Nonlinear Dynamics

, Volume 89, Issue 2, pp 1489–1511 | Cite as

Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters

  • S. Eftekhar Azam
  • S. Mariani
  • N. K. A. Attari
Original Paper


In this paper, an approach based on the synergistic use of proper orthogonal decomposition and Kalman filtering is proposed for the online health monitoring of damaged structures. The reduced-order model of a structure is obtained during an (offline) initial training stage of monitoring; afterward, effective estimations of a possible structural damage are provided online by tracking the evolution in time of stiffness parameters and projection bases handled in the model order reduction procedure. Such tracking is accomplished via two Kalman filters: a first (extended) one to deal with the time evolution of a joint state vector, gathering the reduced-order state and the stiffness terms degraded by damage; a second one to deal with the update of the reduced-order model in case of damage evolution. Both filters exploit the information conveyed by measurements of the structural response to the external excitations. Results are reported for a (pseudo-experimental) benchmark test on an eight-story shear building. Capability and performance of the proposed approach are assessed in terms of tracked variation of the stiffness terms of the reduced-order model, identified damage location and speed-up of the whole health monitoring procedure.


Structural health monitoring (SHM) Reduced-order modeling Damage detection Model updating Kalman filtering Proper orthogonal decomposition (POD) 



Financial support by Fondazione Cariplo through project Safer Helmets is gratefully acknowledged.


  1. 1.
    Glaser, S.D., Li, H., Wang, M.L., Ou, J., Lynch, J.: Sensor technology innovation for the advancement of structural health monitoring: a strategic program of US-China research for the next decade. Smart Struct. Syst. 3(2), 221–244 (2007). doi: 10.12989/sss.2007.3.2.221 CrossRefGoogle Scholar
  2. 2.
    Stallings, J.M., Tedesco, J.W., El-Mihilmy, M., McCauley, M.: Field performance of FRP bridge repairs. J. Bridge Eng. 5, 107–113 (2000)CrossRefGoogle Scholar
  3. 3.
    Aktan, A., Catbas, F., Grimmelsman, K., Tsikos, C.: Issues in infrastructure health monitoring for management. J. Eng. Mech. 126(7), 711–724 (2000). doi: 10.1061/(ASCE)0733-9399. (2000) 126:7(711)CrossRefGoogle Scholar
  4. 4.
    Ko, J.M., Ni, Y.Q.: Technology developments in structural health monitoring of large-scale bridges. Eng. Struct. 27(12), 1715–1725 (2005). doi: 10.1016/j.engstruct.2005.02.021 CrossRefGoogle Scholar
  5. 5.
    Glaser, S.D., Tolman, A.: Sense of sensing: from data to informed decisions for the built environment. J. Infrastruct. Syst. ACSE 14, 4–14 (2008)CrossRefGoogle Scholar
  6. 6.
    Yeum, C.M., Dyke, S.J.: Vision-based automated crack detection for bridge inspection. Comput. Aided Civil Infrastruct. Eng. 30(10), 759–770 (2015). doi: 10.1111/mice.12141 CrossRefGoogle Scholar
  7. 7.
    Memarzadeh, M., Pozzi, M.: Integrated inspection scheduling and maintenance planning for infrastructure systems. Comput. Aided Civil Infrastruct. Eng. (2015). doi: 10.1111/mice.12178 Google Scholar
  8. 8.
    Cho, S., Spencer, B.F.: Sensor attitude correction of wireless sensor network for acceleration-based monitoring of civil structures. Comput. Aided Civil Infrastruct. Eng. 30(11), 859–871 (2015). doi: 10.1111/mice.12147 CrossRefGoogle Scholar
  9. 9.
    Mariani, S., Corigliano, A., Caimmi, F., Bruggi, M., Bendiscioli, P., De Fazio, M.: MEMS-based surface mounted health monitoring system for composite laminates. Microelectron. J. 44(7), 598–605 (2013). doi: 10.1016/j.mejo.2013.03.003 CrossRefGoogle Scholar
  10. 10.
