System Identification Methods for (Operational) Modal Analysis: Review and Comparison

Operational modal analysis deals with the estimation of modal parameters from vibration data obtained in operational rather than laboratory conditions. This paper extensively reviews operational modal analysis approaches and related system identification methods. First, the mathematical models employed in identification are related to the equations of motion, and their modal structure is revealed. Then, strategies that are common to the vast majority of identification algorithms are discussed before detailing some powerful algorithms. The extraction and validation of modal parameter estimates and their uncertainties from the identified system models is discussed as well. Finally, different modal analysis approaches and algorithms are compared in an extensive Monte Carlo simulation study.

[1]  J. Fourier Théorie analytique de la chaleur , 2009 .

[2]  R. Kumaresan,et al.  Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise , 1982 .

[3]  Guido De Roeck,et al.  Reconstructing time-varying wind loads from vibration responses , 2008 .

[4]  Yves Rolain,et al.  Numerically robust frequency domain identification of multivariable systems , 2002 .

[5]  K. Åström Introduction to Stochastic Control Theory , 1970 .

[6]  S. R. Ibrahim,et al.  Modal confidence factor in vibration testing , 1978 .

[7]  J. Rissanen Basis of invariants and canonical forms for linear dynamic systems , 1974, Autom..

[8]  A. Chiuso,et al.  The asymptotic variance of subspace estimates , 2004 .

[9]  Lars Bo Ibsen,et al.  Proceedings of the 1st International Operational Modal Analysis Conference , 2005 .

[10]  Carl Friedrich Gauss Theoria motus corporum coelestium , 1981 .

[11]  G. Yule On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers , 1927 .

[12]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[13]  Julius S. Bendat,et al.  Engineering Applications of Correlation and Spectral Analysis , 1980 .

[14]  朝倉 利光 R. B. Blackman and J. W. Tukey: The Measurement of Power Spectra, Dover Publications, Inc., New York, 1958, 208頁, 13.5×16cm, $1.85 , 1964 .

[15]  Karl Johan Åström,et al.  Numerical Identification of Linear Dynamic Systems from Normal Operating Records , 1965 .

[16]  R. Pintelon System identification , 2001 .

[17]  Leonard Meirovitch,et al.  Elements Of Vibration Analysis , 1986 .

[18]  Dietmar Bauer,et al.  Comparing the CCA Subspace Method to Pseudo Maximum Likelihood Methods in the case of No Exogenous Inputs , 2005 .

[19]  D. Brillinger Time series - data analysis and theory , 1981, Classics in applied mathematics.

[20]  Arjan van der Schaft,et al.  Proceedings of the 30th IEEE Conference on Decision and Control , 1991 .

[21]  G. R. Darbre,et al.  Natural frequencies measured from ambient vibration response of the arch dam of Mauvoisin , 2000 .

[22]  Rik Pintelon,et al.  Uncertainty calculation in (operational) modal analysis , 2007 .

[23]  Β. L. HO,et al.  Editorial: Effective construction of linear state-variable models from input/output functions , 1966 .

[24]  David G. Long,et al.  The probability density of spectral estimates based on modified periodogram averages , 1999, IEEE Trans. Signal Process..

[25]  Karl Johan Åström,et al.  BOOK REVIEW SYSTEM IDENTIFICATION , 1994, Econometric Theory.

[26]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[27]  S. Fassois MIMO LMS-ARMAX IDENTIFICATION OF VIBRATING STRUCTURES—PART I: THE METHOD , 2001 .

[28]  H. Zeiger,et al.  Approximate linear realizations of given dimension via Ho's algorithm , 1974 .

[29]  Gilbert T. Walker,et al.  On Periodicity in Series of Related Terms , 1931 .

[30]  B. Moor,et al.  Model reduction and energy analysis as a tool to detect spurious modes , 2002 .

[31]  Christiaan Heij,et al.  Introduction to mathematical systems theory , 1997 .

[32]  Dietmar Bauer,et al.  Analysis of the asymptotic properties of the MOESP type of subspace algorithms , 2000, Autom..

[33]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[34]  Jan R. Wright,et al.  An eigensystem realization algorithm using data correlations (ERA/DC) for modal parameter identification , 1987 .

