# Development of Active Microvibration Isolation System for Precision Space Payload
## 3 Hybrid Control Design
### Feedforward Control  
In the actual experiment, the noise frequency of the basic platform could not be classified as an ideal white noise, however, it comprises some type of a mixed noise of a specific frequency, which makes it difficult for the feedback loop to suppress these specific noises caused by the base platform when only measuring the acceleration signal of the load platform as the control input. Therefore, feedforward control is selected to optimize the vibration isolation effect by simultaneously measuring the input noise of the vibration source and integrating it into the control signal. The diagram of the hybrid control loop after adding the feedforward controller is as follows:  
According to Equation (4), the influence of the feedforward gain coefficient \(C_{0}\) on vibration transmission is mainly reflected in the numerator of the transfer function. To isolate the vibration with high efficiency, the transfer function of the vibration is expected to be 0, so that \(C_{0}\) can be designed as:  
\[C_{0}(s)=-\frac{ds+k}{s^{2}}, \tag{16}\]  
Multiple integration links are observed to be contained in this controller, and ideal integration is infeasible in practice because of the drift of the low-frequency acceleration signal after integration. For this problem, [PERSON] proposed utilizing the \(n\)-th order weak integrator, which integrates a certain frequency to substitute for the ideal ones [28]. The revised feedforward controller can be expressed as:  
\[C_{0}=-H^{2}_{(\omega,n)}(s)(ds+k), \tag{17}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter** & **K** & **\(\omega_{1}\)(rad)** & **\(\omega_{2}\)(rad)** \\ \hline High-pass & 1.0 & 0.63 & \\ Low-pass & 0.7 & 251.3 & \\ Lead-phase 1 & 8.5 & 50 & 500 \\ Lead-phase 2 & 6.5 & 90.9 & 686.8 \\ Notch 1 & 0.2 & 223.7 & \\ Notch 2 & 0.2 & 461.8 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Filters parameters.  
where \(H_{(\omega,n)}(s)\) is an \(n\)-th order weak integrator, and \(\omega\) is the cut-off frequency. In this study, \(n=2\) and \(\omega=0.2\text{rad}/\text{s}\), and the integrator can be written as  
\[H_{(\omega,n)}(s)=\frac{1-\left(\frac{\omega}{s+\omega}\right)^{n}}{s}=\frac{1- \left(\frac{0.2\cdot 2\times\pi}{s+0.2\times 2\times\pi}\right)^{2}}{s}, \tag{18}\]  
Using the Tustin and describe transformation, the output signal of weak integrator is given as:  
\[H_{n-n\omega}(z)=\frac{2T+2\omega T^{2}+(4\omega T^{2})z^{-1}+(-2T+2\omega T^{ 2})z^{-2}}{(\omega T+2)^{2}+(2\omega^{2}T^{2}-8)z^{-1}+(\omega T-2)^{2}z^{-2}} \tag{19}\]  
According to Equation (4) and Figure 5, the simulation curves of several different controllers are shown in Figure 6. Evidently, the feedback controller has a significant suppression effect, especially on the resonant peak. However, the effect within the low-frequency band before the resonance frequency proved to be insufficient. At this point, the hybrid controller was optimized with the addition of feedforward part.  
Figure 5: Hybrid control loop diagram of active vibration isolation system. The measured basic platform noise \(a_{0}\) passes through the \(n\)-order weak integrator and control gain \(K_{FF}\), and it forms the feedforward control force \(u_{ff}\). At the same time, the measured load platform noise \(a_{1}\) passes through the high-low pass filter, PID, lead-phase filter and notch, and it forms the feedback control force \(u_{fb}\). Finally, they synthesize the hybrid control force \(u\) to drive the voice coil motor to suppress the noise.  
Figure 6: Bode diagram of transmissibility with different control methods.