# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 3 Control Scheme Design
### Observer-Based Inverse Control Compensation Scheme  
The disturbance observer as the most effective compensation control method has been widely used in various applications, which have disturbance, parameter uncertainty, noise signals and system nonlinearity, etc., and has a satisfactory control effect [36; 48]. Thus, in this paper, the disturbance observer cooperating with the feed-forward inverse controller is adopted to the dual-shaking table control system to restrain the system disturbance to obtain good tracking performance, as is shown in Figure 11.  
Figure 11: The feed-forward inverse control combined with the disturbance observer. From Figure 11, it can be seen that the disturbance observer needs to use the close-loop control system inverse model. Also, the cross-coupled control action \(u_{d}(t)\) needs to be separated from the total control signal to avoid the effect on tracking precision. Based on the analysis, this paper proposes a novel simple disturbance observer without the system inverse model and the control \(u_{d}(t)\) also does not need to be separated, as is shown in Figure 12. The proposed control scheme has the same control performance with the original control scheme while simplifying the control system and the control effect is derived as follows. The \(Q(s)\) in Figures 11 and 12 is the low-pass filter to ensure stability and to filter high-frequency noise signals for the control system. The design of \(f_{q}\) not only depends on the frequency range of noise signals but also depends on the robust stability of the control system. Generally, the bigger the frequency width \(f_{q}\), the better the disturbance restraining effect and the worse it is for the system stability. Above all, the design of \(f_{q}\) should make a trade-off between the robust stability, tracking performance and restraining disturbance. It can be designed as [48]  
\[Q(s)=\frac{\frac{1}{2\pi f_{q}}s+1}{\left(2\pi f_{q}\right)^{3}}+\frac{3}{ \left(2\pi f_{q}\right)^{2}}s^{2}+\frac{3}{2\pi f_{q}}s+1 \tag{37}\]  
Combining Equations (32) and (33), the system outputs derived from Figure 11 and from Figure 12 are completely identical and can be written as  
\[ \begin{array}{ll}y^{\prime}(t)&=\frac{\left[1+\Delta\tilde{G}_{d}(s)\right] \left[1+\Delta G_{d}(s)\right]\left[r(t)-Q(s)n(t)\right]}{1+Q(s)\left[\Delta G _{d}(s)+\Delta G_{d}(s)+\Delta G_{d}(s)\right]\Delta G_{d}(s)}\\ &+\frac{\left[1-Q(s)\right]G_{d}(s)\left[1+\Delta G_{d}(s)\right]u_{d}(t)+ \left[1-Q(s)\right]d(t)}{1+Q(s)\left[\Delta G_{d}(s)+\Delta G_{d}(s)+\Delta G_ {d}(s)\right]\Delta G_{d}(s)}\\ \end{array}  \tag{38}\]  
Combining Equation (38), the control system errors from the two control scheme of Figures 11 and 12 are completely identical and can be described as  
\[ \begin{array}{ll}e^{\prime}(t)&=r(t)-y^{\prime}(t)\\ &=\frac{\left[Q(s)-1\right]\left[\Delta\tilde{G}_{d}(s)+\Delta G_{d}(s)+ \Delta\tilde{G}_{d}(s)\right]\chi(t)}{1+Q(s)\left[\Delta G_{d}(s)+\Delta G_{d} (s)+\Delta\tilde{G}_{d}(s)\right]\chi(s)}\\ &+\frac{\left[Q(s)-1\right]G_{d}(s)\left[1+\Delta G_{d}(s)\right]u_{d}(t)+ \left[Q(s)-1\right]d(t)}{1+Q(s)\left[\Delta G_{d}(s)+\Delta G_{d}(s)+\Delta G_ {d}(s)\right]}\\ &+\frac{Q(s)\left[1+\Delta G_{d}(s)\right]\left[1+\Delta G_{d}(s)\right]u(t)}{ 1+Q(s)\left[\Delta G_{d}(s)+\Delta G_{d}(s)+\Delta G_{d}(s)\right]}\\ \end{array}  \tag{39}\]  
From the derived results Equations (38) and (39), it can be demonstrated that the control scheme in Figure 12 has the same control performance with that in Figure 11. Generally, the system noise signals are always high-frequency signals, so the noise signals filtered by the low-pass filter \(Q(s)\) have little effect on the system control precision and can be ignored.