[35]  P. Van Overschee,et al.  Subspace algorithms for the stochastic identification problem , 1991 .

[36]  B. Moore Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[37]  Rik Pintelon,et al.  Identification of Linear Systems: A Practical Guideline to Accurate Modeling , 1991 .

[38]  H. Akaike Markovian Representation of Stochastic Processes by Canonical Variables , 1975 .

[39]  Lennart Ljung,et al.  Some facts about the choice of the weighting matrices in Larimore type of subspace algorithms , 2002, Autom..

[40]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[41]  Dietmar Bauer,et al.  Asymptotic properties of subspace estimators , 2005, Autom..

[42]  P. Durbin,et al.  Statistical Theory and Modeling for Turbulent Flows , 2001 .

[43]  J. Schoukens,et al.  Box-Jenkins identification revisited - Part I: Theory , 2006, Autom..

[44]  J. Schoukens,et al.  Sensitivity of roots to errors in the coefficient of polynomials obtained by frequency-domain estimation methods , 1989 .

[45]  G. De Roeck,et al.  Automated interpretation of stabilization diagrams , 2011 .

[46]  Johan A. K. Suykens,et al.  Identification of positive real models in subspace identification by using regularization , 2003, IEEE Trans. Autom. Control..

[47]  Joseph C. S. Lai,et al.  ARMAX modal parameter identification in the presence of unmeasured excitation—II: Numerical and experimental verification , 2007 .

[48]  Paul A. Durbin,et al.  Fluid dynamics with a computational perspective , 2007 .

[49]  Morten Hartvig Hansen,et al.  Two methods for estimating aeroelastic damping of operational wind turbine modes from experiments , 2006 .

[50]  Carl Friedrich Gauss,et al.  Carl Friedrich Gauss Werke , 2013 .

[51]  E. Parloo,et al.  Sensitivity-based operational mode shape normalisation: Application to a bridge , 2005 .

[52]  Rik Pintelon,et al.  Continuous-time operational modal analysis in the presence of harmonic disturbances—The multivariate case , 2010 .

[53]  P. Billingsley,et al.  Probability and Measure , 1980 .

[54]  Guido De Roeck,et al.  Combined Experimental-Operational Modal Testing of Footbridges , 2010 .

[55]  Pieter Eykhoff,et al.  Trends and progress in system identification , 1981 .

[56]  G. De Roeck,et al.  Vibration based Structural Health Monitoring using output-only measurements under changing environment , 2008 .

[57]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[58]  Joseph C. S. Lai,et al.  ARMAX modal parameter identification in the presence of unmeasured excitation—I: Theoretical background , 2007 .

[59]  B. Peeters,et al.  Stochastic System Identification for Operational Modal Analysis: A Review , 2001 .

[60]  E. Betti Teoria della elasticita , 1872 .

[61]  Max Donath,et al.  American Control Conference , 1993 .

[62]  P. Van Overschee,et al.  A unifying theorem for three subspace system identification algorithms , 1994 .

[63]  James M. W. Brownjohn,et al.  Real-time performance monitoring of tuned mass damper system for a 183 m reinforced concrete chimney , 2010 .

[64]  Nicholas A J Lieven,et al.  Proceedings of the 11th International Modal Analysis Conference , 1993 .

[65]  Bart De Moor,et al.  N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems , 1994, Autom..

[66]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[67]  Bart Peeters,et al.  POLYMAX: A REVOLUTION IN OPERATIONAL MODAL ANALYSIS , 2005 .

[68]  Guido De Roeck,et al.  REFERENCE-BASED STOCHASTIC SUBSPACE IDENTIFICATION FOR OUTPUT-ONLY MODAL ANALYSIS , 1999 .

[69]  R. Clough,et al.  Dynamics Of Structures , 1975 .

[70]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[71]  L. Ljung,et al.  Frequency Domain Maximum Likelihood Identification , 1997 .

[72]  Patrick Dewilde,et al.  Subspace model identification Part 1. The output-error state-space model identification class of algorithms , 1992 .

[73]  B. Moor,et al.  Subspace identification for linear systems , 1996 .

[74]  H. Nyquist Thermal Agitation of Electric Charge in Conductors , 1928 .

