# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 3 Noninteracting Control Design
### Design of the State Feedback Matrix \(F\)  
The modified feedback gains such that the system is both decoupled and stable is considered as follows [12]:  
\[\mathbf{F}=\mathbf{\Delta}^{-1}\bigg{(}\mathbf{A}^{*}+\mathbf{F}\mathbf{s}_{a}\bigg{)} \tag{18}\]  
where \(\mathbf{F}\) is a feedback matrix to be tuned such that the closed-loop system is asymptotically stable, and \(\mathbf{s}_{a}\) is given by the following nonsingular matrix:  
\[\mathbf{S}_{a}=\left[\begin{array}{c}\mathbf{C}_{1}\\ \vdots\\ \mathbf{C}_{1}\mathbf{A}^{x_{1}-1}\\ \mathbf{C}_{2}\\ \vdots\\ \mathbf{C}_{m}\mathbf{A}^{x_{m}-1}\\ \mathbf{W}\end{array}\right]\in R^{n\times n}\text{, with }\left[\begin{array}{c}\mathbf{C}_{1}\\ \vdots\\ \mathbf{C}_{1}\mathbf{A}^{x_{1}-1}\\ \mathbf{C}_{2}\\ \vdots\\ \mathbf{C}_{2}\mathbf{A}^{x_{2}-1}\\ \vdots\\ \mathbf{C}_{m}\mathbf{A}^{x_{m}-1}\end{array}\right]\in\mathbf{R}^{(m+\sum\limits_{i=1}^{ m}(x_{i}-1))\times n}\text{ and }\mathbf{W}\mathbf{B}=\mathbf{O} \tag{19}\]  
Substituting Equation (18) into Equation (16) and neglecting the higher-order term yields the following:  
\[\mathbf{Y}^{*}=-\bigg{(}\mathbf{F}\mathbf{s}_{a}\bigg{)}\mathbf{X}+\mathbf{v} \tag{20}\]  
Therefore, the \(i^{th}\) row of \(\mathbf{Y}^{*}\) is given by the following:  
\[\mathbf{Y}_{i}^{(x_{i})}=-\bigg{(}\mathbf{F}_{i}\mathbf{S}_{a}\bigg{)}\mathbf{X}+v_{i} \tag{21}\]  
where \(\mathbf{F}_{i}^{-}\) is the corresponding \(i^{th}\) row of \(\mathbf{F}\). \(\mathbf{F}_{i}\) can be represented as in Equation (22):  
\[\mathbf{F}_{i}=\left[\begin{array}{ccc}\underbrace{f_{10}}_{F_{11}}&\cdots&f_{1 (x_{1}-1)}\\ \mathbf{F}_{i1}&\end{array}\right.\underbrace{f_{20}}_{F_{20}}&\cdots&\begin{array} []{ccc}f_{2(x_{2}-1)}\\ \mathbf{F}_{i2}\end{array}\right.\cdots&\underbrace{\cdots&f_{m(x_{n}-1)}}_{F_{im} }&f_{w}\bigg{]} \tag{22}\]  
Then, substituting Equations (19) and (22) into Equation (21) yields:  
\[-\mathbf{Y}_{i}^{(x_{i})}=\mathbf{F}_{i1}\mathbf{C}_{1}\left[\begin{array}[] {c}\mathbf{I}\\ \vdots\\ \mathbf{A}^{x_{1}-1}\end{array}\right]\mathbf{X}+\mathbf{F}_{i2}\mathbf{C}_{2}\left[\begin{array} []{c}\mathbf{I}\\ \vdots\\ \mathbf{A}^{x_{2}-1}\end{array}\right]\mathbf{X}+\mathbf{\cdots}+\mathbf{F}_{im}\mathbf{C}_{m} \left[\begin{array}{c}\mathbf{I}\\ \vdots\\ \mathbf{A}^{x_{m}-1}\end{array}\right]\mathbf{X}+\mathbf{f}_{w}\mathbf{W}\mathbf{X}+\mathbf{v}_{i} \tag{23}\] \[=\mathbf{F}_{i1}\Bigg{(}\left[\begin{array}{c}\mathbf{Y}_{1}\\ \vdots\\ \mathbf{Y}_{1}^{(x_{1}-1)}\end{array}\right]+\left[\begin{array}{c}0\\ \vdots\\ \mathbf{C}_{1}\mathbf{B}\mathbf{G}\mathbf{v}^{(x_{1}-2)}\end{array}\right]\Bigg{)}+\mathbf{F}_{i2 }\Bigg{(}\left[\begin{array}{c}\mathbf{Y}_{2}\\ \vdots\\ \mathbf{Y}_{2}^{(x_{2}-1)}\end{array}\right]+\left[\begin{array}{c}0\\ \vdots\\ \mathbf{C}_{2}\mathbf{B}\mathbf{G}\mathbf{v}^{(x_{2}-2)}\end{array}\right]\Bigg{)}+\mathbf{ \cdots}+\mathbf{f}_{w}\mathbf{W}\mathbf{X}+\mathbf{v}_{i}\]If the matrix \(\mathbf{F}\) satisfies the following conditions, it can be considered a solution for the decoupling problem:  
\[\left\{\begin{array}{l}-\\ F_{ij}=0\text{ if }j\
eq i\\ F_{ij}\
eq 0\text{ if }j=i\end{array}\right. \tag{24}\]  
The value of \(\mathbf{F}_{ii}\) can be designed using classical approaches, such as the pole placement technique. Note that the number of the closed-loop poles (\(p\)) can be specified while decoupling the control system, and the number of free parameters (\(f\)) in a decoupling matrix \(\mathbf{\bar{F}}\) is given by [21]:  
\[\begin{array}{l}m+\frac{m}{\sum\limits_{1}^{m}}(x_{i}-1)\leq p\leq n\\ m+\frac{m}{\sum\limits_{1}^{m}}(x_{i}-1)\leq f\end{array} \tag{25}\]