{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T02:37:46Z","timestamp":1776998266010,"version":"3.51.4"},"reference-count":30,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>Recently, a lot of efforts have been devoted to developing more precise Modal Parameter Estimation (MPE) techniques. This is explained by the necessity in civil, mechanical and aerospace engineering of obtaining accurate estimates for the modal parameters of the tested structures, as well as of determining reliable confidence intervals for these estimates. The Non-linear Least Squares (NLS) identification techniques based on Maximum Likelihood (ML) have been increasingly used in modal analysis to improve precision of estimates provided by the Least Squares (LS) based estimators when they are not accurate enough. Apart from providing more accurate estimates, the main advantage of the ML estimators, with regard to their LS counterparts, is that they allow for taking into account not only the measured Frequency Response Functions (FRFs) but also the noise information during the parametric identification process and, therefore, provide the modal parameters estimates together with their uncertainties bounds. In this paper, a new derivation of a Maximum Likelihood Estimator formulated in Pole-residue Modal Model (MLE-PMM) is presented. The proposed formulation is meant to be used in combination with the Least Squares Frequency Domain (LSCF) to improve the precision of the modal parameter estimates and compute their confidence intervals. Aiming at demonstrating the efficiency of the proposed approach, it is applied to two simulated examples in the final part of the paper.<\/jats:p>","DOI":"10.3390\/app9153120","type":"journal-article","created":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T11:39:37Z","timestamp":1564659577000},"page":"3120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model"],"prefix":"10.3390","volume":"9","author":[{"given":"Sandro","family":"Amador","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, Technical University of Denmark (DTU), Building 118, 2800 Kgs. Lyngby, Denmark"}]},{"given":"Mahmoud","family":"El-Kafafy","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussels, Belgium"},{"name":"Department of Mechanical Design, Helwan University, 11975 Cairo, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7878-6224","authenticated-orcid":false,"given":"\u00c1lvaro","family":"Cunha","sequence":"additional","affiliation":[{"name":"ViBest, Faculty of Engineering, University of Porto (FEUP), R. Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"given":"Rune","family":"Brincker","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Technical University of Denmark (DTU), Building 118, 2800 Kgs. Lyngby, Denmark"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,1]]},"reference":[{"key":"ref_1","unstructured":"Guillaume, P., Verboven, P., Vanlanduit, S., Van-der-Auweraer, H., and Peeters, B. (2003, January 3\u20136). A Poly-reference Implementation of the Least-squares Complex Frequency-domain Estimator. Proceedings of the 21st International Modal Analysis Conference (IMACXXI), Kissimmee, FL, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1155\/2004\/523692","article-title":"The PolyMAX frequency-domain method: A new standard for modal parameter estimation?","volume":"1","author":"Peeters","year":"2004","journal-title":"Shock Vib."},{"key":"ref_3","unstructured":"Peeters, B. (2000). System Identification and Damage Detection in Civil Engineering. [Ph.D. Thesis, Katholieke Universiteit Leuven]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1155\/2007\/531739","article-title":"Operational modal analysis for estimating the dynamic properties of a stadium structure during a football game","volume":"14","author":"Peeters","year":"2007","journal-title":"Shock Vib."},{"key":"ref_5","unstructured":"Maia, N., He, L., Lin, S., and To, U. (1998). Theoretical and Experimental Modal Analysis, Research Studies Press Ltd."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.ymssp.2008.04.009","article-title":"Fast calculation of confidence intervals on parameter estimates of least-squares frequency-domain estimators","volume":"23","author":"Guillaume","year":"2009","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1016\/j.ymssp.2009.01.015","article-title":"Fast variance calculation of polyreference least-squares frequency-domain estimates","volume":"23","author":"Guillaume","year":"2009","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1016\/j.ymssp.2007.10.009","article-title":"Uncertainty bounds on modal parameters obtained from stochastic subspace identification","volume":"22","author":"Reynders","year":"2008","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.ymssp.2012.11.011","article-title":"Uncertainty quantification for modal parameters from stochastic subspace identification on multi-setup measurements","volume":"36","author":"Lam","year":"2013","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.ymssp.2015.04.018","article-title":"Uncertainty quantification in operational modal analysis with stochastic subspace identification: Validation and applications","volume":"66\u201367","author":"Reynders","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"El-Kafafy, M., Guillaume, P., De-Troyer, T., and Peeters, B. (2012, January 11\u201313). A Frequency\u2014Domain Maximum Likelihood Implementation using the modal model formulation. Proceedings of the 16th IFAC Symposium on System Identification the International Federation of Automatic Control, Brussels, Belgium.","DOI":"10.3182\/20120711-3-BE-2027.00199"},{"key":"ref_12","unstructured":"El-Kafafy, M. (2013). Design and Validation of Improved Modal Parameter Estimators. [Ph.D. Thesis, Department of Mechanical Engineering, Vrije Universiteit Brussel]."