    Mariani, S., Bruggi, M., Caimmi, F., Bendiscioli, P., De Fazio, M.: Sensor deployment over damage-containing plates: a topology optimization approach. J. Intell. Mater. Syst. Struct. 24, 1105–1122 (2013)CrossRefGoogle Scholar
  11. 11.
    Chan, T.H.T., Yu, L., Tam, H.Y., Ni, Y.Q., Liu, S.Y., Chung, W.H., Cheng, L.K.: Fiber Bragg grating sensors for structural health monitoring of Tsing Ma bridge: background and experimental observation. Eng. Struct. 28(5), 648–659 (2006). doi: 10.1016/j.engstruct.2005.09.018 CrossRefGoogle Scholar
  12. 12.
    Helmi, K., Taylor, T., Zarafshan, A., Ansari, F.: Reference free method for real time monitoring of bridge deflections. Eng. Struct. 103, 116–124 (2015). doi: 10.1016/j.engstruct.2015.09.002 CrossRefGoogle Scholar
  13. 13.
    Hampshire, T.A., Adeli, H.: Monitoring the behavior of steel structures using distributed optical fiber sensors. J. Constr. Steel Res. 53(3), 267–281 (2000)CrossRefGoogle Scholar
  14. 14.
    Gentile, C., Cabboi, A.: Vibration-based structural health monitoring of stay cables by microwave remote sensing. Smart Struct. Syst. 16(2), 263–280 (2015). doi: 10.12989/sss.2015.16.2.263 CrossRefGoogle Scholar
  15. 15.
    Farrar, C.R., Darling, T.W., Migliori, A., Baker, W.E.: Microwave interferometers for non-contact vibration measurements on large structures. Mech. Syst. Signal Process. 13(2), 241–253 (1999). doi: 10.1006/mssp.1998.1216 CrossRefGoogle Scholar
  16. 16.
    Laefer, D.F., Truong-Hong, L., Carr, H., Singh, M.: Crack detection limits in unit based masonry with terrestrial laser scanning. NDTE Int. 62, 66–76 (2014). doi: 10.1016/j.ndteint.2013.11.001 CrossRefGoogle Scholar
  17. 17.
    Breuer, P., Chmielewski, T., Górski, P., Konopka, E.: Application of GPS technology to measurements of displacements of high-rise structures due to weak winds. J. Wind Eng. Ind. Aerodyn. 90(3), 223–230 (2002). doi: 10.1016/S0167-6105(01)00221-5 CrossRefGoogle Scholar
  18. 18.
    Górski, P.: Investigation of dynamic characteristics of tall industrial chimney based on GPS measurements using Random Decrement Method. Eng. Struct. 83, 30–49 (2015). doi: 10.1016/j.engstruct.2014.11.006 CrossRefGoogle Scholar
  19. 19.
    Park, S.W., Park, H.S., Kim, J.H., Adeli, H.: 3D displacement measurement model for health monitoring of structures using a motion capture system. Measurement 59, 352–362 (2015). doi: 10.1016/j.measurement.2014.09.063 CrossRefGoogle Scholar
  20. 20.
    Lee, J.J., Shinozuka, M.: Real-time displacement measurement of a flexible bridge using digital image processing techniques. Exp. Mech. 46(1), 105–114 (2006). doi: 10.1007/s11340-006-6124-2 CrossRefGoogle Scholar
  21. 21.
    Hwa Kim, B.: Extracting modal parameters of a cable on shaky motion pictures. Mech. Syst. Signal Process. 49(1–2), 3–12 (2014). doi: 10.1016/j.ymssp.2014.02.002 CrossRefGoogle Scholar
  22. 22.
    Qarib, H., Adeli, H.: Recent advances in health monitoring of civil structures. Sci. Iran. 21(6), 1733–1742 (2014)Google Scholar
  23. 23.
