# Development of Active Microvibration Isolation System for Precision Space Payload
## 3 Hybrid Control Design  
### Feedback Control  
For a single vibration isolation module, the feedback controller \(C_{1}\) can be designed to be used as a PID controller. \[C_{1}=K_{P}+\frac{K_{I}}{s}+K_{D}s, \tag{5}\]  
where \(K_{P}\) is the proportional term of acceleration; it can change the virtual mass of the system, improve the resonant frequency of the closed-loop control system, and optimize the suppression ability of the high-frequency band to interference, and \(K_{I}\) is the integral term of acceleration, which can provide skyhook damping for the system to enhance the suppression ability of the active vibration isolation system at the resonant frequency. \(K_{D}\) represents the differential term of the acceleration and its trend. Because the feedback loop controls the voice coil motor by measuring the acceleration signal, the output signal needs to change continuously with the input signal to ensure that the vibration of the load platform approaches zero. To improve the tracking and control ability for time-varying acceleration input signals and avoid overshoot caused by excessive gain, a transition process based on a tracking differentiator is added to the feedback controller. Considering the faint acceleration amplitude of microvibration, to avoid the chattering introduced by the differentiator in the solution, a discrete nonlinear tracking differentiator is selected to obtain the tracking signal and its differential value. The expression is as follows:  
\[\left\{ \begin{array}{c}r_{1}(k+1)=r_{1}(k)+hr_{2}(k)\\ r_{2}(k+1)=r_{2}(k)+hfs(r_{1}(k)-v(k),r_{2}(k),\delta,h),\end{array} \right. \tag{6}\]  
where \(h\) is the sampling period, \(v(k)\) is the input signal of the \(h-th\) sequence, \(\delta\) is a parameter that determines the speed of tracking, \(r_{1}(k)\) and \(r_{2}(k)\) is the tracking signal and differential tracking signal of \(v(k)\), respectively. \(fst\) is the optimal control synthesis function of the discrete-time system, which is expressed as:  
\[fst(x_{1},x_{2},\delta,h)=\left\{ \begin{array}{cc}-\delta sign(a)&|a|>d\\ -\delta\frac{a}{d}&|a|\leq d\end{array} \right., \tag{7}\]  
\[a=\left\{ \begin{array}{cc}x_{2}+\frac{a_{0}-d}{2}sign(y)&|y|>d_{0}\\ x_{2}+\frac{y}{h}&|y|\leq d_{0}\end{array} \right., \tag{8}\]  
where \(d=\delta h\), \(d_{0}=hd\), \(y=x_{1}+hx_{2}\), \(a_{0}=\sqrt{d^{2}+8\delta|y|}\). After the tracking differentiator, the control signal in the feedback loop with the recursive notation is given as  
\[u_{C_{1}}(i)=u_{C_{1}}(i-1)+K_{P}[r_{1}(i)-r_{1}(i-1)+\frac{T_{0}}{T_{i}}r_{1 }(i)+T_{d}(r_{2}(i)-r_{2}(i-1))], \tag{9}\]  
where \(T_{0}\) is the sampling period, \(T_{i}\) and \(T_{d}\) are the integration and derivative time constant. Increasing the feedback gain can change the resonant characteristics of the system to improve the vibration suppression effect, however, the exorbitant gain will also cause problems such as signal drift and actuator saturation. Therefore, it is necessary to add a high-pass filter to the loop for filtering. The transfer function of the first-order high-pass filter is as follows:  
\[H_{high-pass}(s)=\frac{s}{s+\omega_{c}}, \tag{10}\]  
where \(\omega_{c}\) is the cut-off frequency. Moreover, the input signal is passed through a low-pass filter to increase the control gain of a specific frequency band through frequency division control, thus focusing on suppressing the noise of the frequency band. The introduction of high- and low-pass filters affects the phase of the system, which results in a decline in the stability of the system. This can be directly observed by calculating the Bode diagram of the open-loop system function. The Bode diagram in Figure 4 reveals that the exorbitant gain leads to an insufficient phase margin and easy oscillation in the open loop. To ensure the stability of the system and increase the phase margin, a lead-lag phase filter is added to the feedback control.