[75]  M. Jansson Asymptotic Variance Analysis of Subspace Identification Methods , 2000 .

[76]  K. Arun,et al.  Balanced approximation of stochastic systems , 1990 .

[77]  J. Kaimal,et al.  Spectral Characteristics of Surface-Layer Turbulence , 1972 .

[78]  Richard Evelyn Donohue Bishop,et al.  An investigation into the theory of resonance testing , 1963, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[79]  Paul Sas,et al.  Modal Analysis Theory and Testing , 2005 .

[80]  T. Söderström,et al.  Instrumental variable methods for system identification , 1983 .

[81]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[82]  Jer-Nan Juang,et al.  An eigensystem realization algorithm for modal parameter identification and model reduction. [control systems design for large space structures] , 1985 .

[83]  Alessandro Chiuso,et al.  Subspace identification by data orthogonalization and model decoupling , 2003, Autom..

[84]  Svend Ole Hansen,et al.  Wind Loads on Structures , 1997 .

[85]  H. Akaike Fitting autoregressive models for prediction , 1969 .

[86]  Poul Henning Kirkegaard,et al.  Identification of Civil Engineering Structures using Vector ARMA Models , 1998 .

[87]  B. Peeters,et al.  Continuous-Time Operational ModalAnalysis- inthePresence ofHarmonic Disturbances , 2008 .

[88]  Rik Pintelon,et al.  Brief paper: Box-Jenkins identification revisited-Part III , 2007 .

[89]  H. Saunders Literature Review : RANDOM DATA: ANALYSIS AND MEASUREMENT PROCEDURES J. S. Bendat and A.G. Piersol Wiley-Interscience, New York, N. Y. (1971) , 1974 .

[90]  E. C. Mikulcik,et al.  A method for the direct identification of vibration parameters from the free response , 1977 .

[91]  M. Verhaegen Subspace model identification Part 2. Analysis of the elementary output-error state-space model identification algorithm , 1992 .

[92]  E. Parloo,et al.  AUTONOMOUS STRUCTURAL HEALTH MONITORING—PART I: MODAL PARAMETER ESTIMATION AND TRACKING , 2002 .

[93]  Filipe Magalhães,et al.  Operational modal analysis and finite element model correlation of the Braga Stadium suspended roof , 2008 .

[94]  H. Van der Auweraer,et al.  Multiple input orthogonal polynomial parameter estimation , 1987 .

[95]  D. Hanson,et al.  Cyclostationarity and the cepstrum for operational modal analysis of mimo systems-Part I Modal parameter identification , 2007 .

[96]  Palle Andersen,et al.  Modal Identification from Ambient Responses using Frequency Domain Decomposition , 2000 .

[97]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1971 .

[98]  R. Fisher 001: On an Absolute Criterion for Fitting Frequency Curves. , 1912 .

[99]  Richard Russell,et al.  A Multi-Input Modal Estimation Algorithm for Mini-Computers , 1982 .

[100]  G. Deodatis Simulation of Ergodic Multivariate Stochastic Processes , 1996 .

[101]  Edwin Reynders,et al.  Damage identification on the Tilff bridge by vibration monitoring using optical fiber strain sensors , 2005 .

[102]  Sun-Yuan Kung,et al.  A new identification and model reduction algorithm via singular value decomposition , 1978 .

[103]  Michel Verhaegen,et al.  Identification of the deterministic part of MIMO state space models given in innovations form from input-output data , 1994, Autom..

[104]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[105]  Daniel Rixen,et al.  Modified ERA method for operational modal analysis in the presence of harmonic excitations , 2006 .

[106]  J. S. Przemieniecki Theory of matrix structural analysis , 1985 .

[107]  R. Kálmán Mathematical description of linear dynamical systems , 1963 .

[108]  N. Wiener Generalized harmonic analysis , 1930 .

[109]  Guido De Roeck,et al.  System identification and modal analysis in structural mechanics , 2012 .

[110]  H. Van der Auweraer,et al.  Discriminating physical poles from mathematical poles in high order systems: use and automation of the stabilization diagram , 2004, Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference (IEEE Cat. No.04CH37510).

