# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 2 System Design and Optimization
### Electro-Hydraulic Servo Valve Design  
For shaking tables system, the servo valves need to have good dynamic and static performance and consistency, especially for the redundant-driven shaking table. The performance of the commercially available servo valve is fixed and it often has poor consistency, which is the main reason causing the differing performances of valve-controlled hydraulic cylinders. To solve this problem, the three-stage electro-hydraulic servo valve [37] with parameter-tunable valve controller is developed, as shown in Figure 3. The servo valve is mainly composed of four parts: pilot valve, built-in valve controller, main valve and built-in linear variable differential transformer (LVDT) to measure the displacement of the main valve. Also, the built-in valve controller has several tunable parameter gains, such as feedback gain, velocity gain and feed-forward gain, etc. to adjust the performance of the servo valve. The basic hydraulic and control schematic is shown in Figure 3c.  
Figure 3: The developed servo valve. (**a**) The three-stage servo valve; (**b**) The servo valve controller; (**c**) The schematic of the three-stage servo valve.  
For the horizontal servo valve, the maximum velocity is 0.5 m/s, so the maximum flow is  
\[Q_{x,y}=A_{x,y}^{d}v_{x,y}=\frac{\pi}{4}D^{2}v_{x,y}=462\;\mathrm{L/min} \tag{4}\]  
where, \(D\) is the diameter of the horizontal cylinder piston.  
Figure 4 shows the maximum capability curve of the shaking tables system, which indicates the relationship between frequency and maximum displacement, velocity and acceleration. The computed servo valve flow in Equation (4) is at the point where the maximum velocity and maximum force occur simultaneously. However, for the shaking table system, whose main force is the inertia force, the maximum acceleration and the maximum velocity cannot be reached at the same time; that is to say, the maximum force and maximum velocity cannot be reached simultaneously. For hydraulic power, the needed flow becomes smaller with the increased force needed and vice versa. So, the hydraulic power does not need the maximum flow 462 L/min at the maximum force moment. Considering Equation (4), we choose the optimal rated flow of horizontal servo valve as 400 L/min. The optimal load matching curves using the 400 L/min valve corresponding to the turning points of Figure 4 are shown in Figure 5a,b, which indicate that the chosen valve can meet the maximum flow requirement with certain allowance to ensure the system performance [38]. The rectangle around the load track curves in Figure 5 is the curve, whose four corner points are the assumptive load track points with the maximum force and maximum velocity simultaneously. Also, the four points provide reference during the design process of the hydraulic power. Besides, the horizontal servo valve is designed as asymmetric to match the asymmetric horizontal hydraulic cylinder.  
Figure 4: The maximum capability curve.  
Figure 5: _Cont._  
For the vertical servo valve, the design process is similar and the rated flow is chosen as 250 L/min. The optimal load matching curves using the 250 L/min valve at turning points of Figure 4 are shown in Figure 5c,d. Furthermore, the servo valve is designed as symmetric to match the symmetric vertical hydraulic cylinder.  
Furthermore, to ensure the consistency and performance, the designed valve controller has several tunable parameters, which contain the offset of pilot valve, the offset of main valve spool, displacement feedback gain, velocity feedback gain and feed-forward gain. By tuning these parameters, the servo valves can have approximately identical performances.