## 2 Theoretical Modeling
### Decentralized Feedback Controllers with Inertial Actuators  
Decentralized feedback controllers with \(S\) feedback loops are introduced to control the vibration and sound radiation of the orthogonally rib stiffened plate. The sketch of the decentralized system is shown in Figure 2. For each feedback loop, the velocity \(v_{i}\) sensed by the sensor is fed back to the controller with control gain \(h_{i}\). Then, the control output \(u_{i}(j\omega)\) is added to the inertial actuators, which produces a force and adds to the ribbed plate to control the vibration of this sensing point. From a practical point of view, the unit is a real time online control system, which means that the plate is first excited, the velocity sensor senses the velocity and feeds this signal into the controller, and the controller then works and sends the control signal to the actuator to control the vibration of the plate. The arrangement of all the feedback loops is shown in Figure 3.  
As for the inertial actuator which can be referenced in [14], Figure 4b shows a lumped parameter model of the concept inertial actuator presented in Figure 4a. Variables with the subscript p refer to the properties of the proof mass, its suspension, and displacement \(w_{p}\). Variables with subscript c refer to the properties of the casing mass, the fixative properties, and the displacement of the casing \(w_{c}\). The force acting between the two masses is due to the current in the coil. If the actuator is driven within its linear range by a constant current amplifier, this force is directly proportional to the control signal \(u(t)\). For the purpose of simulation, the constant of proportionality or force constant is taken to be 1 N/A such that \(f_{i}(t)=u_{i}(t)\). The point force acting on the ribbed plate due to the S identical inertial actuators mounted at the control location is given by [27]:  
\[\mathbf{f}_{S}(t)=k_{c}\left[\mathbf{w}(x_{s},y_{s},t)-\mathbf{w}_{c}(t)\right]+c_{c} \left[\dot{\mathbf{w}}(x_{s},y_{s},t)-\dot{\mathbf{w}}_{c}(t)\right], \tag{7}\]  
where \(\mathbf{w}(x_{s},y_{s},t)=\left[w_{1}(x_{s,1},y_{s,1},t),w_{2}(x_{s,2},y_{s,2},t), \ldots w_{S}(x_{s,S},y_{s,S},t)\right]^{T}\) is the vector of displacement on the locations of the inertial actuators, and \(k_{c}\) and \(c_{c}\) are the stiffness and damping of the casing. \(\mathbf{w}_{c}(t)\) is a column vector that describes displacements of the casing masses, which satisfies the differential equation as [27]. Figure 4: Schematic of the inertial actuator. (**a**) Schematic of a concept inertial actuator; (**b**) inertial actuator model. Figure 3: The arrangement of the feedback loops. Figure 2: The sketch of the decentralized system. \[\ddot{\mathbf{w}}_{c}(t)=-\frac{k_{p}}{m_{c}}\Big{[}\mathbf{w}_{c}(t)-\mathbf{w}_{p}(t)\Big{]}+ \frac{c_{p}}{m_{c}}\Big{[}\dot{\mathbf{w}}_{c}(t)-\dot{\mathbf{w}}_{p}(t)\Big{]}+\frac{1} {m_{c}}\mathbf{u}(t)-\frac{k_{c}}{m_{c}}\Big{[}\dot{\mathbf{w}}_{c}(t)-\dot{\mathbf{w}}(t) \Big{]}, \tag{8}\]  
where \(m_{c}\) is the mass of the casing, \(k_{p}\) and \(c_{p}\) are the stiffness and damping of the proof mass suspension. \(\mathbf{u}(t)\) is the vector of control force, \(\mathbf{u}(t)=\left[u_{1}(t),u_{2}(t),\dots,u_{S}(t)\right]^{T}\). \(\mathbf{w}_{p}(t)\) is a vector of proof mass displacement, which satisfies the following equation [27]:  
\[\ddot{\mathbf{w}}_{p}(t)=-\frac{k_{p}}{m_{p}}\Big{[}\mathbf{w}_{p}(t)-\mathbf{w}_{c}(t) \Big{]}-\frac{c_{p}}{m_{p}}\Big{[}\dot{\mathbf{w}}_{p}(t)-\dot{\mathbf{w}}_{c}(t) \Big{]}-\frac{1}{m_{p}}\mathbf{u}(t), \tag{9}\]  
where \(m_{p}\) is mass of the proof mass. Equations (8) to (10) can be transformed to frequency domain. Then, combining Equations (9) and (10), the displacements of the casing and the proof mass for the S identical inertial actuator \(\mathbf{w}_{c}(t)\) and \(\mathbf{w}_{p}(t)\) satisfy the following matrix equation:  
\[\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]=\left[