## 3 Control of the APIVI Platform
### Parameter Determination for \(H_{\infty}\) Control of APIVI Platform
, \tag{7}\]The relationship between input and output of the closed-loop control system of APIVI platform can be obtained as follows  
\[ \begin{bmatrix}z_{1}\\ z_{2}\\ y\end{bmatrix} =G_{p} \begin{bmatrix}\omega_{B}\\ \omega_{n}\\ u\end{bmatrix} , \tag{8}\]  
The standard H\({}_{\infty}\) robust control model of the APIVI platform is obtained, and the design block diagram of the model is shown in Figure 5.  
In order to prevent the high order of the designed controller from increasing the calculation and implementation difficulty, the weighted function value with low order should be selected. In the frequency range of 10-200 Hz, active control is required to suppress the disturbance and reduce the impact on the sensitive payload. The weighted function value \(W_{1}\) describes the acceleration response of the moving platform and the relative weight between the control target and the input variable. Therefore, \(W_{1}\) is designed to be as large as possible within the desired control effective frequency domain to improve the control performance, and as small as possible outside the desired control effective frequency domain. The weighted function value \(W_{2}\) describes the active control force and the relative weight between the control target and the input variable. \(W_{2}\) is designed to be as small as possible in the desired control effective frequency domain to obtain a larger active control force, and as large as possible outside the desired control effective frequency domain to reduce the output of the actuator. Therefore, in order to protect the piezoelectric ceramics, it is necessary to make \(W_{2}\) greater than \(-77.3\) dB to prevent excessive control. \(W_{n}\) describes a high-pass filter to allow high-frequency noise signals to enter the system. Therefore, the function with smaller weight at low frequency and larger weight at high frequency is selected. The excitation weighted function value \(W_{b}\) is set to \(I_{6\times 6}\). In order to ensure that the controller can have linear output characteristics in the desired effective frequency range, the design \(W_{u}\approx I_{6\times 6}\). According to the requirements of the above weighted function value selection, the value of the weighted function is determined as  
\[W_{1}=\frac{(s+0.1)(s+100)}{(s+1)(s+10)}I_{6\times 6},\ W_{2}=\frac{0.015s^{2}+2.313s+1.5}{s^{2}+100\ s+1000}I_{6\times 6},\ W_{n}=\frac{0.2s+200}{s+20,000}I_{6 \times 6},\ W_{b}=I_{6\times 6},\ W_{u}=\frac{s+8}{s+9}I_{6\times 6}\]  
Bode diagram of the transfer function value of the weighted function of H\({}_{\infty}\) controller of the APIVI platform is shown in Figure 6.  
Figure 5: The general frame of the H\({}_{\infty}\) controller for the six-dimensional orthogonal vibration isolation platform. The weighted function value is substituted into Equation (7), and the minimum value \(\gamma\) of \(\left\|G_{P}\right\|_{\infty}\) is calculated. With iteration, the final \(\gamma=0.9488\) is calculated. For a complex system with high dimensions, the high order controller is difficult to analyze and implement in engineering. Order reduction in the higher-order model is the most effective way to solve problem. Therefore, Hankel norm singular value reduction method was adopted to reduce the order of the controller and screen out a transfer function. The H norm of the model is close to the original controller, and the H norm of the model reduction error is minimized. Hankel singular value distribution of the Ho controller is obtained, as shown in Figure 7. By eliminating the smaller Hankel singular values, the controller order is 10.