## 2 System Modeling and Control Structure
### Parametric System Model  
The standard state-space representation for a typical flexible structure, which comprises \(n\) vibration modes and \(p\) nodes (i.e., possible locations for actuators/sensors and perturbations), is given by:  
\[\dot{\mathbf{x}}_{s}= \mathbf{A}_{s}\mathbf{x}_{s}+\mathbf{B}_{s_{u}}\mathbf{u}+\mathbf{ B}_{s_{w}}\mathbf{w}, \tag{1}\] \[\mathbf{y}= \mathbf{C}_{s}\mathbf{x}_{s}+\mathbf{D}_{s_{u}}\mathbf{u}+\mathbf{ D}_{s_{w}}\mathbf{w},\]  
where \(\mathbf{u}\) represents the control input vector (i.e., the force imparted by the actuators), \(\mathbf{w}\) denotes the perturbation forces, and \(\mathbf{y}\) is the output vector (sensor measurements, typically acceleration). The matrices \(\mathbf{A}_{s}\), \(\mathbf{B}_{s_{u}}\), \(\mathbf{B}_{s_{u}}\), \(\mathbf{C}_{s}\), \(\mathbf{D}_{s_{u}}\), and \(\mathbf{D}_{s_{u}}\) define the state-space model of the structure. When (1) is expressed in modal coordinates, these state-space matrices can be explicitly written as follows [37]:  
\[\begin{split}\mathbf{A}_{s}=\begin{bmatrix}\mathbf{0}& \mathbf{I}\\ -\mathbf{\Omega}^{2}&-2\mathbf{Z}\mathbf{\Omega}\end{bmatrix},\ \mathbf{B}_{s_{u}}=\mathbf{B}_{s_{u}}=\begin{bmatrix} \mathbf{0}\\ \mathbf{\Phi}\end{bmatrix},\\ \mathbf{C}_{s}=\begin{bmatrix}-\mathbf{\Omega}^{2}\mathbf{\Phi}^{T}&-2 \mathbf{Z}\mathbf{\Omega}\mathbf{\Phi}^{T}\end{bmatrix},\mathbf{D}_{s_{u}}= \mathbf{D}_{s_{u}}=\mathbf{\Phi}\mathbf{\Phi}^{T},\end{split} \tag{2}\]  
where \(\mathbf{\Omega}\) and \(\mathbf{Z}\) are diagonal matrices formed by the natural frequencies \(\big{(}[\omega_{1},\cdots,\omega_{n}]\big{)}\) and damping ratios \(\big{(}[\xi_{1},\cdots,\xi_{n}]\big{)}\), respectively, and \(\mathbf{\Phi}\) is an \(n\times p\) matrix. Note that each \(k\)th column is formed by the \(k\)th vibration mode shape. The state vector is defined as \(\mathbf{x}_{s}=[x_{s_{1}},\cdots,x_{s_{n}},\hat{x}_{s_{1}},\cdots,\hat{x}_{s_{ n}}]\), where \([x_{s_{1}},\cdots,x_{s_{n}}]\) are the modal coordinates of the structure and \([\hat{x}_{s_{1}},\cdots,\hat{x}_{s_{n}}]\) are their derivatives.  
The time domain system of Equations (1) and (2) can be transformed into a frequency domain transfer function matrix relationship, given by:  
\[\mathbf{Y}(s)=\mathbf{G}_{m}(s)\big{(}\mathbf{U}(s)+\mathbf{W}(s)\big{)}, \tag{3}\]  
where \(\mathbf{G}_{m}(s)=\Big{[}\mathbf{C}_{s}(s\mathbf{I}-\mathbf{A}_{s})^{-1} \mathbf{B}_{s_{u}}+\mathbf{D}_{s_{u}}\Big{]}\).  
In this work, inertial mass actuators that generate force by accelerating an inertial mass are utilized. These actuators consist of an inertial (or moving) mass attached to a current-carrying coil, moving in a magnetic field. If \(q\) actuators are used, the \(q\times q\) matrix of the actuator linear part can be written as:  
\[\mathbf{G}_{A}(s)=\begin{bmatrix}G_{A,1}(s)&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&G_{A,q}(s)\end{bmatrix}, \tag{4}\]  
where each transfer function \(G_{A,i}(s)\) is defined as:  
\[G_{A,i}(s)=\frac{g_{A,i}s^{2}}{(s^{2}+2\xi_{A,i}\omega_{A,i}s+\omega_{A,i}^{2} )}\frac{\epsilon}{(s+\epsilon)}. \tag{5}\]  
Here, \(g_{A,i}>0\) is the actuator gain parameter, \(\xi_{A,i}\) is the damping ratio, \(\omega_{A,i}\) is the natural frequency of the actuator placed at the \(i^{th}\) node, and \(\epsilon\) models the amplifier dynamics in voltage mode (see [38] for more information).  
With the system model in place, the next section will present the detailed control design of the MIMO IRC scheme employed to demonstrate the superior damping of pedestrian-induced vibrations on a lab-scale civil structure.