# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 3 Analysis of the Stability and the Delay
### Analysis of the Delay
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Pump Speed** \\ **(RPM)** \\ \end{tabular} } & \begin{tabular}{c} **Opening Time of** \\ **Hydraulic Lock** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **The Rising Time** \\ **of Pressure** \\ **Difference (s)** \\ \end{tabular} & \begin{tabular}{c} **The Delay of the** \\ **(s)** \\ \end{tabular} & \
\begin{tabular}{c} **Speed of Piston-** \\ **Cylinder(mm/s)** \\ \end{tabular} &
\begin{tabular}{c} **Product of Speed** \\ **and Delay** \\ \end{tabular} \\ \hline
20 & 2.90/2.92 & 1.12/1.25 & 4.02/4.17 & 1.48/1.30 & 80.4/83.4 \\
40 & 1.28/1.39 & 0.46/0.35 & 1.80/1.74 & 3.64/5.19 & 72/69.6 \\
60 & 0.90/0.93 & 0.19/0.21 & 1.10/1.14 & 11.03/9.22 & 66/68.4 \\
100 & 0.48/0.58 & 0.09/0.12 & 0.64/0.70 & 18.13/17.36 & 64/70 \\
160 & 0.30/0.34 & 0.05/0.06 & 0.36/0.40 & 32.01/29.20 & 57.6/64 \\
200 & 0.22/0.31 & 0.06/0.05 & 0.28/0.36 & 41.31/38.83 & 56/72 \\
500 & 0.10/0.13 & 0.04/0.03 & 0.13/0.16 & 88.17/85.06 & 65/80 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The opening time of the hydraulic lock, the rising time of the pressure difference of the piston-cylinder, and the position delay (experiment/simulation). Figure 23: The simulation and the experimental results of the position output in the open-loop control at the pump speeds of 20, 40, 60, and 100 RPM. (**a**) the simulation results. (**b**) the experimental results. As shown in Figure 24, the opening time of the hydraulic lock is almost unaffected during spring-like loading. Hence, the delay of the position output of the piston-cylinder was almost unchanged. With the movement of the piston-cylinder, the pressure difference increased gradually. With the increase of the pressure difference, the movement speed of the piston-cylinder decreased gradually due to the leakage of the pump and the piston-cylinder. The spring-like loads will affect the dynamic characteristics of the position output, but it will not affect the delay. However, the damping-like load will affect the delay. That is because the piston-cylinder output hydraulic pressure must be greater than the damping (such as friction resistance) before it starts to move, while the spring-like load begins to increase after it starts to move. It is difficult to achieve a quantitative analysis of the delay in the frequency domain caused by the hydraulic lock. Let's define a factor number  
\[N=\int_{0}^{x}A_{amp}\sin(2\pi ft)dt \tag{30}\]  
where \(A_{amp}\) is the amplitude of the pump's speed, and \(f\) is the input frequency. When \(N=65\), as Table 3 shows, we solve for \(x\). The factor number \(N\) indicates the system's amplitude attenuation influenced by the hydraulic lock, and \(x\) indicates the actual delay. If \(x>\frac{1}{2f}\), the output is always 0 because the delay is bigger than the input frequency. The condition \(x>\frac{1}{2f}\) is the same as the condition that the factor number \(N\) is less than 65. When \(N=65\), \(x=\frac{1}{2f}\). Assume that  
\[\tau=\frac{1}{2f}-x \tag{31}\]  
as \(\tau\) grows larger, the influence of the hydraulic lock grows smaller. However, the minimum delay was 0.15 s when using the experimental equipment described in Chapter 2. That is because the maximum speed of the pump was 500 RPM, which means \(A_{amp,\max}=500\). This means that when the frequency is equal to 6.67 Hz, the hydraulic lock cannot be opened, which will also lead the output of the open-loop system to become 0. Figure 24: The simulation results of the position and the pressure when the pump speed is 20 RPM when the load is applied. At the same time, as shown in Figure 15 with its analysis, the spring-like load does not affect the delay. However, the damping-like load will affect the delay time. Therefore, when the damping-like load (such as friction) changes, the limit value of \(N\) is not 65.