# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 1 Introduction  
Over the last years a lot of efforts have been spent in developing more accurate modal parameter estimation techniques. The outcome of these efforts are the development of precise and robust estimators, as, for instance, the poly-reference Least Squares Complex Frequency Domain (pLSCF) [1, 2] formulated in frequency domain and the Stochastic System Identification (SSI) algorithms developed to estimate the modal parameters from the measured responses [3]. Initially formulated for input-output systems and also known by its commercial name PolyMAX [2], the pLSCF was afterwards extended to output-only systems [4]. One of the main advantages of this technique is the possibility of creating clear stabilization diagrams, since the physical poles tend to stabilize faster over the identified model orders when compared to other Modal Parameter Estimation (MPE) algorithms, as, for instance, the poly-reference Least Squares Complex Exponential (pLSCE) (also known as LSCE-Prony) [5]. Later on, the capability of estimating confidence intervals for modal parameter estimates was added both to the LSCF [6] and pLSCF [7] estimators, as well as to the SSI [8, 9, 10] technique. Recently, new improvements were added to the estimation with the pLSCF algorithm to overpass the loss of precision and robustness observed when dealing with very noisy Frequency Response Functions (FRF)s and poorly excited modes [11; 12; 13]. It is verified that, under these circumstances, the algorithm is not so robust and tends to underestimate the damping ratios. This new variant of the algorithm consists of a 2-step approach which was developed to address these issues [14]. In fact, this approach consists of a combination of two different estimators, for example, the pLSCF and the poly-reference Maximum Likelihood Estimator in Modal Model formulation (pMLE-MM) [12; 14; 15; 16; 17] used in a final step of the identification process to increase the precision of the estimates provided by pLSCF as well as to compute the uncertainties bounds on the estimated modal parameters. The combination of these two estimators enhances the MPE by taking advantage of the specific features of each one of them, as for instance, the fast and clear stabilization diagrams provided by the pLSCF and the precision and numeric stability of the pMLE-MM to improve the accuracy of the identified modal parameters and estimate their confidence intervals. More recently, new improvements were added to the combined pLSCF-pMLE-MM by imposing identification constraints based both on the assumption of reciprocity of the identified modal model and of normal (real) mode shape vectors [18]. In this paper, a new ML Estimator (MLE) formulated in Pole-residue Modal Model (MLE-PMM) is proposed to improve the accuracy of the modal parameters estimated with the Least Squares Complex Frequency (LSCF) [1]. Similarly to the combined pLSCF-pMLE-MM, the idea of using the MLE-PMM in combination with the LSCF estimator is to retain and improve multi-reference information provided by the latter in a Non-linear LS sense (NLS) with the former. Similarly to the non-linear optimization technique described in Reference [19], the proposed ML-based approach is formulated with the modal model in pole-residue form. The innovative characteristics of the latter with regard to the former, however, are related to the fact that: (1) both the FRFs and the noise information are taken into account in the modal parameter estimation and, therefore, apart from providing the modal parameter estimates the technique also provides the uncertainties on the estimated modal parameters; (2) the initial values can be calculated by the LSCF estimator that gives good starting values in easy way by means of a clear stabilization chart and (3) the optimization is done by using an advanced technique that enlarges the convergence region and prevents the algorithm from converging to local minima. The paper is basically divided in three parts.