# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 4 Delay Compensation Control  
As Equations (32) and (33) show, once the hydraulic lock characteristics, pump characteristics, pipeline characteristics, and load are determined, there is a one-to-one correspondence between rotational speed and delay time. A delay compensation control method is used to minimize the delay. Figure 25 shows the transfer function diagram. The feedforward control method can also eliminate the delay and the damping of a system with the transfer function \(G_{1}(s)\cdot G_{2}(s)\). So, the delay compensation controller \(G_{CN}(t)\) can be designed. When \(e=0,\dot{e}\
eq 0\), \(G_{CN}(t)\) is enabled and let \(t=0\), then  
\[G_{CN}(t)= \begin{cases}n_{\max}&t\leq t_{0},\dot{e}>0\\ k_{f}&t>t_{0},\dot{e}>0\\ -k_{f}&t>t_{0},\dot{e}<0\\ -n_{\max}&t\leq t_{0},\dot{e}<0\end{cases}  \tag{34}\]  
where \(k_{f}\) is the feedforward values. When \(e=0,\dot{e}\
eq 0\), it means the operation mode changes from A to B. When the operation mode changes, the controller output commands \(n_{\max}\) to the EHA system to minimize the delay. Once the maximum speed \(n_{\max}\) is determined, the minimum delay \(t_{0}\) is determined according to Equations (32) and (33). In this case, the delay compensation controller \(G_{CN}(t)\) can be constructed according to \(n_{\max}\) and \(t_{0}\). Figure 26a shows the simulation results of the EHA system with a hydraulic lock using the delay compensation control method with \(k_{f}=0\), the EHA system with the hydraulic lock not using the delay compensation control method, and the EHA system without the hydraulic lock using the PI control method. Figure 26b shows the experimental results of the EHA system with the hydraulic lock using the delay compensation control method with \(k_{f}=0\), and the EHA system without the hydraulic lock using the PI control method. As the hydraulic lock is placed in the pump, the experimental results lack comparison with  
Figure 25: The transfer function diagram of the delay compensation method. the EHA system without the hydraulic lock. The input signal is a 15 mm amplitude, 40 mm mean level, and 0.6 Hz frequency sine wave. And the proportional and integral coefficients are \(k_{p}=20,k_{i}=0\). The result shows that using the delay compensation control method can reduce the influence of the hydraulic lock. In more detail, the delay occurs when the derivative of the position \(\dot{x}=0\). A delay compensation control command is outputted to ensure the delay reaches the minimum value of \(t_{0}\). In this paper, \(n_{\text{max}}=500\) and \(t_{0}=0.15\) s, which is analyzed in Section 3, and it can be obtained by Equations (32) and (33). It can be seen from Figure 26 that the position output amplitude of the system using the delay compensation control method is very closed to the system without a hydraulic lock, which in the simulation was approximately 10.5 mm and in the experiment it was approximately 10.08 mm. In the BHL-EHA system, when the position command is at the stagnation point of the sine wave, if the delay compensation control method is not used, there will be a delay of approximately 0.35 s, which resulted in the amplitude of only 8 mm in the simulation and only 7.6 mm in the experiment. After using the controller, the amplitude error of the system was reduced from 49.3% to 32.8%. The phase lag was approximately 5\({}^{\circ}\) compared with the system that uses the delay compensation control method in the experiment. Figure 27a,b show the simulation results using the delay compensation control method, with the same command of Figures 18a and 18b, respectively. In Figure 27a, \(k_{f}=250\) and in Figure 27b, \(k_{f}=125\). Figure 27c,d show the results when the sine load is applied, and the load is 10 Hz, 20 bar. Figure 27e shows the result of using the same command and control parameters as (a) when the temperature is 0 \({}^{\circ}\)C. Figure 27e shows the result of using the same command and control parameters as Figure 27a when the load of 5 MPa is applied.