# A High-Precision Active Vibration Isolation Control System: Experimental Study
## 3 Controller Design
### Design of Robust Feedback Control  
The control input \(\mathbf{U}_{\mathbf{k}}\) of the totally decoupled system \([G_{ii}(s)]\) is designed in this section. In this case, the feedforward compensator compensates for the mutual interferences, but at the same time, together with the other disturbance \(\mathbf{F}_{\mathbf{t}}\) and ground vibration \(\mathbf{X}_{\mathbf{b}}\), acts as an uncertainty factor affecting the decoupled system. As mentioned, the mixed-sensitivity \(H_{\infty}\) control is applied due to its efficiency and effectiveness in dealing with uncertainties in a high-order system. Figure 2 shows the configuration of the mixed-sensitivity \(H_{\infty}\) control for the 6-DoF AVIS and Equation (8) shows the corresponding feedback control law.  
\[\mathbf{U}_{\mathbf{k}}(s)=-\mathbf{K}(s)\mathbf{Y}(s) \tag{8}\]  
where the controller matrix \(\mathbf{K}(s)\) is obtained in the form of Equation (9):  
\[\mathbf{K}(s)=diag\big{\{}K_{x}(s),K_{y}(s),K_{z}(s),K_{\theta x}(s),K_{\theta y}( s),K_{\theta z}(s)\big{\}} \tag{9}\]  
in which \(K_{j}\), \(j=x,y,z,\theta_{x},\theta_{y},\theta_{z}\), are the controllers of the interference-free system. The mixed-sensitivity problem can be explained as finding the rational function controller \(\mathbf{K}(s)\) ensuring the stabilization of the closed-loop system and minimizing its \(H_{\infty}\) norm defined by [20; 21]:  
\[\left\|\begin{array}{c}\mathbf{W}_{1}(s)\mathbf{K}(s)\mathbf{T}(s)\\ \mathbf{W}_{2}(s)\mathbf{S}(s)\\ \mathbf{W}_{3}(s)\mathbf{T}(s)\\ \end{array}\right\|_{\infty} \tag{10}\]  
In Equation (10), \(\mathbf{S}(s)=\left(\mathbf{I}_{6\times 6}+[G_{ii}(s)]\mathbf{K}(s)\right)^{-1}\) is the sensitivity function and the complementary sensitivity function is given by \(\mathbf{T}(s)=\begin{bmatrix}G_{ii}(s)\big{\rvert}\mathbf{(I}_{6\times 6}+[G_{ii}(s)]\mathbf{K}(s) )^{-1}\end{bmatrix}\)[20; 21]. \(\mathbf{W}_{1}(s),\mathbf{W}_{2}(s)\), and \(\mathbf{W}_{3}(s)\) are weighting functions of the 6-DoF AVIS's mixed-sensitivity  
Figure 2: Schematic drawing of the proposed control with fictitious weighted outputs.  
\(H_{\infty}\) control. Those frequency-dependent weighting functions must be proper and analytic. They are given in the form of  
\[\mathbf{W}_{i}(s)=diag\big{\{}W_{ix}(s),W_{iy}(s),W_{iz}(s),W_{i\theta x}(s),W_{i \theta y}(s),W_{i\theta z}(s)\big{\}} \tag{11}\]  
in which \(i=1\sim 3\).  
As discussed, using the feedforward compensators \(\big{[}G_{ii}(s)^{-1}G_{in}(s)\big{]}\mathbf{U}_{\text{p}}(s)\), mutual interferences among the AVIS's DoF are ruled out. However, although the direct disturbance compensators help overcome the mutual interferences, those compensators also contribute to the disturbances affecting the system. Hence, the weighting function \(\mathbf{W}_{1}(s)\) is chosen to correspond to the control input \(\mathbf{U}_{\mathbf{k}}(s)\). The total disturbances are suppressed by the weighting function \(\mathbf{W}_{2}(s)\). The weighting function \(\mathbf{W}_{3}(s)\) is chosen to correspond to the output of the system. The weighting functions are selected based on the system's characteristics and the designer's experience, and they must exceed the corresponding system's maximum singular value. Finally, the control law of the 6-DoF AVIS is obtained from (6) and (8) as follows:  
\[\mathbf{U}_{\text{p}}(s)=-\mathbf{K}(s)\mathbf{Y}(s)-\Big{[}G_{ii}(s)^{-1}G_{ij}(s)\Big{]} \mathbf{U}_{\text{p}}(s) \tag{12}\]