    Bursi, O.S., Kumar, A., Abbiati, G., Ceravolo, R.: Identification, model updating, and validation of a steel twin deck curved cable-stayed footbridge. Comput. Aided Civil Infrastruct. Eng. 29(9), 703–722 (2014). doi: 10.1111/mice.12076 CrossRefGoogle Scholar
  24. 24.
    Fuggini, C., Chatzi, E., Zangani, D.: Combining genetic algorithms with a meso-scale approach for system identification of a smart polymeric textile. Comput. Aided Civil Infrastruct. Eng. 28(3), 227–245 (2013). doi: 10.1111/j.1467-8667.2012.00789.x CrossRefGoogle Scholar
  25. 25.
    Moaveni, B., Conte, J.P., Hemez, F.M.: Uncertainty and sensitivity analysis of damage identification results obtained using finite element model updating. Comput. Aided Civil Infrastruct. Eng. 24(5), 320–334 (2009). doi: 10.1111/j.1467-8667.2008.00589.x CrossRefGoogle Scholar
  26. 26.
    Moaveni, B., Behmanesh, I.: Effects of changing ambient temperature on finite element model updating of the Dowling Hall Footbridge. Eng. Struct. 43, 58–68 (2012)CrossRefGoogle Scholar
  27. 27.
    Farrar, C.R., Doebling, S.W., Nix, D.A.: Vibration-based structural damage identification. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 359(1778), 131–149 (2001)CrossRefMATHGoogle Scholar
  28. 28.
    Haritos, N., Owen, J.S.: The use of vibration data for damage detection in bridges: a comparison of system identification and pattern recognition approaches. Struct. Health Monit. 3(2), 141–163 (2004). doi: 10.1177/1475921704042698 CrossRefGoogle Scholar
  29. 29.
    Farrar, C.R., Worden, K.: Structural Health Monitoring: A Machine Learning Perspective. Wiley Publishing, London (2012)CrossRefGoogle Scholar
  30. 30.
    Amezquita-Sanchez, J.P., Adeli, H.: Signal processing techniques for vibration-based health monitoring of smart structures. Arch. Comput. Methods Eng. 23(1), 1–15 (2016). doi: 10.1007/s11831-014-9135-7 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Dervilis, N., Worden, K., Cross, E.: On robust regression analysis as a means of exploring environmental and operational conditions for SHM data. J. Sound Vib. 347, 279–296 (2015)CrossRefGoogle Scholar
  32. 32.
    Spiridonakos, M.D., Chatzi, E.N., Sudret, B.: Polynomial Chaos expansion models for the monitoring of structures under operational variability. ASCE ASME J. Risk Uncertain. Eng. Syst. Part A Civil Eng. 2(3), B4016003 (2016)Google Scholar
  33. 33.
    Reynders, E., Wursten, G., De Roeck, G.: Output-only structural health monitoring in changing environmental conditions by means of nonlinear system identification. Struct. Health Monit. 13(1), 82–93 (2014)CrossRefGoogle Scholar
  34. 34.
    Yang, J., Lin, S.: Identification of parametric variations of structures based on least squares estimation and adaptive tracking technique. J. Eng. Mech. 131(3), 290–298 (2005). doi: 10.1061/(ASCE)0733-9399. (2005) 131:3(290)CrossRefGoogle Scholar
  35. 35.
    Van Overschee, P., De Moor, B.: Subspace Identification for Linear Systems: Theory-Implementation-Applications. Springer, New York (1996)CrossRefMATHGoogle Scholar
  36. 36.
    Van Overschee, P., De Moor, B.: N4SID: subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 30(1), 75–93 (1994). doi: 10.1016/0005-1098(94)90230-5 MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Chin-Hsiung, L., Jian-Huang, W., Yi-Cheng, L., Pei-Yang, L., Shieh-Kung, H.: Structural damage diagnosis based on on-line recursive stochastic subspace identification. Smart Mater. Struct. 20(5), 055004 (2011)CrossRefGoogle Scholar
  38. 38.
