e}L_{q}i_{q}, \tag{1}\]  
Figure 1: Schematic structure of BHL-EHA.  
Figure 2: Structure diagram of PMSM.  
\[u_{q}=R_{s}i_{q}+L_{q}\frac{di_{q}}{dt}-\omega_{e}L_{d}i_{d}+\omega_{e}\Psi_{f}, \tag{2}\]  
where \(u_{d}\), \(u_{q}\), \(i_{d}\), \(i_{q}\) is the voltage (V) and the current (A) of the motor under q-axis and d-axis, respectively. The inductance of the q-axis and the d-axis are \(L_{d}\) and \(L_{q}\). The electromagnetic angular velocity is \(\omega_{e}\).  
The \(i_{d}=0\)[35], and the electromagnetic torque is:  
\[T_{e}=n\Psi_{f}i_{q}, \tag{3}\]  
and the angular velocity is:  
\[\frac{d}{dt}\omega_{m}=\frac{1}{J}(T_{e}-T_{f}-F\omega_{m}-T_{m}) \tag{4}\]  
\[\frac{d\theta}{dt}=\omega_{m} \tag{5}\]  
according to Equations (1)-(5), the model of PMSM is as shown in Figure 3. A PMSM component in SIMULINK is used. The Universal Bridge1 is the FSTPL. The Tm in Figure 3 is the torque of the pump, and the constant 500 in the figure is the rotational speed that is used to test the model.  
The PMSM's parameters is shown in Table 1. The parameters of the PMSM are set according to the PMSM that will be used in the experimental facility. The rated speed affects the working limit of the motor. Since the delay of the BHL-EHA system is particularly obvious at low motor speed, the maximum speed of the motor is limited to 500 RPM to fully display this phenomenon. Figure 4 shows the result of the motor's simulation and the experiment.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Quantity/Unit** \\ \hline Stator resistance & 0.045 \(\Omega\) \\ Moment coefficient & 0.363 N-m/A \\ Rotor flux & 0.0605 V-S \\ Nominal voltage & 300 V \\ L\({}_{\rm ss}\) inductance & 3.9 mH \\ Moment of inertia & 0.672 \(\times\) 10\({}^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ Number of poles & 2 \\ Friction coefficient & 0 \\ Rated speed & 500 RPM \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of PMSM.  
Figure 3: Open-loop model of PMSM.  
As shown in Figure 4, in the experiment, the overshoot is approximately 150 RPM, while in the simulation it is approximately 100 RPM. The setting time is approximately 0.10 s in the experiment, while in the simulation it is 0.08 s. It indicates that the simulation can reflect the experimental facility's characteristics, and the control method of the experiment can achieve a good result.