[111]  Jan C. Willems,et al.  In Control, Almost from the Beginning Until the Day After Tomorrow , 2007, Eur. J. Control.

[112]  M. Gevers,et al.  A personal view of the development of system identification: A 30-year journey through an exciting field , 2006, IEEE Control Systems.

[113]  P. Welch The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms , 1967 .

[114]  K. Bathe Finite Element Procedures , 1995 .

[115]  Guido De Roeck,et al.  Estimation of impulse responses: a novel method and its use in experimental modal analysis , 2008 .

[116]  Yves Rolain,et al.  Identification of linear systems with nonlinear distortions , 2003, Autom..

[117]  Rune Brincker,et al.  Proceedings of the 10th International Modal Analysis Conference , 1993 .

[118]  Edward R. Dougherty,et al.  Random processes for image and signal processing , 1998, SPIE / IEEE series on imaging science and engineering.

[119]  E. Peter Carden,et al.  Operational modal analysis of resiliently mounted marine diesel generator/alternator , 2011 .

[120]  Yves Rolain,et al.  Numerically robust transfer function modeling from noisy frequency domain data , 2005, IEEE Transactions on Automatic Control.

[121]  Bart Peeters,et al.  System identification and damage detection in civil engineering , 2000 .

[122]  T. Söderström Brief paper: Test of pole-zero cancellation in estimated models , 1975 .

[123]  P. Guillaume,et al.  The PolyMAX Frequency-Domain Method: A New Standard for Modal Parameter Estimation? , 2004 .

[124]  E. Parloo,et al.  Improved modal parameter estimation for lowly damped systems using non-parametric exponential windowing techniques , 2005 .

[125]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[126]  Bart De Moor,et al.  A note on persistency of excitation , 2005, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[127]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[128]  Guido De Roeck,et al.  Reference-based combined deterministic–stochastic subspace identification for experimental and operational modal analysis , 2006 .

[129]  Richard S. Pappa,et al.  A consistent-mode indicator for the eigensystem realization algorithm , 1992 .

[130]  E. Parloo,et al.  On the influence of the parameter constraint on the stability of the poles and the discrimination capabilities of the stabilisation diagrams , 2005 .

[131]  Akira Mita,et al.  Challenges in developing confidence intervals on modal parameters estimated for large civil infrastructure with stochastic subspace identification , 2009 .

[132]  L. Hermans,et al.  MODAL TESTING AND ANALYSIS OF STRUCTURES UNDER OPERATIONAL CONDITIONS: INDUSTRIAL APPLICATIONS , 1999 .

[133]  Dietmar Bauer,et al.  Consistency and asymptotic normality of some subspace algorithms for systems without observed inputs , 1999, Autom..

[134]  Thomas Kailath,et al.  Linear Systems , 1980 .

[135]  Yves Rolain,et al.  Box-Jenkins identification revisited - Part II: Applications , 2006, Autom..

[136]  Andreas Johann Felber,et al.  Development of a hybrid bridge evaluation system , 1994 .

[137]  E. Gilbert Controllability and Observability in Multivariable Control Systems , 1963 .

[138]  C. Jordan,et al.  Mémoire sur les formes bilinéaires. , 1874 .

[139]  P. Guillaume,et al.  Operational modal analysis for estimating the dynamic properties of a stadium structure during a football game , 2007 .

[140]  Rik Pintelon,et al.  Box-Jenkins identification revisited - Part III: Multivariable systems , 2007, Autom..

[141]  Prasenjit Mohanty,et al.  Operational modal analysis in the presence of harmonic excitation , 2004 .

[142]  Carmelo Gentile,et al.  Ambient vibration testing of historic masonry towers for structural identification and damage assessment , 2007 .

[143]  H. Akaike Stochastic theory of minimal realization , 1974 .

[144]  Rik Pintelon,et al.  Continuous-Time Noise Modeling From Sampled Data , 2006, IEEE Transactions on Instrumentation and Measurement.

[145]  Patrick Guillaume,et al.  Maximum likelihood identification of modal parameters from operational data , 1999 .

[146]  E. Parloo,et al.  SENSITIVITY-BASED OPERATIONAL MODE SHAPE NORMALISATION , 2002 .

