## 2 Methodology
### Formulation for Modal Parameter Identification
\right], \tag{19}\]  
\[\mathbf{\hat{Y}}_{\bar{l}}=\left[\mathbf{\overline{Y}}_{\bar{l}}\ \ \ \mathbf{\overline{Y}}_{\bar{l}+1}\ \ \ \cdots\ \ \mathbf{\overline{Y}}_{\bar{l}+N-1}\right], \tag{20}\]  
\[\mathbf{\hat{Z}}_{\bar{l}}=\left[\mathbf{Z}_{\bar{l}}\ \ \ \mathbf{Z}_{\bar{l}+1}\ \ \ \cdots\ \ \ \mathbf{Z}_{\bar{l}+N-1}\right], \tag{21}\]\[\mathbf{F}_{\overline{l}}=\left[\overline{\mathbf{F}}_{\overline{l}}\quad\overline{ \mathbf{F}}_{\overline{l}+1}\quad\cdots\quad\overline{\mathbf{F}}_{\overline{l }+N-1}\right], \tag{22}\]  
\[\overline{\mathbf{Y}}_{\overline{l}}=\left(\mathbf{Y}_{m,\overline{l}}^{ \overline{l}}\quad\mathbf{Y}_{m,\overline{l},\overline{l}+1}^{\overline{l}} \quad\cdots\quad\mathbf{Y}_{m,\overline{l},\overline{l}+s-1}^{\overline{l}} \right)^{\overline{l}}, \tag{23}\]  
\[\overline{\mathbf{F}}_{\overline{l}}=\left(\mathbf{F}_{m,\overline{l}}^{ \overline{l}}\quad\mathbf{F}_{m,\overline{l},\overline{l}+1}^{\overline{l}} \quad\cdots\quad\mathbf{F}_{m,\overline{l},\overline{l}+s-1}^{\overline{l}} \right)^{\overline{l}}. \tag{24}\]  
To deal with the time-varying system, the present approach introduced a weighting function, as expressed in Equation (25), to precisely identify the modal parameters corresponding to the midpoint of the time window \(t\in\left[\overline{l}\Delta t,\ \left(\overline{l}+2L\right)\Delta t\right]\) :  
\[w(\widetilde{l})=e^{-\left(\left(l_{m}-\overline{l}\right)\Delta t/d\right)^{ 2}}, \tag{25}\]  
where \(l_{m}\) is the middle point of \(\left[\overline{l},\ \overline{l}+2L\right],\overline{l}\in\left[\overline{l},\ \overline{l}+2L\right],\) and \(d\) is a specified parameter. The following are now defined:  
\[\widetilde{\mathbf{Y}}_{\overline{l}}=\left[w(\overline{l})\overline{\mathbf{ Y}}_{\overline{l}}\quad w(\overline{l}+1)\overline{\mathbf{Y}}_{\overline{l}+1 }\quad\cdots\quad w(\overline{l}+N-1)\overline{\mathbf{Y}}_{\overline{l}+N-1 }\right], \tag{26}\]  
\[\widetilde{\mathbf{Z}}_{\overline{l}}=\left[w(\overline{l})\mathbf{Z}_{ \overline{l}}\quad w(\overline{l}+1)\mathbf{Z}_{\overline{l}+1}\quad\cdots \quad w(\overline{l}+N-1)\mathbf{Z}_{\overline{l}+N-1}\right], \tag{27}\]  
\[\widetilde{\mathbf{F}}_{\overline{l}}=\left[w(\overline{l})\overline{\mathbf{ F}}_{\overline{l}}\quad w(\overline{l}+1)\overline{\mathbf{F}}_{\overline{l}+1 }\quad\cdots\quad w(\overline{l}+N-1)\overline{\mathbf{F}}_{\overline{l}+N-1 }\right]. \tag{28}\]  
Equation (29) can be readily derived from Equation (17) through elementary matrix multiplication operations. \[\widetilde{\mathbf{Y}}_{\overline{l}}=\mathbf{\Gamma}_{s}\widetilde{\mathbf{ Z}}_{\overline{l}}+\mathbf{H}_{s}\widetilde{\mathbf{F}}_{\overline{l}}. \tag{29}\]  
The measured \(\overline{\mathbf{Y}}_{\overline{l}}\) and \(\widetilde{\mathbf{F}}_{\overline{l}}\) carry greater weights in formulating Equation (29) as \(\overline{l}\) approaches the midpoint of the considered window. Multiplying Equation (29) by an orthogonal projection matrix \(\mathbf{O}^{\perp}=\mathbf{I}-\widetilde{\mathbf{F}}_{\overline{l}}^{ \overline{l}}\left(\widetilde{\mathbf{F}}_{\overline{l}}\widetilde{\mathbf{F}}_ {\overline{l}}\right)^{-1}\widetilde{\mathbf{F}}_{\overline{l}}\) onto the null space of \(\widetilde{\mathbf{F}}_{\overline{l}}\) leads to  
\[\mathbf{\hat{Z}}_{\bar{l}}=\left[\mathbf{Z}_{\bar{l}}\ \ \ \mathbf{Z}_{\bar{l}+1}\ \ \ \cdots\ \ \ \mathbf{Z}_{\bar{l}+N-1}\right], \tag{21}\]0  
To mitigate the impact of noise on the estimation of coefficient matrices in Equations (4) and (6), instrumental variables \(\mathbf{P}\)[34] and a weighting matrix \(\mathbf{W}_{c}\)[53] are introduced and applied to Equation (30), leading to the following equation:  
\[\mathbf{\hat{Z}}_{\bar{l}}=\left[\mathbf{Z}_{\bar{l}}\ \ \ \mathbf{Z}_{\bar{l}+1}\ \ \ \cdots\ \ \ \mathbf{Z}_{\bar{l}+N-1}\right], \tag{21}\]1  
where  
\[\mathbf{P}=\left[ \begin{matrix}\widetilde{\mathbf{F}}_{p}\\ \widetilde{\mathbf{Y}}_{p}\end{matrix}