# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 3 Control Scheme Design
### Feed-Forward Inverse Control  
The designed inverse model controller as the feed-forward controller of the control system can improve the system control precision. However, the identified system model cannot accurately describe the actual transfer system due to the undetermined system model error with Equation (20), system nonlinearity and the RLS algorithm convergence accuracy, etc. From Equation (31), it can be seen that the designed inverse model is the only approximately inverse of the identified model. There are the system noise signals, the disturbance from the other shaking table and external disturbance, which can also affect the tracking control precision of the shaking table system.  
The feed-forward inverse model controller designed in the former section is applied to the control system considering various system errors and disturbance factors, as shown in Figure 10. The \(\Delta G_{d}(s)\) in Figure 10 is the model error between the real system model and the identified system model, \(u_{d}(t)\) is the cross-coupled control disturbance and \(d(t)\) is the compound disturbance, which contains the disturbance due to the specimen from the other shaking table and external disturbance source.  
Figure 10: The proposed control scheme for dual-shaking table system.  
We define the relationship between the real system model \(G_{d}(s)\) and the identified system model \(\hat{G}_{d}(s)\) as  
\[G_{d}(s)=\hat{G}_{d}(s)\left[\mathbb{1}+\Delta G_{d}(s)\right] \tag{32}\]  
From Equation (31), it can be seen that the designed inverse model \(\hat{G}_{d}^{-1}(s)\) is only approximate to the real inverse model of the identified model \(\hat{G}_{d}(s)\). We define the designed system model error as \(\Delta\hat{G}_{d}(s)\) and the relationship of \(\Delta\hat{G}_{d}(s)\), \(\hat{G}_{d}^{-1}(s)\) and \(\hat{G}_{d}(s)\) is defined as  
\[G_{s}(s)=\hat{G}_{d}(s)\hat{G}_{d}^{-1}(s)=1+\Delta\hat{G}_{d}(s) \tag{33}\]  
Combining Equations (32) and (33) and Figure 10, it can be obtained that  
\[\begin{array}{ll}y(t)&=\hat{G}_{d}^{-1}(s)\hat{G}_{d}(s)\left[\mathbb{1}+ \Delta G_{d}(s)\right]r(t)+\hat{G}_{d}(s)\left[\mathbb{1}+\Delta G_{d}(s) \right]u_{d}(t)+d(t)\\ &=\left[\mathbb{1}+\Delta\hat{G}_{d}(s)\right]\left[\mathbb{1}+\Delta G_{d}(s) \right]r(t)+\hat{G}_{d}(s)\left[\mathbb{1}+\Delta G_{d}(s)\right]u_{d}(t)+d(t )\end{array} \tag{34}\]  
According to Equation (34) and Figure 10, the control system error \(e(t)\) is  
\[\begin{array}{ll}e(t)&=r(t)-y(t)\\ &=-\left[\Delta\hat{G}_{d}(s)+\Delta G_{d}(s)+\Delta\hat{G}_{d}(s)\Delta G_{d }(s)\right]r(t)-\hat{G}_{d}(s)\left[\mathbb{1}+\Delta G_{d}(s)\right]u_{d}(t)- d(t)\end{array} \tag{35}\]  
When the identified system model error \(\Delta\hat{G}_{d}(s)\) and the designed system inverse model error \(\Delta\hat{G}_{d}(s)\) are both zeros, that is to say, there is no identified and designed errors with the dual-shaking table system model, then the Equation (35) becomes  
\[e(t)=\hat{G}_{d}(s)u_{d}(t)+d(t) \tag{36}\]  
From Equation (35), it can be seen that due to the cross-coupled effect of the two shaking tables, identified model error, the designed inverse model error and the disturbance action, the control precision of the system with feed-forward inverse controller compensation is affected. Equation (36) demonstrates that the system control error still exists even if the identified system model and designed system inverse model are completely accurate.