# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 3 Analysis of the Stability and the Delay
### Analysis of the Delay
The delay is approximately 3 s. Figure 19 shows that the pressure at the input of the piston-cylinder also started to increase at 3 s.  
At approximately 4.2 s, the pressure at the input of the piston-cylinder reached 0.6 bar, and the piston-cylinder started to move. At the same time, the input pressure of the hydraulic lock \(P_{inA}\) reached 4.6 bar. It satisfied the condition that Equation (19) expresses. As the flow coefficient \(C_{q}\) is a function of the Reynolds number Re, and Re is related to the velocity and the hydraulic diameter, so Re and \(C_{q}\) are different at different pump speeds. Figure 21 shows flow coefficients and Figure 22 shows the Reynolds numbers of the hydraulic lock at pump speeds of 20 RPM, 40 RPM, 100 RPM, 160 RPM, 200 RPM, and 500 RPM. As shown in Figure 22, the Reynolds number is larger at a higher speed. The maximum Reynolds number is 450. When the piston-cylinder started to move, the opening area and the flow rate of the hydraulic lock remained stable, and the Reynolds number also remained stable. At 160 RPM and above, the Reynolds number suddenly decreased and it soon remained stable because the flow rate decreased due to the movement of the piston-cylinder to the limit position. When the piston-cylinder reached the limit position, there was still a low flow rate due to leakage, so the Reynolds number will not be 0. Figure 21: The flow coefficients of the hydraulic lock. Figure 22: The Reynolds numbers of the hydraulic lock. Figure 23 shows the simulation and the experimental results of the position output in the open-loop control at a different rotational speed. It can be seen from Figure 23 that when the pump speed was low to 20 RPM, the position started to change at 4.2 s. When the pump speed was 40 RPM, 60 RPM, or 100 RPM, the position started to change at 1.7 s, 1.2 s, and 0.7 s, respectively. The larger the flow rate, the faster the pressure change, which resulted in a lower delay. The delay of the simulation and the experiment can nearly agree. Table 4 shows more details about the opening time of the hydraulic lock, the rising time of the pressure difference of the piston-cylinder, and the position delay. The detailed data comparison between the experiment and the simulation is also shown in Table 4.  
Table 4 indicates that the sum of the opening time of the hydraulic lock and the rising time of the pressure difference was equal to the delay of the position output. The results conform to the analysis of the delay. The hydraulic lock started to open when Equation (21) was satisfied. The product of the pump speed and the delay of position was approximately 65 to 85. This product was equivalent to the integral of flow over time. The difference of the product at different pump speeds may have been due to leakage, which needs to be studied in the future. At the same time, Table 3 also shows that the experimental results are consistent with the simulation results, and the delay length of the two is almost the same. Figure 24 shows the position, inlet pressure, and outlet pressure of the piston-cylinder under spring-like loads at 20 RPM.