# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 4 Delay Compensation Control
Comparing Figures 18 and 27, the delay compensation control method combined with feedforward control achieves excellent control results. When the sinusoidal input is applied, the output position tracks the input very well. The amplitude error is within 3% and the phase lag is 0\({}^{\circ}\). Compared with the method without feedforward, this method reduces the amplitude error by 29.8% and corrects the phase lag of 10\({}^{\circ}\). Compared with the normal PID controller, this method reduces the amplitude error by 46.3% and it corrects the phase lag of 15\({}^{\circ}\). Figure 26: The simulation and the experimental results of the system using the delay compensation method with a typical PI controller and with the EHA system without the hydraulic lock. (**a**) the simulation results. (**b**) the experimental results. It can also be seen from Figure 27e that the change of temperature and load has little effect on the delay compensation control method. Periodically changing load will cause a slight jitter of position output, but it will not affect the delay. According to Equations (32) and (33), the temperature change will change the effective bulk modulus \(\beta\), thus changing  
Figure 27: The simulation results in sine control using the delay compensation control method. (**a**) \(k_{f}=250\) with 0.63 Hz. (**b**) \(k_{f}=125\) with 0.25 Hz. (**c**) \(k_{f}=250\) with 0.63 Hz and 10 Hz sine load applied. (**d**) \(k_{f}=125\) with 0.25 Hz and 10 Hz sine load applied. (**e**) \(k_{f}=250\) with 0.63 Hz and temperature 0 \({}^{\circ}\)C. (**f**) \(k_{f}=250\) with 0.63 Hz and 5 MPa load applied. the delay \(t_{0}\). However, the \(t_{0}\) change caused by \(\beta\) is very small, so the control performance is almost not influenced, which ensures the robustness of the controller. Figure 27f shows the result of the delay compensation control method when 5 MPa load is applied. According to Equations (32) and (33), with \(n_{\text{max}}\) unchanged, the increase of the damping-like load will increase \(t_{0}\). As a result, the load increases the delay. In the positive half cycle of the sine command, the load direction is opposite to the command. In the negative half cycle of the sine command, the load direction is the same as the command. Therefore, the positive half cycle tracking performance is weakened, while the negative half cycle tracking performance is enhanced. However, compared with the result without the compensation control method (as shown in Figure 19, even without load), it can achieve a good control effect. However, to improve the control effect, the controller should be modified according to Equation (33) when the load changes  
\[G_{CN}(t)= \begin{cases}n_{\text{max}}&t\leq t_{0,1},\dot{e}>0\\ k_{f}&t>t_{0,1},\dot{e}>0\\ -k_{f}&t>t_{0,2},\dot{e}<0\\ -n_{\text{max}}&t\leq t_{0,2},\dot{e}<0\\ \end{cases}  \tag{35}\]  
where \(t_{0,1}\) is the delay time after adding the opposite load, calculated according to Equations (32) and (33) and \(t_{0,2}\) is the delay time after adding the aiding load. Figure 28 shows the control effect under a 5 MPa unidirectional load after the controller parameters are modified. By comparing Figure 28 with Figure 27f, although Figure 27f shows good robustness of the controller, a better control effect can be achieved in the positive half cycle of the sine command after controller parameter correction. Figure 29a,b show the experimental results using the delay compensation control method, with the same command of Figure 18a,b, respectively. In Figure 29a, the proportional coefficient is 20 and the \(k_{f}=250\), while in Figure 29b, the proportional coefficient is 10 and the \(k_{f}=125\). The experimental results show that the delay compensation control method can strengthen the tracking performance of the BHL-EHA system and completely compensate for the amplitude difference of sinusoidal control; the amplitude of the position output is even 3% higher than the command amplitude. Figure 28: The simulation result after the controller parameters are modified.