# 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)  
Once the poles are estimated with the LSCF technique and the modal residues with the LSFD algorithm, these parameters can be optimized by means of the ML estimator formulated using the Modal Model in Pole-residue form. The idea behind the MLE-PMM algorithm is to optimize the initial modal parameters given, for instance, by the LSCF and the LSFD estimators by adjusting these values so that FRF equation in Pole-residue Modal Model (4) is fitted to the measured FRF. Compared to the LS-based approaches with uncertainty estimation capabilities, the ML-based algorithms have the advantage of taking into account both the measured FRFs and the noise information at once during the parametric identification process and thus, apart from providing the more accurate estimates for the modal parameters, they also allows for computation of confidence intervals for these estimates directly from the Fisher information matrix. Furthermore, the ML approach herein introduced is formulated in continuous-time with the Pole-residue Modal Model, meaning that it does not suffer from numerical ill-conditioning which is a well-known identification problem of the rational fraction polynomial models in Laplace domain. The optimization of the starting parameters with the MLE-PMM is accomplished by minimizing the following (negative) log-like cost function  
\[l(\Theta)_{\text{ML-PMM}}=\sum_{k=1}^{N_{o}N_{i}}\sum_{f=1}^{N_{f}}\left|E_{k} (\Theta,\omega_{f})\right|^{2} \tag{6}\]  
with \(E_{k}(\Theta,\omega_{f})\) denoting the equation error given as follows:  
\[E_{k}(\Theta,\omega_{f})=\frac{\left|\hat{H}_{k}(\Theta,\omega_{f})-H_{k}( \omega_{f})\right|}{\sigma_{H_{k}(\omega_{f})}} \tag{7}\]  
where \(H_{k}(\omega_{f})\) and \(\sigma_{H_{k}(\omega_{f})}\) are, respectively, the measured FRF and its standard deviation and \(H_{k}(\Theta,\omega_{f})\) is the modelled FRF; \(k\) (\(k=1,\,2,\cdots,\,N_{o}N_{i}\)) denotes any of the \(N_{o}N_{i}\) elements of the measured and estimated FRF matrices; \(N_{f}\) represents the number of frequency lines and \(\omega_{f}=2\pi f\) the angular frequency evaluated at frequency line \(f\). The parameter \(\Theta\) is a column vector containing the invariants of the pole residue model to be optimized by the ML algorithm and is given as follows  
\[\Theta=\left[\theta_{1}\quad\theta_{2}\quad\cdots\quad\theta_{N_{o}N_{i}} \quad\theta_{\lambda}{}^{T}\right]\in\mathbb{R}^{1\times(2N_{m}+4)(N_{o}N_{i} )+2N_{m}} \tag{8}\]  
with  
\[\theta_{k}= \begin{bmatrix}\text{Re}\left([Res]_{1k}\right)&\cdots&\text{Re} \left([Res]_{Nu_{k}}\right)&\text{Re}\left([LR]_{k}\right)&\text{Re}\left([UR] _{k}\right)&\cdots\\ \cdots&\text{Im}\left([Res]_{1k}\right)&\cdots&\text{Im}\left([Res]_{N_{m}k} \right)&\text{Im}\left([LR]_{k}\right)&\text{Im}\left([UR]_{k}\right)\end{bmatrix}  \in\mathbb{R}^{1\times(2N_{m}+4)} \tag{9}\]  
and  
\[\theta_{\lambda}= \begin{bmatrix}f_{n_{1}}&\cdots&f_{n_{N_{m}}}&\xi_{n_{1}}& \cdots&\xi_{n_{N_{m}}}\end{bmatrix} \in\mathbb{R}^{1\times 2N_{m}} \tag{10}\]  
The ML estimates of the invariants of the modal model is given by minimizing the cost function (6) with respect to parameter \(\Theta\). Similarly to the MLE formulated in Common Denominator Model (MLE-CDM) described in Reference [21], this minimization is accomplished by means of a Gauss-Newton optimization algorithm combined with Levenberg-Marquardt approach to guarantee a continuous reduction of the cost function over the performed iterations. In case of real coefficients, each Gauss-Newton iteration is performed in two steps:  
1.