    Chatzis, M., Chatzi, E., Smyth, A.W.: An experimental validation of time domain system identification methods with fusion of heterogeneous data. Earthq. Eng. Struct. Dyn. 44(4), 523–547 (2015). doi: 10.1002/eqe.2528 CrossRefGoogle Scholar
  39. 39.
    Moaveni, B., He, X., Conte, J., Restrepo, J., Panagiotou, M.: System identification study of a 7-story full-scale building slice tested on the UCSD-NEES shake table. J. Struct. Eng. 137(6), 705–717 (2010). doi: 10.1061/(ASCE)ST.1943-541X.0000300 CrossRefGoogle Scholar
  40. 40.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960)CrossRefGoogle Scholar
  41. 41.
    Julier, S.J., Uhlmann, J.K.: A new extension of the Kalman filter to nonlinear systems. In: International Symposium on Aerospace/Defence, Sensing, Simulation and Controls, vol. 26, p. 32. Orlando (1997)Google Scholar
  42. 42.
    Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F RadarSignal Process. 140(2), 107–113 (1993)CrossRefGoogle Scholar
  43. 43.
    Chatzi, E.N., Smyth, A.W.: Particle filter scheme with mutation for the estimation of time-invariant parameters in structural health monitoring applications. Struct. Control Health Monit. 20(7), 1081–1095 (2013)CrossRefGoogle Scholar
  44. 44.
    Li, B.: Multiple-model Rao-Blackwellized particle CPHD filter for multitarget tracking. Nonlinear Dyn. 79(3), 2133–2143 (2014). doi: 10.1007/s11071-014-1799-x CrossRefGoogle Scholar
  45. 45.
    Eftekhar Azam, S., Mariani, S.: Dual estimation of partially observed nonlinear structural systems: a particle filter approach. Mech. Res. Commun. 46, 54–61 (2012)CrossRefGoogle Scholar
  46. 46.
    Chatzi, E.N., Smyth, A.W., Masri, S.F.: Experimental application of on-line parametric identification for nonlinear hysteretic systems with model uncertainty. Struct. Saf. 32(5), 326–337 (2010)CrossRefGoogle Scholar
  47. 47.
    Eftekhar Azam, S.: Online Damage Detection in Structural Systems. Springer Briefs in Applied Sciences and Technology. Springer, Berlin (2014)CrossRefGoogle Scholar
  48. 48.
    Eftekhar Azam, S., Mariani, S.: Investigation of computational and accuracy issues in POD-based reduced order modeling of dynamic structural systems. Eng. Struct. 54, 150–167 (2013)CrossRefGoogle Scholar
  49. 49.
    Kerschen, G., Golinval, G.C.: Physical interpretation of the proper orthogonal modes using the singular value decomposition. J. Sound Vib. 249, 849–865 (2002)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Corigliano, A., Dossi, M., Mariani, S.: Model order reduction and domain decomposition strategies for the solution of the dynamic elastic-plastic structural problem. Comput. Methods Appl. Mech. Eng. 290, 127–155 (2015). doi: 10.1016/j.cma.2015.02.021 MathSciNetCrossRefGoogle Scholar
  51. 51.
    Kerschen, G., Golinval, J.-C., Vakakis, A.F., Bergman, L.A.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41(1–3), 147–169 (2005)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Lu, K., Yu, H., Chen, Y., Cao, Q., Hou, L.: A modified nonlinear POD method for order reduction based on transient time series. Nonlinear Dyn. 79(2), 1195–1206 (2014). doi: 10.1007/s11071-014-1736-z CrossRefMATHGoogle Scholar
  53. 53.
    Lu, K., Jin, Y., Chen, Y., Cao, Q., Zhang, Z.: Stability analysis of reduced rotor pedestal looseness fault model. Nonlinear Dyn. 82(4), 1611–1622 (2015). doi: 10.1007/s11071-015-2264-1 MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Zhao, X., Shang, P.: Principal component analysis for non-stationary time series based on detrended cross-correlation analysis. Nonlinear Dyn. 84(2), 1033–1044 (2015). doi: 10.1007/s11071-015-2547-6 MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Liang, Y.C., Lin, W.Z., Lee, H.P., Lim, S.P., Lee, K.H., Sun, H.: Proper orthogonal decomposition and its applications-part II: model reduction for mems dynamical analysis. J. Sound Vib. 256(3), 515–532 (2002). doi: 10.1006/jsvi.2002.5007 CrossRefGoogle Scholar
  56. 56.
