# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 4 Conclusions  
In this paper, a novel ML-based estimator formulated in pole-residue modal model is proposed. Just like any other ML-based approach, the MLE-PMM introduced in this paper requires a good starting guess to assure convergence over the performed Gauss-Newton iterations. Despite the fact that any LS-based identification technique can be used to provide the starting guess, the proposed ML approach is combined with the LSCF estimator to retain the multi-reference information and improve the accuracy of the LS estimates. The idea of combining the ML-PMM with LSCF estimator aims at taking advantage of the particular features of each estimators, that is, (i) the fast modal parameter estimation obtained with the latter; and (ii) the numerical stability of the former. It was verified that, when combined with the LSCF estimator, the MLE-PMM is capable of improving the accuracy of the LSCF estimates and providing confidence intervals for the optimized estimates.  
The main advantage of the proposed MLE-PMM with regards to the MLE-MM is that, due to the reduced dimension of the system matrices, it is significantly less time consuming, as demonstrated in the first application example presented in Section 3.1. Another distinguishing feature of the derived ML approach is that, due to the parametrization used in its formulation, the variances of the natural frequencies and damping coefficients are estimated directly from the Jacobian matrix. Therefore, the use of additional linearization formulas is avoided if one is only interested on these variances. The main disadvantages, however, are in fact, related to the limitations of modal model in pole residue form. These disadvantages include: (1) difficulty of distinguishing between close spaced modes; (2) difficulty in accurately synthesizing the FRF matrix once the mode shapes and modal participation factor vectors are estimated from the modal residues by means of the singular value decomposition; and (3) difficulty in fully optimizing the LSCF modal parameter estimates used as starting guess in case of closely spaced modes.  
When dealing with reasonably or well separated modes, however, it is observed that MLE-PMM performs very well when compared to it modal model based counterpart. Two application examples were used to validate the proposed ML-based estimator. The first consisted of well separated whereas the second of closely spaced modes. From the simulations with former, it was demonstrated the ability of the MLE-PMM technique to significantly reduce the bias error of the LSCF estimates and predict the uncertainties for the optimized estimates. This is demonstrated by the results presented in Tables 4 and 5. From the simulations with the latter, it was verified (as shown in Tables 8 and 9) that, even in case of closely spaced modes, the proposed MLE-PMM was capable of reducing the bias error of the LSCF estimates and providing reliable predictions for the confidence intervals of the optimized estimates.  
**Acknowledgments:** The authors acknowledge: (1) all the support received from Centre for Oil and Gas of the Technical University of Denmark (DTU)/Danish Hydrocarbon Research and Technology Centre (DHRTC) to the first author; (2) the support provided by the Faculty of Engineering of University of Porto (FEUP); and (3) the financial support provided by the family of the first author, without which it would not have been possible to derive the ML-based approach presented in this paper.  
**Conflicts of Interest:** The authors declare no conflicts of interest.