## 2 System Modeling and Control Structure
### Proposed Control Scheme  
The general control scheme is depicted in Figure 1, illustrating two primary components. The first component encompasses the dynamics of the flexible structure, represented by \(\mathbf{G}_{m}(s)\) (see Equations (1)-(3)). The second component constitutes the AVC system, denoted as \(\mathbf{K}_{m}(s)\). This therm comprises the shakers' dynamics, denoted as \(\mathbf{G}_{A}(s)\) (see Equation (5)), and the controller dynamics, denoted as \(\mathbf{K}(s)\). In addition, there are the output of the shakers (\(\mathbf{U}(s)\)), the perturbations (\(\mathbf{U}(s)\)), which in this work will be the shakers (open loop) and a large-sled impulse hammer (closed loop), the accelerometers' outputs (\(\mathbf{Y}(s)\)), the control voltage of the shaker amplifiers (\(\mathbf{V}(s)\)), and the control reference (\(\mathbf{R}\)), which in this application is zero. This representation delineates the interplay between the structure's inherent dynamics and the active vibration control system, emphasizing their integral roles in mitigating vibrations and enhancing structural performance.  
The MIMO AVC system, represented by the transfer function \(\mathbf{G_{m}}(s)\), embodies multiple IRC units. The IRC controller, initially introduced in [39], stands out as a straightforward, robust, and highly effective method for managing vibrations in intelligent structures equipped with sensors and actuators collocated in proximity. Renowned for its versatility, this controller has found wide application across diverse domains, showcasing its prowess in real-world scenarios such as damping oscillations in automobiles [40] and controlling vibrations in wind turbines [41].  
Originating from its single-input single-output (SISO) variant proposed for human-induced vibrations in [34], the IRC framework has evolved to accommodate the complexities of MIMO systems. In the pioneering work of [36], the adaptation to the MIMO domain was introduced, where adjustments were made to incorporate direct feed-through elements into the collocated system, thus furnishing a comprehensive MIMO version of the IRC controller. This extension not only broadened the applicability of the IRC technique, but also enhanced its efficacy in addressing complex multivariable vibration control challenges, marking a significant advancement in the field. This extension was configured in the MIMO version as the following \(q\times q\) diagonal matrix:  
\[\mathbf{K}(s)=\begin{bmatrix}K_{1}(s)&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&K_{q}(s)\end{bmatrix}, \tag{6}\]  
where \(K_{i}(s)=\gamma_{i}/(s-d_{i}\gamma_{i})\) is the IRC configured for node \(i\). In this specific instance of MIMO IRC, it is important to note that only the acceleration measured at the node where the actuator is positioned is considered. Similar to the SISO case [34], controller stability can be achieved as the phase of each transfer function \(K_{i}(s)G_{A,i}(s)\) begins at 180 degrees. Notably, the actuator features two zeros at the origin, steadily decreasing until reaching \(-180\) degrees. It is worth mentioning that a forthcoming study will address the formal demonstration of stability. This observation underscores the robustness of the MIMO IRC approach in effectively mitigating vibrations and enhancing the structural performance, thereby laying the groundwork for further advancements and insights in the field of active vibration control.  
With the theoretical control design explained, the next section presents the detailed design of the MIMO IRC scheme within the common framework established. The employed performance indices chosen to quantify closed-loop performance, as well as the control parameter optimization results are also presented.