    Ruotolo, R., Surace, C.: Using svd to detect damage in structures with different operational conditions. J. Sound Vib. 226(3), 425–439 (1999). doi: 10.1006/jsvi.1999.2305 CrossRefGoogle Scholar
  57. 57.
    Vanlanduit, S., Parloo, E., Cauberghe, B., Guillaume, P., Verboven, P.: A robust singular value decomposition for damage detection under changing operating conditions and structural uncertainties. J. Sound Vib. 284(3–5), 1033–1050 (2005). doi: 10.1016/j.jsv.2004.07.016 CrossRefGoogle Scholar
  58. 58.
    Galvanetto, U., Violaris, G.: Numerical investigation of a new damage detection method based on proper orthogonal decomposition. Mech. Syst. Signal Process. 21(3), 1346–1361 (2007). doi: 10.1016/j.ymssp.2005.12.007 CrossRefGoogle Scholar
  59. 59.
    Shane, C., Jha, R.: Proper orthogonal decomposition based algorithm for detecting damage location and severity in composite beams. Mech. Syst. Signal Process. 25(3), 1062–1072 (2011). doi: 10.1016/j.ymssp.2010.08.015 CrossRefGoogle Scholar
  60. 60.
    Mariani, S., Ghisi, A.: Unscented Kalman filtering for nonlinear structural dynamics. Nonlinear Dyn. 49(1–2), 131–150 (2007)CrossRefMATHGoogle Scholar
  61. 61.
    Hughes, T.J.R.: The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover, New York (2000)MATHGoogle Scholar
  62. 62.
    Corigliano, A., Mariani, S.: Parameter identification in explicit structural dynamics: performance of the extended Kalman filter. Comput. Methods Appl. Mech. Eng. 193, 3807–3830 (2004)CrossRefMATHGoogle Scholar
  63. 63.
    Sirovich, L.: Turbulence and the dynamics of coherent structures. I-coherent structures. II-symmetries and transformations. III-dynamics and scaling. Q. Appl. Math. 45(1), 573–590 (1987)CrossRefMATHGoogle Scholar
  64. 64.
    Liang, Y.C., Lee, H.P., Lim, S.P., Lin, W.Z., Lee, K.H., Wu, C.G.: Proper orthogonal decomposition and its applications-part I: theory. J. Sound Vib. 252(3), 527–544 (2002). doi: 10.1006/jsvi.2001.4041 CrossRefMATHGoogle Scholar
  65. 65.
    Butcher, E.A., Al-Shudeifat, M.A.: An efficient mode-based alternative to principal orthogonal modes in the order reduction of structural dynamic systems with grounded nonlinearities. Mech. Syst. Signal Process. 25(5), 1527–1549 (2011)CrossRefGoogle Scholar
  66. 66.
    Al-Shudeifat, M.A., Butcher, E.A.: Order reduction of forced nonlinear systems using updated LELSM modes with new Ritz vectors. Nonlinear Dyn. 62(4), 821–840 (2010)CrossRefMATHGoogle Scholar
  67. 67.
    Kappagantu, R., Feeny, B.: An “optimal” modal reduction of a system with frictional excitation. J. Sound Vib. 224(5), 863–877 (1999)CrossRefGoogle Scholar
  68. 68.
    Al-Shudeifat, M.A., Butcher, E.A.: On the dynamics of a beam with switching crack and damaged boundaries. J. Vib. Control 19(1), 1077546311428640 (2013)CrossRefGoogle Scholar
  69. 69.
    Feeny, B., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211(4), 607–616 (1998)CrossRefGoogle Scholar
  70. 70.
