# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 3 Noninteracting Control Design  
### Conditions for a Decoupled Performance  
The decoupling process consists of compensating a given system so that each system input independently controls a corresponding system output [9]. The system inputs refer to the references fed to the system. The proposed controller is designed based on a typical integral-type servosystem given by the block diagram illustrated in Figure 2 [20]. The feedback control law is in the following form:  
\[\mathbf{U}=-\mathbf{F}\mathbf{X}+\mathbf{G}\mathbf{v}\quad\left(\mathbf{F}\in\mathbf{R}^{m\times n},G\in \mathbf{R}^{m}\right) \tag{10}\]  
with \(\mathbf{v}\) a pseudo-state vector defined by the integration of the control error. \(\mathbf{F}\) and \(\mathbf{G}\) are controller gains to be designed. The system dynamics with the feedback control input is represented by:  
\[\dot{\mathbf{X}}=\left(\mathbf{A}-\mathbf{B}\mathbf{F}\right)\mathbf{X}+\mathbf{B}\mathbf{G}\mathbf{v}+\mathbf{D}_{ \mathbf{x}} \tag{11}\]  
\(\mathbf{Y}=\mathbf{C}\mathbf{X}\)  
Let \(\mathbf{C}_{i}\) denote the \(i^{th}\) row of the matrix \(\mathbf{C}\), and let  
\[x_{i}=\left\{ \begin{array}{c}\min\Bigl{\{}j\Bigl{|}\mathbf{C}_{i}\mathbf{A}^{j-1}\mathbf{ B}\
eq 0\Bigr{\}}\ (j=1,2,\ldots,n)\\ n\ \text{if}\ \mathbf{C}_{i}\mathbf{A}^{j-1}\mathbf{B}=0\ \text{for all}\ j\end{array} \right. \tag{12}\]  
The above denotations are characterized by the following:  
\[ \begin{array}{c}\mathbf{C}_{i}\left(\mathbf{A}-\mathbf{B}\mathbf{F}\right)^{k}=\mathbf{C}_{i} \mathbf{A}^{k}\ \ \ \ \ \ k=0,1,\ldots,x_{i}-1\\ \mathbf{C}_{i}\left(\mathbf{A}-\mathbf{B}\mathbf{F}\right)^{k}=\mathbf{C}_{i}\mathbf{A}^{x_{i}-1}(\bm {A}-\mathbf{B}\mathbf{F})^{k-x_{i}+1}\ k=x_{i},x_{i}+1,\ldots,n\end{array}  \tag{13}\]  
Consider the \(i^{th}\) row of the output vector, denoted by \(Y_{i}\), then the following relations between its time-derivatives and the states are obtained as follows:  
\[ \begin{array}{c}Y_{i}=\mathbf{C}_{i}\mathbf{X}\\ \dot{Y}_{i}=\mathbf{C}_{i}\dot{\mathbf{X}}=\mathbf{C}_{i}\mathbf{A}\mathbf{X}+\mathbf{C}_{i}\mathbf{B}\mathbf{G }\mathbf{v}\\ \ddot{Y}_{i}=\mathbf{C}_{i}\mathbf{A}\dot{\mathbf{X}}+\mathbf{C}_{i}\mathbf{B}\mathbf{G}\dot{\mathbf{v}}\\ =\mathbf{C}_{i}\mathbf{A}^{2}\mathbf{X}+\mathbf{C}_{i}\mathbf{B}\mathbf{G}\dot{\mathbf{v}}\\ \vdots\\ \gamma_{i}^{(x_{i})}=\mathbf{C}_{i}\mathbf{A}^{x_{i}-1}\dot{\mathbf{X}}+\mathbf{C}_{i}\mathbf{B} \mathbf{G}\mathbf{v}^{(x_{i}-1)}\\ =\left(\mathbf{C}_{i}\mathbf{A}^{x_{i}}-\mathbf{C}_{i}\mathbf{A}^{x_{i}-1}\mathbf{B}\mathbf{F}\right) \mathbf{X}+\mathbf{C}_{i}\mathbf{A}^{x_{i}-1}\mathbf{B}\mathbf{G}\mathbf{v}+\mathbf{C}_{i}\mathbf{B}\mathbf{G}\bm {v}^{(x_{i}-1)}\end{array}  \tag{14}\]  
Furthermore, let the following expressions be used:  
\[\mathbf{Y}^{\prime}=\left[ \begin{array}{cccc}Y_{1}^{(x_{1})}&Y_{2}^{(x_{2})}& \ldots&Y_{m}^{(x_{m})}\\ \end{array} \right]^{T}\!\!,\]  
\[\mathbf{A}^{*}=\left[ \begin{array}{c}\mathbf{C}_{1}\mathbf{A}^{x_{1}}\\ \mathbf{C}_{2}\mathbf{A}^{x_{2}}\\ \vdots\\ \mathbf{C}_{m}\mathbf{A}^{x_{m}}\end{array} \right]\!,\ \mathbf{\Delta}=\left[ \begin{array}{c}\mathbf{C}_{1}\mathbf{A}^{x_{1}-1} \mathbf{B}\\ \mathbf{C}_{2}\mathbf{A}^{x_{2}-1}\mathbf{B}\\ \vdots\\ \mathbf{C}_{m}\mathbf{A}^{x_{m}-1}\mathbf{B}\end{array} \right]\!,\ \mathbf{g}=\left[ \begin{array}{c}\mathbf{C}_{1}\mathbf{B}\mathbf{G}\mathbf{v}^{(x_{1}-1)} \\ \mathbf{C}_{2}\mathbf{B}\mathbf{G}\mathbf{v}^{(x_{2}-1)}\\ \vdots\\ \mathbf{C}_{m}\mathbf{B}\mathbf{G}\mathbf{v}^{(x_{m}-1)}\end{array} \right] \tag{15}\]  
Then, from Equations (14) and (15), one can obtain the following:  
\[\mathbf{Y}^{\prime}=\left(\mathbf{A}^{*}-\mathbf{\Delta}\mathbf{F}\right)\mathbf{X}+\mathbf{\Delta}\bm {G}\mathbf{v}+\mathbf{g} \tag{16}\]  
If the higher-order term \(\mathbf{g}\) is neglected, with \(\mathbf{\Delta}\mathbf{G}\) resulting in a diagonal matrix and \(\left(\mathbf{A}^{*}-\mathbf{\Delta}\mathbf{F}\right)=\mathbf{O}\), then the decoupling can be achieved. That is, \(x_{i}\)-order time-derivation  
Figure 2: An integral-type servosystem for the 6-DoF AVIS.