# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 2 Structure of the Combined LSCF-MLE-PMM
### The MLE-PMM (2 rd Step)
Solve the normal equations \[\text{Re}\left(J_{i}^{H}J_{i}\right)(\Delta\Theta_{i})=-\text{Re}\left(J_{i }^{H}E_{i}\right)\quad\text{for}\quad\Delta\Theta_{i}\] (11)
Compute an update of the previous solution  
\[\Theta_{i+1}=\Theta_{i}+\Delta\Theta_{i} \tag{12}\]  
with \(E_{i}=E(\Theta_{i})\) representing the equation error, \(J_{i}=\frac{\partial E(\Theta_{i})}{\partial\Theta_{i}}\) the Jacobian matrix and \(\Delta\Theta_{i}\) the perturbation on the parameters \(\Theta\) evaluated at the \(i\)th iteration. The error equation or so-called residual vector is given by  
\[E(\Theta_{i})=\left\{ \begin{array}{c}E_{1}(\Theta_{i})\\ E_{2}(\Theta_{i})\\ \vdots\\ E_{N_{o}N_{i}}(\Theta_{i})\end{array} \right\}\in\mathbb{C}^{N_{f}N_{o}N_{i} \times 1},\quad E_{k}(\Theta_{i})=\left\{ \begin{array}{c}\frac{\theta_{k}( \omega_{1},\Theta_{i})-H_{k}(\omega_{1})}{\sigma_{k_{i}(\omega_{1})}}\\ \frac{H_{k}(\omega_{2},\Theta_{i})-H_{k}(\omega_{1})}{\sigma_{k_{i}(\omega_{ 1})}}\\ \vdots\\ \frac{H_{k}(\omega_{N_{f}},\Theta_{i})-H_{k}(\omega_{N_{f}})}{\sigma_{k_{i}( \omega_{N_{f}})}}\end{array} \right\}\in\mathbb{C}^{N_{f}\times 1},\quad k=1,2, \cdots,N_{o}N_{i} \tag{13}\]  
and the Jacobian matrix by  
\[J= \begin{bmatrix}Y_{1}&0&\cdots&0&X_{1}\\ 0&Y_{2}&\cdots&0&X_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&Y_{N_{o}N_{i}}&X_{N_{o}N_{i}}\end{bmatrix}  \tag{14}\]  
with \(Y_{k}\in\mathbb{C}^{N_{f}\times 2(N_{m}+2)}\) containing the derivatives of the equation error (13) with respect to the real and imaginary parts of the \(k\)th element of the residue matrices \([Res]_{r}\), and to the real and imaginary parts of the \(k\)th element of the upper and lower residual terms, \([LR]\) and \([UR]\); and \(X_{k}\in\mathbb{C}^{N_{f}\times 2N_{m}}\) the derivatives with respect to the natural frequencies and damping ratios. The entries \(Y_{k}\) and \(X_{k}\) are computed, respectively, by  
\[Y_{k}= \begin{bmatrix}\frac{\partial E_{k}(\Theta)}{\partial\text{Re}([Res]_{ 1k})}&\cdots&\frac{\partial E_{k}(\Theta)}{\partial\text{Re}\big{(}[Res]_{N_{ m}\pm}\big{)}}&\frac{\partial E_{k}(\Theta)}{\partial\text{Re}([LR]_{k})}&\frac{ \partial E_{k}(\Theta)}{\partial\text{Re}([UR]_{k})}&\cdots\\ \cdots&\frac{\partial E_{k}(\[PERSON])}{\partial\text{Im}([Res]_{1k})}&\cdots& \frac{\partial E_{k}(\Theta)}{\partial\text{Im}\big{(}[Res]_{N_{m}\pm}\big{)} }&\frac{\partial E_{k}(\[PERSON])}{\partial\text{Im}([LR]_{k})}&\frac{\partial E _{k}(\[PERSON])}{\partial\text{Im}([UR]_{k})}\end{bmatrix}  \tag{15}\]  
and  
\[X_{k}= \begin{bmatrix}\frac{\partial E_{k}(\[PERSON])}{\partial f_{n_{1}}}&\frac {\partial E_{k}(\Theta)}{\partial f_{n_{2}}}&\cdots&\frac{\partial E_{k}( \Theta)}{\partial f_{n_{N_{m}}}}&\frac{\partial E_{k}(\Theta)}{\partial\xi_{ n_{1}}}&\frac{\partial E_{k}(\Theta)}{\partial\xi_{n_{2}}}&\cdots&\frac{\partial E_{k}( \Theta)}{\partial\xi_{n_{N_{m}}}}\end{bmatrix}  \tag{16}\]  
with the partial derivatives in each entry computed according the expressions found in Appendix C. By taking advantage of the block structure of the Jacobian matrix, the normal Equation (11) are rewritten as follows  
\[ \begin{bmatrix}R_{1}&0&\cdots&0&S_{1}\\ 0&R_{2}&\cdots&0&S_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&R_{N_{o}N_{i}}&S_{N_{o}N_{i}}\\ S_{1}{}^{T}&S_{2}{}^{T}&\cdots&S_{N_{o}N_{i}}{}^{T}&\sum_{k=1}^{N_{o}N_{i}}T_{ k}\end{bmatrix}\begin{Bmatrix}\Delta\theta_{1}\\ \Delta\theta_{2}\\ \vdots\\ \Delta\theta_{N_{o}N_{i}}\\ \Delta\theta_{\lambda}\end{Bmatrix} =- \begin{Bmatrix}\text{Re}\left(Y_{1}^{H} E_{1}\right)\\ \text{Re}\left(Y_{2}^{H}E_{2}\right)\\ \vdots\\ \text{Re}\left(Y_{N_{o}N_{i}}^{H}E_{N_{o}N_{i}}\right)\\ \sum_{k=1}^{N_{o}N_{i}}\text{Re}\left(X_{k}^{H}E_{k}\right)\end{Bmatrix}  \tag{17}\]  
with  
\[R_{k}=\text{Re}\left(Y_{k}^{H}Y_{k}\right)\in\mathbb{R}^{2(N_{m}+2)\times 2 (N_{m}+2)}\]  
\[S_{k}=\text{Re}\left(Y_{k}^{H}X_{k}\right)\in\mathbb{R}^{2(N_{m}+2)\times 2 N_{m}}\]  
\[T_{k}=\text{Re}\left(X_{k}^{H}X_{k}\right)\in\mathbb{R}^{2N_{m}\times 2N_{m}}\]