# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 3 Noninteracting Control Design
### The LMI-Based Design of the Feedback Matrix \(\mathbf{G}\)  
Since several elements of the state feedback matrix are neglected in order to identify the decoupled control law, the system performance may be affected. The matrix \(\mathbf{G}\) from Equation (17) should be modified to achieve the desired performances and its modified expression is given by:  
\[\mathbf{G}=\mathbf{\Delta}^{-1}\mathbf{\overline{G}} \tag{26}\]  
with \(\mathbf{\overline{G}}\), a diagonal matrix, chosen so that the decoupling characteristics are preserved. Substituting \(\mathbf{F}\) (Equation (18)) and \(\mathbf{G}\) (Equation (26)) into the system model (Equation (11)), the closed-loop system is obtained as follows:  
\[\begin{array}{l}\dot{\mathbf{\bar{X}}}=\left[\begin{array}{l}\dot{\mathbf{ \bar{X}}}\\ \dot{\mathbf{\bar{v}}}\end{array}\right]=\underbrace{\left[\begin{array}{l} \mathbf{A}-\mathbf{B}\bigg{(}\mathbf{\Delta}^{-1}\mathbf{A}^{*}+\mathbf{F} \mathbf{S}_{a}\bigg{)}&\mathbf{O}_{6}\\ -\mathbf{C}&\mathbf{O}_{6}\end{array}\right]}_{\mathbf{\bar{X}}}+\underbrace{ \left[\begin{array}{l}\mathbf{B}\mathbf{\Delta}^{-1}\\ \mathbf{O}_{6}\end{array}\right]}_{\mathbf{\bar{B}}}\underbrace{\left[ \begin{array}{l}\mathbf{O}_{12\times 6}&\mathbf{G}\\ \mathbf{\bar{G}}\end{array}\right]}_{\mathbf{\bar{K}}}+\underbrace{\left[ \begin{array}{l}\mathbf{D}_{\mathbf{\bar{r}}}\\ \mathbf{r}\end{array}\right]}_{d}\end{array} \tag{27}\]  
If there exists a \(\mathbf{Z}\in\mathbf{R}^{6\times 18}\) and a \(\mathbf{Q}\in\mathbf{R}^{18\times 18}(>0)\) that follow, respectively, the following forms:  
\[\begin{array}{l}\mathbf{Z}=\left[\begin{array}{l}\mathbf{O}_{6\times 12}& \mathbf{Z}_{3}\end{array}\right]\\ \mathbf{Q}=\left[\begin{array}{l}\mathbf{Q}_{11}&\mathbf{Q}_{12}&\mathbf{Q}_{ 13}\\ \mathbf{Q}_{21}&\mathbf{Q}_{22}&\mathbf{Q}_{23}\\ \mathbf{O}_{6}&\mathbf{O}_{6}&\mathbf{Q}_{33}\end{array}\right],\mathbf{Q}_{ ij}\in\mathbf{R}^{6\times 6},\\ \mathbf{Z}_{3}\text{ and }\mathbf{Q}_{33}\text{ are diagonal matrices.}\end{array} \tag{28}\]  
If there exist \(\gamma>0\) such that the following LMI is feasible:  
\[\left[\begin{array}{cc}\widetilde{\mathbf{A}}\mathbf{Q}+\mathbf{Q}\widetilde {\mathbf{A}}^{-T}+\widetilde{\mathbf{B}}\mathbf{Z}+\mathbf{Z}^{T}\widetilde{ \mathbf{B}}^{-T}+\mathbf{I}&\mathbf{Q}\widetilde{\mathbf{C}}^{-T}\\ \widetilde{\mathbf{C}}\mathbf{Q}&-\gamma^{2}\mathbf{I}\end{array}\right]\leq 0 \tag{29}\]  
Then, the state feedback gain, given by:  
\[\mathbf{\widetilde{K}}=\mathbf{Z}\mathbf{Q}^{-1} \tag{30}\]  
ensures that the \(\mathbf{L}_{2}\) gain of the system described by \(\sup\limits_{\|d\|_{2}\
eq 0}(\|\mathbf{\bar{Y}}\|_{2}/\|d\|_{2})\) is less than \(\gamma\)[22]. Minimizing \(\gamma\) over the variables \(\mathbf{Z}\) and \(\mathbf{Q}\) results in the smallest upper bound on the \(L_{2}\) gain of the system.  
Moreover, from Equations (27) and (30), the following is deduced:  
\[\begin{split}\mathbf{Z}=\widetilde{\mathbf{K}}\mathbf{Q}=\begin{bmatrix}& \overset{-}{\mathbf{O}}_{6\times 12}&\mathbf{G}\mathbf{Q}_{33}\end{bmatrix}\\ &\Rightarrow\ \mathbf{G}=\mathbf{Z}_{3}\mathbf{Q}_{33}^{-1}\end{split} \tag{31}\]  
and since both \(\mathbf{Z}_{3}\) and \(\mathbf{Q}_{33}\) are diagonal matrices, \(\overline{\mathbf{G}}\) is also diagonal. Thus, the decoupled characteristics are guaranteed.