## 2 Theoretical Modeling Approach  
The control system is schematically shown in Figure 1. The upper ribbed plate A and lower ribbed plate B are orthogonally rib stiffened plates with one rib along the horizontal and another along the vertical directions. The rib parallel with \(x\)-axis is the horizontal rib, and the one parallel with \(y\)-axis is the vertical rib. For the upper ribbed plate A, the vertical and horizontal ribs are located at \(x=x_{a}\) and \(y=y_{a}\), respectively. The lower ribbed plate B has the vertical rib located at \(x=x_{b}\) and the horizontal rib located at \(y=y_{b}\). The boundary conditions of the base plate and the ribs are all clamped. The remaining parts of the system (the rigid walls of four sides, and the control source) are basically the same as that of the single ribbed case [8]. The upper plate is excited by primary excitation of an oblique incident plane wave. A controllable point source is applied to control the sound field of the cavity, which indirectly suppresses the sound transmission.  
The beam/plate interface is modeled as a nonslip line connection, in which a pair of coupling force \(F\) and moment \(M_{m}\) is considered to simulate the beam/plate coupling effects, as shown in Figure 1b. Under the excitation of an oblique incident plane wave [22] and the coupling effects of the ribs, the equations that the transverse bending displacements of the base plate A and B (\(w_{a}\), \(w_{b}\)) satisfy are similar to that of the single ribbed case [8]. However, another coupling force and moment induced by the horizontal rib should be added into the right side of the equations [8]. Taking the ribbed plate A as an example, the equation that the bending displacements \(w_{a}\) of the base plate A satisfies can be expressed as:  
\[\begin{split} D_{a}\
abla^{4}w_{a}+\rho_{a}h_{a}\frac{\partial^{2 }w_{a}}{\partial t^{2}}=f_{p}(\mathbf{r},t)+F_{a,p}\delta(x-x_{a})+M_{m,a,p} \delta^{\prime}(x-x_{a})\\ +F_{a,h}\delta(y-y_{a})+M_{m,a,h}\delta^{\prime}(y-y_{a})-p( \mathbf{r},t)\end{split} \tag{1}\]  
where \(\
abla^{4}\) denotes the Laplacian, \(D_{a}\) are the flexural rigidity of the base plates, \(D_{a}=E_{a}h_{a}{}^{3}/12(1-\sigma_{a}^{2})\). \(E_{a}\), \(h_{a}\) and \(\sigma_{a}\) are the Young's modulus, thickness and Poisson's ratio of the base plate, respectively. \(\rho_{a}\) is the density the base plates. \(F_{a,v}\) and \(M_{m,a,v}\) are the coupling force and moment for the vertical rib. \(F_{a,h}\) and \(M_{m,a,h}\) are the coupling force and moment for the horizontal rib. \(f_{p}(\mathbf{r},t)\) is the primary excitation of the oblique incident plane wave, and \(\mathbf{r}=(x,y,z)\) is the random spatial position. \(p(\mathbf{r},t)\) is the sound pressure in the air cavity.  
The sound pressure \(p(\mathbf{r},t)\) in the cavity also satisfies the inhomogeneous wave equation when a point source with controllable amplitude \(Q_{s}\) is introduced, which is presented in Ref. [8]. By solving the coupled equations of the system, the modal amplitude vectors of the base plate A and B (\(\mathbf{q}_{a}\),\(\mathbf{q}_{b}\)), and the cavity (\(\mathbf{P}\)) can be calculated. Then, the systematic response can be obtained. It is noteworthy that variable \(Q_{s}\) remains undetermined in these expressions of the modal amplitude. The system response before control can be obtained when \(Q_{s,a}(\omega)=0\). For the controlled condition, the response continually depends on the variable \(Q_{s,a}(\omega)\). The optimal amplitude \(Q_{s,a,opt}(\omega)\) can be obtained by minimizing the objective function of the radiated power of the ribbed plate B. The detailed derivation  
Figure 1: Sketch of the model. (**a**) Active sound insulation double-panel system. (**b**) Sketch of coupling effects in connection line between the base plate and the rib.  
process is similar to Ref. [8], where the coupling force and moment induced by the horizontal rib is added into the relative vectors and matrices.