# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 3 Control Scheme Design
### Model Inverse Controller Design  
As is shown in Figure 9, in the proposed control scheme, the control of each shaking table is a displacement DOF close-loop system and the proposed ASMC is used to reduce the synchronization error of dual shaking tables through the cross-coupled compensation control. With synchronization control, the performances of the two shaking tables are very similar, and the coupling effect of the specimen on each shaking table control system due to the synchronization error is considered as disturbance. This section is aimed to improve the tracking performance of each shaking table displacement close-loop system through the outer loop controller using the proposed observer-based inverse model controller. So, for each shaking table, the tracking control scheme is the same. Thus, the following parts introduce the tracking control scheme for only one shaking table for simplicity, and it is applicable to the other one. In this section, the RLS algorithm is applied to identify the system close-loop transfer function, which will be used to design the system inverse model controller. The RLS algorithm is referred in Eweda [30]. We define that the discretization transfer function of 1# shaking table close-loop system \(G_{d}(s)\) is \(G_{d}(z)\), then the auto regressive moving model of \(G(z)\) can be described as  
\[A(z^{-1})y(t)=B(z^{-1})u(t-d)+\xi(t) \tag{20}\]  
where, \(y(t)\) is the system output sequences and there is \(A(z^{-1})=1+a_{1}z^{-1}+...+a_{n}z^{-n}\); \(u(t)\) is the system input sequences and there is \(B(z^{-1})=b_{0}+b_{1}z^{-1}+...+b_{m}z^{-m}\) and \(\xi(t)\) is a random white noise sequence with zero mean. Writing Equation (20) into least squares form gives  
\[y(t)=\hat{\varphi}(t)\hat{\theta}+\xi(t) \tag{21}\]  
where \(\varphi(t)=[-y(t-1),...,-y(t-n),u(t-d),...,-u(t-d-m)]\), \(\hat{\theta}=[a_{1},...,a_{n},b_{0},...,b_{m}]\). The noise signal \(\xi(t)\) as the estimator can be written as  
\[\hat{\xi}(t)=y(t)-\hat{y}(t)=y(t)-\hat{\varphi}^{T}(t)\hat{\theta}(t) \tag{22}\]  
where \(\hat{\theta}=[\hat{a}_{1},...,\hat{a}_{n},\hat{b}_{0},...,\hat{b}_{m}]\), \(\hat{\varphi}(t)=[-\hat{y}(t-1),...,-\hat{y}(t-n),\hat{u}(t-d),...,-\hat{u}(t -d-m)]\). To obtain system parameters \(\hat{\theta}\), the RLS algorithm employed can be put as [27]  
\[ \begin{array}{c}\hat{\theta}(t)=\hat{\theta}(t-1)+K(t)[y(t)-\hat{\varphi}^{ T}(t)\hat{\theta}(t-1)]\\ K(t)=\frac{P(t-1)\hat{\varphi}(t)}{1+\hat{\varphi}^{T}(t)P(t-1)\hat{\varphi}(t)} \\ P(t)=[I-K(t)\hat{\varphi}^{T}(t)]P(t-1)\end{array}  \tag{23}\]  
Ideally, the exact system inverse model can be obtained and allows to accurately track the system's desired trajectory. However, for the identified system model in Equation (20), there are always zeros outside the unit circle due to sampling holder, transport delay and sensors and actuators being physically non-collocated, etc. [47]. The direct inversion of the identified system model with zeros outside the unit circle is inappropriate and the system is unstable when the direct inversion is used as the feed-forward controller. To solve this problem, a stable approximate inverse model is designed instead of the exact inverse. The ZPETC proposed by [PERSON] [33] is a successfully and widely used method to design the inverse controller of the non-minimum phase system and is adopted to design the inverse controller of the identified dual-shaking table system model. Assuming that the identified system transfer function can be expressed as  
\[\hat{G}_{d}(z)=\frac{N_{+}(z)N_{-}(z)}{M(z)} \tag{24}\]  
where \(N_{+}(z)\) is factors with stable zeros, \(N_{-}(z)\) is factors with unstable zeros. The system approximate inverse model can be designed as  
\[\hat{G}_{d}^{-1}(z)=\frac{M(z)N_{-}(z^{-1})}{z^{p}N_{+}(z)N_{-}^{2}(1)} \tag{25}\]  
where \(p\) is the number of unstable zeros of the identified system transfer function.