## 1 Introduction
### Modeling of the Isolation Units  
Let the isolation unit between the double-layer plate be a linear hybrid element of active and passive control with negative feedback, whose transfer characteristics are expressed by the four-terminal parameters:  
\[ \begin{bmatrix}\mathbf{F}_{\rm E1}\\ \mathbf{U}_{n}^{t}\end{bmatrix} = \begin{bmatrix}\mathbf{S}_{11}&\mathbf{S}_{12}\\ \mathbf{S}_{21}&\mathbf{S}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{F}_{\rm Eb}\\ \mathbf{U}_{n}\end{bmatrix}  \tag{29}\]  
where \(\mathbf{F}_{\rm Ei}\) and \(\mathbf{F}_{\rm Eb}\) are the column vectors of the force of the active actuator acting on the upper and lower plates, respectively; \(\mathbf{U}_{n}^{t}\) and \(\mathbf{U}_{n}\) are the column vectors of the velocity of the isolation device at the connection with the upper and lower plates, respectively; \(\mathbf{S}_{ij}(i,j=1,2)\) is the four-terminal parameters matrix of the transfer characteristic, and:  
\[\left\{ \begin{aligned} S_{11}=\Big{(}\mathbf{Z}_{11}+\mathbf{B}_{11}\mathbf{B}_{ 21}^{-1}\Big{)}\bigl{(}\mathbf{I}+\mathbf{B}_{21}\mathbf{Z}_{21}\bigr{)}^{-1}\mathbf{B}_{21}\\ S_{12}=\mathbf{B}_{12}-\mathbf{B}_{11}\mathbf{B}_{21}^{-1}\mathbf{B}_{21}+\mathbf{Z}_{12}+ \Big{(}\mathbf{Z}_{11}+\mathbf{B}_{11}\mathbf{B}_{21}^{-1}\Big{)}\bigl{(}\mathbf{I}+\mathbf{B}_{21} \mathbf{Z}_{21}\bigr{)}^{-1}\bigl{(}\mathbf{B}_{22}-\mathbf{B}_{21}\mathbf{Z}_{22}\bigr{)}\\ S_{21}=\bigl{(}\mathbf{I}+\mathbf{B}_{21}\mathbf{Z}_{21}\bigr{)}^{-1}\mathbf{B}_{21}\\ S_{22}=\bigl{(}\mathbf{I}+\mathbf{B}_{21}\mathbf{Z}_{21}\bigr{)}^{-1}\bigl{(}\mathbf{B}_{22} -\mathbf{B}_{21}\mathbf{Z}_{22}\bigr{)}\end{aligned} \right. \tag{30}\]  
In the above formula, \(\mathbf{I}\) is the unit matrix; \(\mathbf{B}_{ij}(i,j=1,2)\) is the four-terminal parameters matrix of the passive isolated element, and for the ideal elastic element, \(\mathbf{B}_{11}=\mathbf{B}_{22}=\mathbf{I}\), \(\mathbf{B}_{12}=0\), \(\mathbf{B}_{21}=\mathrm{j}\omega\cdot\mathrm{diag}\bigl{[}1/K_{1},1/K_{2},\ldots,1/K _{N_{\rm I}}\bigr{]}\), \(K_{l}(l=1,2,\ldots,N_{\rm I})\) is the complex stiffness of the passive isolated element, \(\mathbf{K}=\mathrm{diag}\left[K_{1},K_{2},\ldots,K_{N_{\rm I}}\right]\) is the system stiffness matrix; \(\mathbf{Z}_{ij}(i,j=1,2)\) is the equivalent impedance of the active isolating element, and the following equations hold:  
\[\left\{ \begin{aligned} \mathbf{Z}_{11}=\mathbf{Z}_{21}=\frac{\mathbf{G}_{4}}{ \mathrm{j}\omega}+\mathbf{G}_{5}+\mathrm{j}\omega\mathbf{G}_{6}\\ \mathbf{Z}_{12}=\mathbf{Z}_{22}+\mathrm{j}\omega\mathbf{M}_{\rm a}\\ \mathbf{Z}_{22}=\frac{\mathbf{G}_{1}-\mathbf{G}_{6}}{\mathrm{j}\omega}+(\mathbf{G}_{2}-\mathbf{G} _{5})+\mathrm{j}\omega(\mathbf{G}_{3}-\mathbf{G}_{6})\end{aligned} \right. \tag{31}\]where \(M_{\rm a}={\rm diag}\big{[}m_{\rm a1},m_{\rm a2},m_{\rm a3}\ldots,m_{\rm aN_{1}} \big{]}\) is the diagonal matrix of the active actuator masses, and \(m_{\rm al}\) is the mass of the \(l\)th actuator; \(G_{1}\), \(G_{2}\), and \(G_{3}\) are the feedback parameter matrices of absolute displacement, velocity, and acceleration at the node where each isolation unit is coupled to the lower plate, respectively, and \(G_{4}\), \(G_{5}\), and \(G_{6}\) are the feedback parameter matrices of relative displacement, velocity, and acceleration which the above node relatives to the node where each isolation unit is coupled to the upper plate, respectively, and:  
\[\mathbf{G}_{k}=\left[