[147]  J. Suykens,et al.  Subspace identification of Hammerstein systems using least squares support vector machines , 2005 .

[148]  J. Schoukens,et al.  Study of the LTI relations between the outputs of two coupled Wiener systems and its application to the generation of initial estimates for Wiener-Hammerstein systems , 2008, Autom..

[149]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[150]  John William Strutt,et al.  Some General Theorems relating to Vibrations , 1871 .

[151]  Rik Pintelon,et al.  Consistent Impulse-Response Estimation and System Realization From Noisy Data , 2008, IEEE Transactions on Signal Processing.

[152]  Wallace E. Larimore,et al.  Canonical variate analysis in identification, filtering, and adaptive control , 1990, 29th IEEE Conference on Decision and Control.

[153]  R. Pintelon,et al.  Fast calculation of confidence intervals on parameter estimates of least-squares frequency-domain estimators , 2009 .

[154]  Y. G. Tsuei,et al.  Complex Mode Indication Function and its Applications to Spatial Domain Parameter Estimation , 1988 .

[155]  J. Juang Applied system identification , 1994 .

[156]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[157]  Palle Andersen,et al.  Modal Analysis of an Offshore Platform using Two Different ARMA Approaches , 1996 .

[158]  R. Kumaresan On the zeros of the linear prediction-error filter for deterministic signals , 1983 .

[159]  Bart De Moor,et al.  Subspace algorithms for the stochastic identification problem, , 1993, Autom..

[160]  William Thomson,et al.  VI. Mechanical integration of the general linear differential equation of any order with variable coefficients , 1876, Proceedings of the Royal Society of London.

[161]  Gene F. Franklin,et al.  Digital control of dynamic systems , 1980 .

[162]  J. Bendat,et al.  Random Data: Analysis and Measurement Procedures , 1987 .

[163]  N. L. Johnson,et al.  Breakthroughs in Statistics , 1992 .

[164]  Geert Degrande,et al.  Finite element modelling of a silo based on experimental modal analysis , 2006 .

[165]  J. Schoukens,et al.  Frequency domain system identification using arbitrary signals , 1997, IEEE Trans. Autom. Control..

[166]  Richard W. Longman,et al.  Linear system identification via an asymptotically stable observer , 1991 .

[167]  Karim Abed-Meraim,et al.  Blind system identification , 1997, Proc. IEEE.

[168]  Hirotugu Akaike,et al.  MODERN DEVELOPMENT OF STATISTICAL METHODS , 1981 .

[169]  E. I. Jury,et al.  Theory and application of the z-transform method , 1965 .

[170]  Mats Viberg,et al.  Subspace-based methods for the identification of linear time-invariant systems , 1995, Autom..

[171]  E. Parloo,et al.  Combined Deterministic-Stochastic Frequency-Domain Subspace Identification for Experimental and Operational Modal Analysis , 2004 .

[172]  Kenneth J. Pope,et al.  Blind Signal Separation II. Linear, Convolutive Combinations: II. Linear, Convolutive Combinations , 1996, Digit. Signal Process..

[173]  Thomas G. Carne,et al.  The inception of OMA in the development of modal testing technology for wind turbines , 2010 .

[174]  Rik Pintelon,et al.  On the frequency scaling in continuous-time modeling , 2005, IEEE Transactions on Instrumentation and Measurement.

[175]  Nuno M. M. Maia,et al.  Theoretical and Experimental Modal Analysis , 1997 .

[176]  Charles C. Kennedy,et al.  Use of Vectors in Vibration Measurement and Analysis , 1947 .

[177]  Rik Pintelon,et al.  Uncertainty bounds on modal parameters obtained from stochastic subspace identification , 2008 .

[178]  Bart De Moor,et al.  Algorithms for deterministic balanced subspace identification , 2005, Autom..

[179]  M. Deistler,et al.  On the Impact of Weighting Matrices in Subspace Algorithms , 2000 .

[180]  Richard W. Longman,et al.  Identification of linear systems by an asymptotically stable observer , 1992 .

[181]  K. Åström,et al.  Numerical Identification of Linear Dynamic Systems from Normal Operating Records , 1966 .

[182]  Gene H. Golub,et al.  Matrix computations , 1983 .