# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 3 Analysis of the Stability and the Delay
### Analysis of Stability
Due to the small proportional coefficient, which leads \(N(\circ)=0\), the oscillation of the system was also suppressed. Figure 14: The result of the position in closed-loop step control with the large proportional coefficient. (**a**) the proportional coefficient is 160. (**b**) the proportional coefficient is 120. Figure 15: The results of the position in the closed-loop step simulation and the experiment with the different proportional coefficients. The BHL-EHA system comprises the proportional component, inertial component, delay component, and integral component. As the inertial component was the motor with a closed-loop speed controller, the time parameter of the inertial component was small, which is shown in Figure 4. So, in the low-frequency range, the open-loop frequency characteristics of BHL-EHA were mainly affected by the proportion link, the delay component, and the integral link. However, the open-loop performance of the BHL-EHA system shown in Figure 16 was low. This was due to the maximum speed of the pump being limited to 500 RPM, so the performance of the BHL-EHA system was low. Figure 16: The position output in the open-loop sinusoidal simulation. **(a)** 1 Hz, 500 RPM. **(b)** 1 Hz, 250 RPM. **(c)** 0.5 Hz, 500 RPM. **(d)** 0.5 Hz, 250 RPM. Figure 16 is the open-loop sinusoidal simulation and the experimental results, and in the open-loop control, the input was the speed of the motor and the output was the position of the piston. The figure shows that the simulation results are consistent with the experimental results, which further shows the accuracy of the simulation model. There was a delay when the speed of the motor was 0, that is when the phase of sinusoidal speed was 0 or \(-180^{\circ}\). It can be seen in the figure that the output of the piston position is a cosine wave, which means that the piston-cylinder was an integral component and the integral link had the most influence. When the frequency changed, the phase lag hardly changed. The phase lag was approximately \(-90^{\circ}\), which was the same as the phase lag of the integral link. The delay did not affect the phase. Figure 17 shows the closed-loop simulation of the sinusoidal position output control with different proportional coefficients. The lower the frequency of the sinusoidal control signal, the better were the control effects. The higher the proportional coefficient, the better were the control results. On the one hand, increasing the proportional coefficient could increase the cut-off frequency of the system. On the other hand, when the motor commutated, the delay influenced the sinusoidal position output control. The delay was smaller when the proportional coefficient was larger at the same error. Figure 18a shows the simulation results with a proportional coefficient of 20 and Figure 18b with a proportional coefficient of 10. Figure 18c shows the experimental results with a proportional coefficient of 20 and Figure 18d with a proportional coefficient of 10. It can be seen from Figure 18 that the experimental results are consistent with the simulation results. Figure 18a,c show that when the frequency of the input command was 0.63 Hz, the amplitude of output displacement was approximately 40 mm, which was approximately \(\frac{1}{\sqrt{2}}\) of the input that was 60 mm. Moreover, when the proportional coefficient was 10, the closed-loop system's gain margin was 0.25 Hz, which can be seen in Figure 18b,d. Figure 17: The result of the position in the closed-loop sinusoidal simulation with the different proportional coefficients. It indicates that the cut-off frequency of the closed-loop system with a proportional coefficient of 20 was 0.6 Hz, while the cut-off frequency was 0.25 Hz when the proportional coefficient was 10. Table 3 shows the cut-off frequency of the closed-loop system in different proportional coefficients by simulation. It can be seen that with the increase of the proportional coefficient, the cut-off frequency increased.