# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 3 Analysis of the Stability and the Delay
### Analysis of Stability
So, in \(G(s)\), \(l=1,m=0,n=1\) the transfer function of the system is  
\[G_{f}(s)=\frac{N(\circ)kk_{p}s+N(\circ)kk_{i}}{T_{1}s^{3}+s^{2}+N(\circ)kk_{p} s+N(\circ)kk_{i}} \tag{24}\]  
and the characteristic Equations of the closed-loop system can be written as  
\[T_{1}s^{3}+s^{2}+N(\circ)kk_{p}s+N(\circ)kk_{i}=0 \tag{25}\]  
The Rouse criterion shows that the system is stable only if \(k_{p}>T_{1}k_{i}\). Due to the transfer function \(N(\circ)\), when \(P_{inA}\) or \(P_{inB}\) is not large enough, the output is 0, making the error not 0. The integral control will lead to the accumulation of errors. It may lead to instability and oscillation. To satisfy the Rouse criterion, and to avoid the accumulation of errors, let us assume \(k_{i}=0\), and the zero and the pole of the transfer function could cancel at \(s=0\), and the other pole is  
\[s_{1,2}=\frac{-1\pm\sqrt{1-4N(\circ)kk_{p}}}{-2T_{1}} \tag{26}\]  
if \(k_{p}\) is not big enough, the motor's rotational speed is low when the error is small, and the check valve opening is almost zero. This means \(N(\circ)=0\) and \(s_{1}=s_{2}=\frac{-1}{-2T_{1}}\), thus the system is stable and it will not oscillate. If \(k_{p}\) is too big, the rotation output is high even when the error is small. It means \(N(\circ)=1\) and \(1-4N(\circ)kk_{p}<0\), thus the system is stable but it will oscillate. Figure 14 shows the closed-loop step control simulation and the experimental result with the substantial coefficient proportional controller. In Figure 14a, the proportional coefficient is as large as 160, and the system has apparent oscillation. In Figure 14b, the proportional coefficient is as large as 120, but the oscillation of the system is eventually suppressed. The simulation and the experimental results show the same stability characteristics. Since the piston cylinder is an integral component, if there was no hydraulic lock and the proportional coefficient was appropriate, the steady-state error would be eliminated. The oscillation of the system is due to the larger proportional coefficient. Equation (29) shows that if \(N(\circ)\
eq 0\), the oscillation occurs, and if \(N(\circ)=0\), the oscillation suppresses. The pressure between the piloted check valve's input and output is essential to \(N(\circ)\). Due to leakage, when the pump speed is not high enough, \(N(\circ)=0\). When the proportional coefficient is small, the pump speed is low and \(N(\circ)=0\). So, the addition of the hydraulic lock reduces the oscillation of the system. Therefore, the appropriate proportional coefficient is essential. The steady-state error is required to be within 1 mm to meet the requirements of position control in MEA, and corresponding to a 1 mm error, the proportional coefficient is 20. When the pump speed is 20 RPM, the output displacement of the piston-cylinder can be ensured, although there is a delay of 4.5 s. If the pump speed is lower than 20 RPM, it cannot be guaranteed that there is a position output, which leads to a steady-state error of approximately 1 mm. Figure 15 shows the closed-loop simulation and the experimental results of step control of position when the proportional coefficient is 10 and 20. The results show that with the proportional coefficients 10 and 20, the delay was approximately 0.15 s, and the rising times were 1 s and 1.5 s, respectively. At the same time, the simulation and the experimental results are consistent in both the dynamic and the steady-state, and the error of the dynamic state was less than 3%, while the error of the steady-state was less than 0.5%. The proportional coefficient had a noticeable influence on the dynamic characteristics of step control. The delay was the same because the controller output saturation was 500 RPM. The steady-state error was approximately 1.2 mm with the proportional coefficient 10. When the proportional coefficient was 20, the steady-state error was suppressed, which satisfied the analysis above.