# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 3 Control Scheme Design
### Synchronization Controller Design  
To obtain a satisfactory synchronization control precision of dual-shaking table system, this paper proposes a novel ASMC based on adaptive reaching law to cross-coupling control strategy, as is shown in Figure 9. The attitude DOF control signal is compensated by the synchronization control signal, which is processed by the proposed ASMC synchronization controller according to the attitude feedback error of two shaking tables. For convenience, we define the left shaking table in Figure 1b as 1# shaking table and the right one as 2# shaking table, and use subscript \(i\) to represent the \(i\)th table in the following parts, \(i\) = 1,2. The attitude tracking error, velocity tracking error and acceleration tracking error of each shaking table can be defined as  
\[ \begin{array}{l}\mathbf{q}_{ie}=\mathbf{q}_{ir}-\mathbf{q}_{i}\\ \tilde{\mathbf{q}}_{ie}=\tilde{\mathbf{q}}_{ir}-\tilde{\mathbf{q}}_{i}\\ \tilde{\mathbf{q}}_{ie}=\tilde{\mathbf{q}}_{ir}-\tilde{\mathbf{q}}_{i}\end{array}  \tag{14}\]  
where \(\mathbf{q}_{ir}\) is the reference attitude (DOF) signal, \(\mathbf{q}_{i}\) is the feedback attitude signal and there is \(\mathbf{q}_{1r}=\mathbf{q}_{2r}=\mathbf{q}_{r}\). The attitude synchronization error, velocity synchronization error and acceleration synchronization error can be defined as  
\[ \begin{array}{l}\mathbf{q}_{ise}=(-1)^{i}(\mathbf{q}_{2}-\mathbf{q}_{1})\\ \hat{\mathbf{q}}_{ise}=(-1)^{i}(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{1})\\ \tilde{\mathbf{q}}_{ise}=(-1)^{i}(\tilde{\mathbf{q}}_{2}-\tilde{\mathbf{q}}_{1})\end{array}  \tag{15}\]The synchronization control sliding mode surface is designed as  
\[\mathbf{s}_{is}(t)=\left(\frac{d}{dt}+\lambda_{i}\right)^{2}\mathbf{q}_{ise}=\ddot{\mathbf{q} }_{ise}+\mathbf{c}_{2}\dot{\mathbf{q}}_{ise}+\mathbf{c}_{i1}\mathbf{q}_{ise} \tag{16}\]  
Using the exponential reaching law to design the control law can not only improve the dynamic quality of the reaching phase of the sliding mode, but also reduce the system chattering by choosing the appropriate reaching law parameters [46]. So, this paper applies the SMC based on exponential reaching law. The traditional exponential reaching law is  
\[\dot{\mathbf{s}}_{is}(t)=-\mathbf{k}\mathbf{s}_{is}-\beta sign(\mathbf{s}_{is}) \tag{17}\]  
where, \(\mathbf{k}_{s6\times 6}\) and \(\beta_{s6\times 6}\) are the diagonal matrixes with positive constants. This reaching law can be divided into two parts: \(\dot{\mathbf{s}}_{is}=-\mathbf{k}_{s}\mathbf{s}_{is}\) is the exponential reaching part, whose solution is \(\mathbf{s}_{is}=\mathbf{s}_{is}(0)e^{-ht}\), and \(\dot{\mathbf{s}}_{is}=-\beta_{s}sign(\mathbf{s}_{is})\) is the constant speed reaching part. As the coefficient matrixes \(\mathbf{k}_{s}\) and \(\beta_{s}\) are constant matrixes, the system does not have the ability of adaptive tuning. So, for different system state variables, the convergence performance cannot achieve the optimal effect. To solve this problem, this paper proposes an adaptive variable speed reaching law  
\[\dot{\mathbf{s}}_{is}(t)=-\mathbf{k}_{s}(1+c\,|\,\mathbf{s}_{is}\,|\,1^{1/\gamma})\mathbf{s}_{is }-\beta_{s}(1+\,|\,\mathbf{s}_{is}\,|\,1^{\gamma}/c)sign(\mathbf{s}_{is}) \tag{18}\]  
where, \(c\) and \(\gamma\) are positive constants, and \(0<\gamma<1\). The proposed reaching law can tune the reaching speed adaptively according to the distance between the system state variables and the sliding mode surface. When the system state is far away from the sliding mode surface, \(|\,\mathbf{s}_{is}\,|\,1\) is quite large and the reaching law can be approximated to \(\dot{\mathbf{s}}_{is}(t)=-\mathbf{k}_{s}(1+c\,|\,\mathbf{s}_{is}\,|\,1^{1/\gamma})\mathbf{s}_{is}\), which has a far higher speed compared with the traditional exponential reaching law and can largely reduce the reaching time.