## 3 Algebraic and Online Modal Parameters Identification
 \tag{12}\]  
where the vector \(\Theta=[\begin{matrix}a_{2n_{def}-1}\cdots\hat{a}_{1}&\hat{a}_{0}\end{bmatrix} ^{T}\) contains the estimated coefficients of the characteristic polynomial and \(\Delta\) is the determinant of the matrix \(G\) and \(\Delta_{i}\), which are partial determinants used for the Cramer's rule. The matrix \(G\in R^{2n_{def}\times 2n_{def}}\) and the vector \(V\in R^{2n_{def}}\) are given by  
\[G=\left[ \begin{matrix}g_{11}&g_{12}&\cdots&g_{12n_{def}}\\ \xi_{21}&g_{22}&\cdots&g_{22n_{def}}\\ \vdots&&\ddots\\ g_{2n_{def}1}&g_{n_{def}2}&\cdots&g_{2n_{def}2n_{def}}\end{matrix} \right],V= \left[ \begin{matrix}v_{1}\\ v_{2}\\ \vdots\\ v_{2n_{def}}\end{matrix} \right] \tag{13}\]with entries defined by  
\[g_{11} = (-1)^{2n_{def}}\int_{t_{0}}^{2n_{def}}(\Delta t)^{(2n_{def})}x_{i}(t)\] \[g_{12} = \sum_{k=0}^{1}(-1)^{2n_{def}-k}\frac{(2n_{def})!(1)!}{k!(2n_{def}- k)!(1-k)!}\int_{t_{0}}^{(2n_{def}-1+k)}(\Delta t)^{2n_{def}-k}x_{i}(t)\] \[\vdots = \vdots \tag{14}\] \[g_{1,2n_{def}-1} = \sum_{k=0}^{2n_{def}-2}\beta_{k}\int_{t_{0}}^{(2+k)}(\Delta t)^{( 2n_{def}-k)}x_{i}(t)\] \[g_{1,2n_{def}} = \sum_{k=0}^{2n_{def}-1}\gamma_{k}\int_{t_{0}}^{(1+k)}(\Delta t)^{( 2n_{def}-k)}x_{i}(t)\]  
where \(x_{i}(t)\) is a single available output measurement. Notice that none of Equation (15)'s expressions depend on the initial conditions of the considered output \(x_{i}(t)\). In (15), \(\Delta t=t-t_{0}\), and the notation \(\int_{t_{0}}^{(k)}\varphi(t)\) represents iterated integrations of the form  
\[\int_{t_{0}}^{t}\int_{t_{0}}^{\tau_{1}}\cdots\int_{t_{0}}^{\tau_{k-1}}\varphi( \tau_{k})d\tau_{k}\cdots d\tau_{1} \tag{15}\]  
where \(k\) is a positive integer. The constants \(\beta_{k}\), \(\gamma_{k}\) used in (15) are defined as  
\[\beta_{k}=(-1)^{2n_{def}-k}\frac{(2n_{def})!(2n_{def}-2)!}{k!(2n_{def}-k)!(2n_{ def}-2-k)!} \tag{16}\]  
\[\gamma_{k}=(-1)^{2n_{def}}\frac{(2n_{def})!(2n_{def}-1)!}{k!(2n_{def}-k)!(2n_{ def}-1-k)!}\]  
Finally, the first element \(v_{1}\) of the vector \(V=[v_{1}\;\;v_{2}\cdots v_{n_{def}}]\), defined in Equation (13), is obtained as follows  
\[v_{1}=-\sum_{k=0}^{2n_{def}-1}\lambda_{k}\int_{t_{0}}^{(k)}(\Delta t)^{(2n_{ def}-k)}x_{i}(t) \tag{17}\]  
where the constants \(\lambda_{k}\) are calculated by using the expression  
\[\lambda_{k}=(-1)^{2n_{def}-k}\frac{(2n_{def})!(2n_{def})!}{k!(2n_{def}-k)!(2n_{ def}-k)!} \tag{18}\]  
The iterated integrations of Equations (15) and (17) lead to the rest of the entries or components of the matrix \(G\) and the vector \(V\)  
\[g_{12} = \sum_{k=0}^{1}(-1)^{2n_{def}-k}\frac{(2n_{def})!(1)!}{k!(2n_{def}- k)!(1-k)!}\int_{t_{0}}^{(2n_{def}-1+k)}(\Delta t)^{2n_{def}-k}x_{i}(t)\]0  
with \(i=2,\ldots,2n_{def}\) and \(j=1,\ldots,2n_{def}\). It is possible to estimate the desired coefficients by using Cramer's rule. The iterated integration \(m\) times of the numerator and denominator leads to smooth estimations with the consequent slower convergence of the calculations. Let us rewrite Equation (12) with no changes in the final result as  
\[e^{-\gamma(t-t_{0})}|\Delta|\hat{a}_{k}=e^{-\gamma(t-t_{0})}\left[ \begin{array} []{c}|\Delta_{1}|\\ |\Delta_{2}|\\ \vdots\\ |\Delta_{n}|\end{array} \right] \tag{20}\]\[\hat{a}_{k}=\frac{e^{-\gamma(t-t_{0})}\int_{t_{0}}^{(m)}|\Delta_{k-1}|}{e^{-\gamma( t-t_{0})}\int_{t_{0}}^{(m)}|\Delta|},\quad k=1,2,\cdots,2n_{dof}-1 \tag{21}\]  
where \(|\cdot|\) denotes absolute value, \(m\) is an integer, \(\hat{\cdot}\) denotes estimated, and \(\gamma_{1}>0\) is a low-pass filter coefficient used to smooth the online estimations [28]. Thus, from the algebraic estimation of the coefficients \(\hat{a}_{k}\), it is possible to obtain the components of the characteristic polynomial and, hence the natural frequencies and damping ratios, as reported in [26]  
\[\hat{r}_{i}=\xi_{i}+j\hat{\omega}_{di},\;\;\;\hat{r}_{i}^{*}=\xi_{i}-j\hat{ \omega}_{di},\quad i=1,2,\cdots,n_{dof} \tag{22}\]  
here, \(\hat{\zeta}_{i}\) and \(\hat{\omega}_{di}\) are the estimates of the damping factors and damped natural frequencies, respectively.