## 3 Control of the APIVI Platform
### Robust Control of APIVI Platform with Uncertain Parameters  
In order to suppress the micro-vibration of space-sensitive payloads, it is necessary to consider the uncertain factors, such as the machining precision of payloads or the deviation between the design value and the actual value of stiffness and damping caused by aging components. A controller was designed to make the system work normally under the condition of parameter perturbation and achieve the control goal. Robustness analysis was added to stability analysis by \(\mu\) synthesis theory, so that the singular value of the system can meet certain requirements even in the worst case. \(A\) system with uncertain structure is shown in Figure 8, where \(P\) is the generalized system, \(v\) is the control input, \(r\) is the measurement output, and \(\Delta\) is the set of partitioned diagonal matrices. Figure 6: Weighted function’s Bode diagram of H\({}_{\infty}\) controller for APIVI platform. Figure 7: Hankel matrix singular value distribution diagram of H\({}_{\infty}\) controller for APIVI platform. \(\mu\) is defined as a structural singular value  
\[\mu_{\Delta}=\frac{1}{\min\{\sigma_{\max}(\Delta)|\Delta\in\Delta,\det(E-P\Delta)= 0\}}, \tag{9}\]  
Assuming the stiffness and damping of the APIVI platform have 5% parameter perturbations, the system control model with parameter perturbations is established. The state space Equation (3) can be rewritten as  
\[ \begin{cases}\dot{x}=Ax+B\lambda\\ t=Cx+D\lambda\end{cases} , \tag{10}\]  
where \(B= \begin{pmatrix}\text{B}_{1}&\text{B}_{2}\end{pmatrix} ,C= \begin{pmatrix}C_{1} \\ C_{2}\end{pmatrix} ,t= \begin{pmatrix}z\\ y\end{pmatrix} ,\lambda= \begin{pmatrix}\omega\\ \mathbf{u}\end{pmatrix} \). Parametric perturbations of the spring and damping cause matrix A to change. Assuming \(\text{A}=\text{A}_{0}+\text{A}_{\Delta}\), \(\text{A}_{0}\) is a matrix composed of design parameters, and \(\text{A}_{\Delta}\) is a matrix composed of perturbation parameters. \(\text{A}_{\Delta}=\text{A}_{\text{k}}\eta_{\text{k}}+\text{A}_{\text{c}}\eta_ {\text{lc}}\), where \(\text{K}=\text{K}_{0}+0.05\text{K}_{0}\eta_{\text{lc}},\text{C}=\text{C}_{0}+ 0.05\text{C}_{0}\eta_{\text{lc}}\), \(|\eta_{\text{k}}|<1\), \(|\eta_{\text{c}}|<1\). According to matrix A obtained in Equation (1), \(\text{A}_{\text{k}}= \begin{bmatrix}0_{6\times 6}&\text{E}_{6\times 6}\\ -0.1\text{M}(\mathbf{q})^{-1}\text{K}_{0}(\mathbf{q})&0_{6\times 6}\end{bmatrix} \),  
\[\text{A}_{\text{c}}= \begin{bmatrix}0_{6\times 6}&0_{6\times 6}\\ 0_{6\times 6}&-0.1\text{M}(\mathbf{q})^{-1}\text{C}_{0}(\mathbf{q})\end{bmatrix} .\]  
Defining \(\text{S}= \begin{bmatrix}\text{A}&\text{B}\\ \text{C}&\text{D}\end{bmatrix} =\text{S}_{0}+\text{S}_{\text{K}}+\text{S}_{ \text{C}}\), by using linear fraction transformation to decompose \(S\), the structural parameter uncertainty problem can be transformed into matrix uncertainty. \[S_{k}= \begin{bmatrix}A_{k}&0_{12\times 18}\\ 0_{18\times 12}&0_{18\times 18}\end{bmatrix} = \begin{bmatrix}E_{k}\\ F_{k}\end{bmatrix}\begin{bmatrix}G_{k}&H_{k}\end{bmatrix} , \tag{11}\]  
\[\text{where }\text{E}_{\text{k}}= \begin{bmatrix}0_{6\times 6}\\ -0.1\text{M}(\mathbf{q})^{-1}\text{K}_{0}(\mathbf{q})\end{bmatrix} ,\text{F}_{ \text{k}}= \begin{bmatrix}\text{E}_{6\times 6}\\ 0_{6\times 6}\\ 0_{6\times 6}\end{bmatrix} ,\text{G}_{\text{k}}= \begin{bmatrix}0_{6 \times 6}&\text{E}_{6\times 6}\end{bmatrix} ,\text{H}_{\text{k}}= \begin{bmatrix} \text{E}_{6\times 6}\\ 0_{6\times 6}\\ 0_{6\times 6}\end{bmatrix} ^{\mathsf{T}}.\]  
\[S_{\text{c}}= \begin{bmatrix}A_{\text{c}}&0_{12\times 18}\\ 0_{18\times 12}&0_{18\times 18}\end{bmatrix} = \begin{bmatrix}E_{\text{c}}\\ F_{\text{c}}\end{bmatrix}\begin{bmatrix}G_{\text{c}}&H_{\text{c}}\end{bmatrix} , \tag{12}\]  
\[\text{where }\text{E}_{\text{c}}= \begin{bmatrix}0_{6\times 6}\\ -0.1\text{M}(\mathbf{q})^{-1}\text{K}_{0}(\mathbf{q})\end{bmatrix} ,\text{F}_{ \text{k}}= \begin{bmatrix}0_{6\times 6}\\ 0_{6\times 6}\\ 0_{6\times 6}\end{bmatrix} ,\text{G}_{\text{c}}=