},{"key":"ref_13","unstructured":"Peeters, B., El-Kafafy, M., and Guillaume, P. (2012, January 17\u201319). The new PolyMAX Plus method: Confident modal parameter estimation even in very noisy cases. Proceedings of the 2012 Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/j.ymssp.2013.01.013","article-title":"Fast maximum-likelihood identification of modal parameters with uncertainty intervals: A modal model\u2013based formulation","volume":"37","author":"Peeters","year":"2013","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.ymssp.2014.02.011","article-title":"Fast maximum-likelihood identification of modal parameters with uncertainty intervals: A modal model formulation with enhanced residual term","volume":"48","author":"Guillaume","year":"2014","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.engstruct.2016.12.060","article-title":"High spatial resolution modal identification of a stadium suspension roof: Assessment of the estimates uncertainty and of modal contributions","volume":"135","author":"Diord","year":"2017","journal-title":"Eng. Struct."},{"key":"ref_17","unstructured":"El-kafafy, M., Guillaume, P., Peeters, B., Marra, F., and Coppotelli, G. (February, January 30). Advanced frequency-domain modal analysis for dealing with measurement noise and parameter uncertainty. Proceedings of the 19th International Modal Analysis Conference (IMACXXX), Jacksonville, FL, USA."},{"key":"ref_18","first-page":"567","article-title":"Constrained maximum likelihood modal parameter identification applied to structural dynamics","volume":"72\u201373","author":"Peeters","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_19","unstructured":"Balm\u00e8s, E. (1996, January 12\u201315). Frequency domain identification of structural dynamics using the pole-residue parametrization. Proceedings of the 1996 International Modal Analysis Conference (IMAC), Orlando, FL, USA."},{"key":"ref_20","unstructured":"Heylen, W., Lammens, S., and Sas, P. (1998). Modal Analysis Theory and Testing, Mechanical Engineering, Katholieke Universiteit Leuven. [2nd ed.]."},{"key":"ref_21","unstructured":"Guillaume, P., Verboven, P., and Vanlanduit, S. (1998, January 16\u201318). The new PolyMAX Plus method: Confident modal parameter estimation even in very noisy cases. Proceedings of the 23rd International Seminar on Modal Analysis, Leuven, Belgium."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Cauberghe, B. (2004). Applied Frequency-Domain System Identification in the Field of Experimental and Operational Modal Analysis. [Ph.D. Thesis, Department of Mechanical Engineering, Vrije Universiteit].","DOI":"10.1115\/ESDA2004-58542"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Pintelon, R., and Schoukens, J. (2001). System Identification: A Frequency Domain Approach, IEEE Press. [1st ed.].","DOI":"10.1002\/0471723134"},{"key":"ref_24","unstructured":"Schoukens, J., and Pintelon, R. (1991). System Identification: A Frequency Domain Approach, Pergamon Press. [1st ed.]."},{"key":"ref_25","unstructured":"Kailath, T. (1998). Linear Systems, Prentice-Hall. [1st ed.]."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2359","DOI":"10.1016\/j.ymssp.2006.11.007","article-title":"Uncertainty calculation in (operational) modal analysis","volume":"21","author":"Pintelon","year":"2007","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_27","unstructured":"Verboven, P. (2002). Frequency-Domain System Identification for Modal Analysis. [Ph.D. Thesis, Department of Mechanical Engineering, Vrije Universiteit Brussel]."},{"key":"ref_28","unstructured":"B\u00f6swald, M., G\u00f6ge, D., F\u00fcllekrug, U., and Govers, Y. (2006, January 18\u201320). A review of experimental modal analysis methods with respect to their applicability to test data of large aircraft. Proceedings of the International Conference on Noise and Vibration Engineering, Leuven, Belgium."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"e2006","DOI":"10.1002\/stc.2006","article-title":"Automated modal tracking in a football stadium suspension roof for detection of structural changes","volume":"24","author":"Diord","year":"2017","journal-title":"Struct. Control Health Monit."},{"key":"ref_30","unstructured":"Amador, S.D.R. (2015). Uncertainty Quantification in Operational Modal Analysis and Continuous Monitoring of Special Structures. [Ph.D. Thesis, Department of Civil Engineering, Faculty of Engineering of University of Porto]."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/9\/15\/3120\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:12:09Z","timestamp":1760188329000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/9\/15\/3120"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,1]]},"references-count":30,"journal-issue":{"issue":"15","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["app9153120"],"URL":"https:\/\/doi.org\/10.3390\/app9153120","relation":{},"ISSN":["2076-3417"],"issn-type":[{"value":"2076-3417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,1]]}}}