## 1 Introduction
### Modeling of the Isolation Units
\begin{array}{cccc}\frac{k_{\rm 3}{\rm e}^{i\omega \tau_{l}^{\prime}}}{R_{1}(1+i\omega\tau_{1})}g_{11}^{(k)}&\frac{k_{\rm 3}{\rm e}^{i \omega\tau_{1}^{\prime}}}{R_{1}(1+i\omega\tau_{1})}g_{12}^{(k)}&\cdots&\frac{ k_{\rm 3}{\rm e}^{i\omega\tau_{1}^{\prime}}}{R_{1}(1+i\omega\tau_{1})}g_{1 {\rm N_{1}}}^{(k)}\\ \frac{k_{\rm 2}{\rm e}^{i\omega\tau_{2}^{\prime}}}{R_{2}(1+i\omega\tau_{2})}g_{2 1}^{(k)}&\frac{k_{\rm 2}{\rm e}^{i\omega\tau_{2}^{\prime}}}{R_{2}(1+i\omega \tau_{2})}g_{22}^{(k)}&\cdots&\frac{k_{\rm 2}{\rm e}^{i\omega\tau_{2}^{\prime}}}{R_{ 2}(1+i\omega\tau_{2})}g_{2{\rm N_{1}}}^{(k)}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{k_{\rm 2N_{1}}{\rm e}^{i\omega\tau_{N_{1}}^{\prime}}}{R_{N_{1}}\big{(}1+i \omega\tau_{N_{1}}\big{)}}g_{N1}^{(k)}&\frac{k_{\rm 2N_{1}}{\rm e}^{i\omega\tau_{N_{1}}^{ \prime}}}{R_{N_{1}}\big{(}1+i\omega\tau_{N_{1}}\big{)}}g_{N2}^{(k)}&\cdots& \frac{k_{\rm 2N_{1}}{\rm e}^{i\omega\tau_{N_{1}}^{\prime}}}{R_{N_{1}}\big{(}1+i \omega\tau_{N_{1}}\big{)}}g_{N_{1}{\rm N_{1}}}^{(k)}\end{array} \right] \tag{32}\]  
where \(\tau_{l}=L_{l}/R_{l}\) is the electrical time constant of the \(l\)th active actuator; \(L_{l}\) and \(R_{l}\) are the inductance and resistance of its control circuit, respectively; \(\tau_{l}^{\prime}\) is the delay compensation generated by the controller for the \(l\)th active actuator; \(k_{sl}\) is the power amplification gain parameter of the \(l\)th active actuator; and \(g_{rs}^{(k)}\) (\(r\), \(s=1,2,\ldots,N_{\rm I}\); \(k=1,2,\ldots,6\)) is the controller gain of the \(r\)th actuator relative to the feedback signal taken from the sth mounting node. According to the principle of acoustic vibration isolation and the law of mass action, when \(m_{\rm al}\
eq 0\), the vibration mass will increase, lowering the fundamental frequency of the system, expanding the effective sound insulation band to lower frequencies and increasing the sound insulation volume. However, the mass of the active actuator is much smaller than the mass of the acoustic cavity and the double-layer plate, such that the effect is almost negligible on the sound insulation of the system, so the \(M_{\rm a}\) can be approximated to 0 in the actual simulation. For each active actuator, the equivalent complex stiffness:  
\[ \begin{split}\Delta k^{*}&={\rm j}\omega\cdot\frac{ k_{\rm 3}{\rm e}^{i\omega\tau_{l}^{\prime}}}{R_{1}(1+i\omega\tau_{1})}g_{rs}^{(k)}=\frac{k_ {\rm 3}\omega^{2}\tau_{g_{rs}^{(k)}}}{R_{1}\sqrt{1+(\omega\tau_{1})^{2}}}\Big{(}1+{ \rm j}{\rm j}{\rm\frac{1}{\omega\tau_{1}}}\Big{)}{\rm e}^{i\omega\tau_{l}^{ \prime}}\\ &=\Delta k_{0}(1+{\rm j}\Lambda\eta_{0}){\rm e}^{i\omega\tau_{l}^{ \prime}}=\Delta k_{0}\sqrt{1+(\Delta\eta_{0})^{2}}\cdot{\rm e}^{i(\arctan \Delta\eta_{0}+\omega\tau_{l}^{\prime})}\end{split}  \tag{33}\]  
where \(\Delta\eta_{0}\) denotes the loss factor of the elastic support. Since the value of \(\omega\) is extremely large, \(\Delta\eta_{0}\) can be approximated as 0. Thus, the equivalent stiffness:  
\[\Delta k={\rm Re}\{\Delta k^{*}\}=\Delta k_{0}\sqrt{1+(\Delta\eta_{0})^{2}} \cdot\cos\big{(}\arctan\Delta\eta_{0}+\omega\tau_{l}^{\prime}\big{)} \tag{34}\]  
Since the time lag \(\omega\tau_{l}^{\prime}\) of the active actuator is extremely small, it tends to 0, it can be neglected in the actual simulation calculation. The time lag does not improve the control effect. When \(\omega\tau_{l}^{\prime}\) is large enough, the control of the active actuator on the amount of stiffness change does not achieve the expected effect, reducing the sound insulation performance of the system. In extreme cases of inappropriate control, it can even cause system instability. Therefore, the time lag cannot be artificially added to the active actuator, except for the almost negligible self-contained time lag.