# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 4 Simulation Studies  
### Implementation  
The specifications of the AVIS are listed in Table 1, and the matrix gains of the system model were obtained accordingly. With these values, the matrix \(\mathbf{\Delta}\) was computed and confirmed to be nonsingular. Thus, the noninteracting controller was designed. The state feedback matrix \(\mathbf{F}\), given in Equation (18), was identified. Using the Robust Control Toolbox in Matlab, the LMI given by Equation (29) with the predefined conditions from Equation (28) was solved for the AVIS with a minimal value of \(\gamma\) of 0.08. \(\overline{\mathbf{G}}\) was identified afterward from Equation (31), and subsequently, \(\mathbf{G}\) was obtained from Equation (26). The two control matrices are presented in Equation (32).  
Moreover, another state feedback controller was designed to provide a fair assessment of the control system proposed in this paper. This controller was also designed based on  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(a\) [m] & 0.4 & \(m\) [kg] & 288 & \(l_{yy}\) [m] & 0.53 \\ \(b\) [m] & 0.6 & \(l_{xx}\) [kg.m\({}^{2}\)] & 34.8 & \(l_{xx}\) [m] & 0.015 \\ \(c\) [m] & 0.2 & \(l_{yy}\) [kg.m\({}^{2}\)] & 15.6 & \(l_{xx}\) [m] & 0.37 \\ \(k_{x},k_{y},k_{z}\) [N/m] & 14,180 & \(l_{zz}\) [kg.m\({}^{2}\)] & 49.92 & \(l_{xy}\) [m] & 0.53 \\ \(c_{x},\,c_{y},\,c_{z}\) [Ns/m] & 100 & \(l_{xx}\) [m] & 0.29 & \(l_{yz}\) [m] & 0.015 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of the controlled AVIS.  
the integral-type servo system, illustrated in Figure 2. The state-space representation of the closed-loop system is given in Equation (33).  
\[\begin{split}\dot{\vec{\mathbf{X}}}=\left[\begin{array}{c}\dot{\vec{ \mathbf{X}}}\\ \dot{\vec{\mathbf{v}}}\end{array}\right]=\underbrace{\left[\begin{array}{cc}\mathbf{A} &\mathbf{O}_{6}\\ -\mathbf{C}&\mathbf{O}_{6}\end{array}\right]}_{\mathbf{A}}\vec{\mathbf{X}}+\underbrace{\left[ \begin{array}{c}\mathbf{B}\\ \mathbf{O}_{6}\end{array}\right]}_{\mathbf{B}}\mathbf{U}+\underbrace{\left[\begin{array} []{c}\mathbf{D}_{x}\\ \mathbf{r}\end{array}\right]}_{\mathbf{d}}\\ \dot{\vec{\mathbf{Y}}}=\vec{\mathbf{C}}\vec{\mathbf{X}}\\ \mathbf{U}=-\vec{\mathbf{F}}\vec{\mathbf{X}}+\vec{\mathbf{G}}\vec{\mathbf{v}}\\ \end{split} \tag{33}\]  
Accordingly, the following LMIs were considered:  
\[\left[\begin{array}{cccc}\mathbf{AQ}+\mathbf{QA}^{\mathbf{T}}+\mathbf{BZ}+\mathbf{Z}+\mathbf{Z}^{- \mathbf{T}}\mathbf{B}^{\mathbf{T}}+\mathbf{I}&\mathbf{QC}^{\mathbf{T}}\\ \dot{\vec{\mathbf{Y}}}-\dot{\vec{\mathbf{C}}}\vec{\mathbf{X}}&\dot{\vec{\mathbf{C}}}\vec{\mathbf{ Q}}&\dot{\vec{\mathbf{Y}}}-\dot{\vec{\mathbf{Y}}}^{2}\mathbf{I}\end{array}\right]\leq 0 \tag{34}\]  
\(\vec{\mathbf{Z}}>0\), \(\mathbf{Z}\in R^{6\times 18}\)  
\(\vec{\mathbf{Q}}>0\), \(\mathbf{Q}\in R^{18}\)  
The feedback control gains were obtained such as \(\left[\begin{array}{cc}\mathbf{F}&\mathbf{G}\\ \dot{\vec{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{ }}}}}}}}}}}}}}}} -\end{array}\right]=\mathbf{Z}\mathbf{Q}^{-1}\). Their values are presented in Equation (35).  
\[\begin{split}\mathbf{F}=\left[\begin{array}{cccc}23055996.125&0&0& 0&-451615.356&0\\ 0&23037296.982&0&343167.966&0&0\\ 0&0&23020562.484&0&0&0\\ 0&-48722.485&0&331246.305&0&0\\ 52733.158&0&0&0&226948.799&0\\ 0&0&0&0&0&500955.474\\ 66660.618&0&0&0&-1121.874&0\\ 0&66606.378&0&863.757&0&0\\ 0&0&66560.178&0&0&0\\ 0&-145.165&0&803.471&0&0\\ 156.715&0&0&0&526.273&0\\ 0&0&0&0&0&1231.046\end{array}\right]\end{split} \tag{35}\]  
\[\begin{split}\mathbf{G}=\left[\begin{array}{cccc}242833691.013&0& 0&0&-4787897.103&0\\ 0&242637163.198&0&3633362.131&0&0\\ 0&0&242641475.407&0&0&0\\ 0&-508721.625&0&3530889.292&0&0\\ 550890.757&0&0&0&2420898.234&0\\ 0&0&0&0&0&5336827.751\end{array}\right]\end{split}\]