# Adaptive Deterministic Vibration Control of a Piezo-Actuated Active-Passive Isolation Structure
## 3 Design of Adaptive Active Vibration Controller
### Test Results of \(H_{\infty}\) Control  
In the following, the well-known \(H_{\infty}\) optimal feedback control approach is first designed and evaluated. The \(H_{\infty}\) control algorithm for the suppression of external disturbance can be realized by selecting the appropriate weighting functions [18]. In this paper, the \(H_{\infty}\) controller is designed using the augw and hinfsyn functions in MATLAB based on the identified model. Considering the deterministic vibration disturbance with frequencies [35-70-105] Hz, the weighting functions \(W_{e}\) on the output \(e\) and \(W_{u}\) on the control input \(u\) are selected as:  
\[W_{e}=\frac{0.001567z^{6}-0.004701z^{4}+0.004701z^{2}-0.001567}{z^{6}-5.409z^{ 5}+12.82z^{4}-15z^{3}+10.38z^{2}-3.863z+0.6041} \tag{30}\]  
\[W_{u}=\frac{0.007772z^{6}-0.04587z^{5}+0.1136z^{4}-0.1509z^{3}+0.1136z^{2}-0.0 4587z+0.007772}{z^{6}-5.409z^{5}+12.28z^{4}-15z^{3}+10.38z^{2}-3.863z+0.6041} \tag{31}\]  
Figure 8: The Bode diagram of the identified model. Figure 7: Experimental setup system (**a**) overall distribution of the system, (**b**) local amplification of the piezo-actuated active–passive isolation structure. The experimental analysis of the frequency response of the system's primary path is shown in Figure 12. The solid line represents the vibration transmissibility curve of the open-loop (passive) state. Although under the action of the passive unit, the isolation structure has an obvious suppression effect on the high-frequency vibration disturbance; it can be seen that the isolation structure has a limited isolation effect on the low-frequency vibration disturbance, and even the vibration will be suddenly amplified near the system natural frequency. The dashed line represents the vibration transmissibility curve of the closed-loop system with only the inner-loop LQG controller when the adaptive algorithm is turned off. The state feedback gain \(F\) and the observer gain \(L\) are obtained using the dldqr and Kalman function in MATLAB, respectively. Compared with the open-loop state, under the action of the inner-loop LQG controller, the band-limited local loop shaping of the closed system has successfully attenuated the vibration transmissibility curve around the system's natural frequency. As mentioned in the introduction, the reaction and momentum wheel assemblies of the on-orbit spacecraft will generate variable low-frequency deterministic disturbances. Then, the Youla parameterized controller which is formulated by augmenting the LQG controller will be further used to perfectly suppress the unknown deterministic residual vibration. In the following cases of the experiment, the \(Q\) parameter in the controller is chosen as \(Q(z)=\left( \begin{smallmatrix}6\\ \beta=1\end{smallmatrix} \theta_{q}z^{1-q}\right)F(z)\), where the \(F(z)\) is a low-pass filter with the corner frequency at 150 Hz. In the adaption algorithm, the U-D factorization algorithm is used to improve the numerical properties of the algorithm in (28)-(29), and the initial conditions are \(P(0)=1000I_{6\times 6}\) and \(\lambda(k)=0.99\). Deterministic residual vibration suppression  
Two experiment cases are carried out to verify the effectiveness of the adaptive active vibration controller. For case one, a deterministic disturbance at three different frequencies is suppressed to verify the effectiveness of the regulator to multifrequency deterministic residual vibration. The attenuation performance for the deterministic disturbance with step changes in frequencies is evaluated in case two. **Case one:** deterministic disturbance at three different frequencies  
According to the adaptive regulator already designed, \(n_{q}=6\) indicates that three different frequencies of deterministic vibration disturbances can be suppressed. Then a deterministic residual vibration is excited by 35 Hz, 70 Hz, and 105 Hz sinusoidal signals with frequency doubling property which is widely found in spacecraft.