## 3 Control of the APIVI Platform  
### Parameter Determination for \(H_{\infty}\) Control of APIVI Platform  
Aiming at the problem of vibration control in the APIVI platform, a controller was designed to stabilize the closed-loop control system and minimize the transfer function Heo of from external disturbances to sensitive payloads. The acceleration response of the moving platform and the active control force provided by the integrated active and passive actuator based on piezoelectric ceramics are defined as the output signals of the entire closed-loop control system, the measured quantity is the response of the moving platform, and the disturbance (excitation signal) received by the lower platform is the external input signal. The generalized coordinates of the six-dimensional orthogonal vibration isolation platform are defined as \(q=\left(x,y,z,\alpha,\beta,\gamma\right)^{T}=x_{A}\), where \(x_{A}\) is state variable. The state vector of the system is \(X=\left[x_{A}^{T}\quad\tilde{x}_{A}^{T}\right]\), the input vector is \(\omega=\left[\omega_{B}^{T}\quad\omega_{n}^{T}\right]\), \(\omega_{B}\) is disturbed signal, and \(\omega_{n}\) is the noise signal of the sensor. The control vector is \(u=Fa\), \(Fa\) is active actuator dynamic output force. Measured output vector is \(y=\tilde{y}+\omega_{n}\), \(\tilde{y}\) is represents the acceleration signal on the platform without considering the influence of sensor noise. Figure 4: The APIVI platform for sensitive instruments. Figure 3: The schematic diagram of the support pole. \(z= \begin{bmatrix}z_{1}&z_{2}\end{bmatrix} ^{T}\) indicates the output signal of the control system, \(z_{1}\) is the acceleration response of the moving platform, and \(z_{2}\) is the control force provided by an integrated active and passive actuator. The state space expression of the AIPVI platform can be derived as [30]  
\[\dot{X}=AX+B_{1}\omega+B_{1}u, \tag{1}\]  
\[\text{where}\quad A= \begin{bmatrix}0_{6\times 6}&E_{6\times 6}\\ -M(q)^{-1}K(q)&-M(q)^{-1}C(q)\end{bmatrix} ,B_{1}\quad=\quad \begin{bmatrix}0_{6 \times 6}&0_{6\times 6}\\ M(q)^{-1}E_{6\times 6}&0_{6\times 6}\end{bmatrix} ,\]  
\[B_{2}= \begin{bmatrix}0_{6\times 6}\\ M(q)^{-1}I\end{bmatrix} .\]  
Assuming \( \begin{bmatrix}0_{6\times 6}&E_{6\times 6}\end{bmatrix} =H\), the measured quantity can be derived as  
\[y= \begin{bmatrix}0_{6\times 6}&E_{6\times 6}\end{bmatrix} \dot{x}+ \begin{bmatrix}0_{6 \times 6}&E_{6\times 6}\end{bmatrix}\begin{bmatrix}\omega_{B}\\ \omega_{n}\end{bmatrix} =HAx+(HB_{1}+H)\omega+HB_{2}u=C_{2}x+D_{21}\omega+D_{21}u, \tag{2}\]  
\[\text{where}\quad C_{2}= \begin{bmatrix}-M(q)^{-1}K(q)&-M(q)^{-1}C(q)\end{bmatrix} ,\quad D_{21}= \begin{bmatrix}M(q)^{-1}E_{6\times 6}&E_{6\times 6}\end{bmatrix} ,\]  
\[D_{22}=M(q)^{-1}I.\]  
State space expression of AIPVI platform can be derived as  
\[ \begin{cases}\dot{x}=Ax+B_{1}\omega+B_{2}u\\ z=C_{1}x+D_{11}\omega+D_{12}u\\ y=C_{2}x+D_{21}\omega+D_{22}u\end{cases} , \tag{3}\]  
Laplace transform Equation (1),  
\[sX(s)=AX(s)+B_{1}W(s)+B_{2}U(s), \tag{4}\]  
and assuming \(Q=(sI-A)^{-1}\),  
\[X=QB_{1}W+QB_{2}U, \tag{5}\]  
\[\text{substitute Equation (\ref{eq:A2})}\]  
\[ \begin{split} Y&=(HAQB_{1}+HB_{1}+H)W+(HAQB_{2}+HB_{2})U\\ &=(HAQB_{1}+HB_{1})W+(HAQB_{2}+HB_{2})U+WU\\ &=(HAQB_{1}+HB_{1})\Bigg{[}\begin{bmatrix}0\\ E\end{bmatrix}W_{B}+(HAQB_{1}+HB_{1})H^{T}W_{n}+(HAQB_{2}+HB_{2})U+HW\\ &=G_{1}W_{B}+G_{2}U+W_{n}\\ &\text{where }(HAQB_{1}+HB_{1})H^{T}=0,G_{1}=(HAQB_{1}+HB_{1}) \begin{bmatrix}0\\ E\end{bmatrix}+E,G_{2}=HAQB_{2}+HB_{2}.\end{split}  \tag{6}\]  
\[\text{Substitute }Z= \begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix} ^{T}= \begin{bmatrix}Y&U\end{bmatrix} ^{T}\text{into Equation (\ref{eq:A2})}.\]  
By adding weight functions \(W_{1},\ W_{2},\ W_{N},\ W_{b},\ W_{u}\) to \(Z_{1},\ Z_{2},\ W_{N},\ W_{B},\ W_{B},\ W_{B},\ W_{B},\ W_{B},\ W_{B},\ W_{B}\), and \(U,\ W_{i}\) generalized matrix can be obtained with simplification. \[G_{p}= \begin{bmatrix}W_{1}G_{1}W_{b}&W_{N}&W_{1}G_{2}W_{u}\\ 0&0&W_{2}\\ G_{1}W_{b}&W_{N}&G_{2}W_{u}\end{bmatrix}