    Han, C.S., Feeny, B.: Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures. J. Vib. Control 8(1), 19–40 (2002)MATHGoogle Scholar
  71. 71.
    Feeny, B.: On proper orthogonal co-ordinates as indicators of modal activity. J. Sound Vib. 255(5), 805–817 (2002)CrossRefGoogle Scholar
  72. 72.
    Bryson, A., Johansen, D.: Linear filtering for time-varying systems using measurements containing colored noise. IEEE Trans. Autom. Control 10(1), 4–10 (1965). doi: 10.1109/TAC.1965.1098063 MathSciNetCrossRefGoogle Scholar
  73. 73.
    Geist, M., Pietquin, O.: Kalman filtering and colored noises: the (autoregressive) moving-average case. In: IEEE Workshop on Machine Learning Algorithms, Systems and Applications (MLASA 2011), Honolulu, United States. pp. 1–4 (2011)Google Scholar
  74. 74.
    Grewal, M.S., Andrews, A.P.: Kalman Filtering: Theory and Practice Using MATLAB, 4th edn. Wiley Publishing, London (2011)MATHGoogle Scholar
  75. 75.
    Wan, E.A., Nelson, A.T.: Dual Extended Kalman Filter Methods. In: Haykin, S. (ed.) Kalman Filtering and Neural Networks. Wiley Publishing, London (2001)Google Scholar
  76. 76.
    Capellari, G., Eftekhar Azam, S., Mariani, S.: Damage detection in flexible plates through reduced-order modeling and hybrid particle-Kalman filtering. Sensors 16(1), 2 (2016). doi: 10.3390/s16010002
  77. 77.
    Roffel, A.J., Narasimhan, S.: Extended Kalman filter for modal identification of structures equipped with a pendulum tuned mass damper. J. Sound Vib. 333(23), 6038–6056 (2014). doi: 10.1016/j.jsv.2014.06.030
  78. 78.
    Reif, K., Gunther, S., Yaz, E., Unbehauen, R.: Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans. Autom. Control 44(4), 714–728 (1999). doi: 10.1109/9.754809 MathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    Sharma, G., Agarwala, A., Bhattacharya, B.: A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA. Comput. Struct. 128, 31–37 (2013)CrossRefGoogle Scholar
  80. 80.
    De Callafon, R.A., Moaveni, B., Conte, J.P., He, X., Udd, E.: General realization algorithm for modal identification of linear dynamic systems. J. Eng. Mech. 134(9), 712–722 (2008)CrossRefGoogle Scholar
  81. 81.
    Krajcinovic, D.: Damage mechanics. Mech. Mater. 8(2–3), 117–197 (1989)CrossRefGoogle Scholar
  82. 82.
    Corigliano, A., Dossi, M., Mariani, S.: Domain decomposition and model order reduction methods applied to the simulation of multiphysics problems in MEMS. Comput. Struct. 122, 113–127 (2013)CrossRefGoogle Scholar
  83. 83.
    Brand, M.: Fast low-rank modifications of the thin singular value decomposition. Linear Algebra Appl. 415, 20–30 (2006)MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    Bittanti, S., Savaresi, S.M.: On the parameterization and design of an extended Kalman filter frequency tracker. IEEE Trans. Autom. Control 45(9), 1718–1724 (2000)CrossRefMATHGoogle Scholar
  85. 85.
    Kontoroupi, K., Smyth, A.W.: Online noise identification for joint state and parameter estimation of nonlinear systems. ASCE ASME J. Risk Uncertain. Eng. Syst. 2(3), B4015006 (2016). doi: 10.1061/AJRUA6.0000839 Google Scholar
  86. 86.
    Yuen, K.-V., Liang, P.F., Kuok, S.C.: Online estimation of noise parameters for Kalman filter. Struct. Eng. Mech. 47(3), 361–381 (2013)CrossRefGoogle Scholar
  87. 87.
    Yuen, K.-V., Kuok, S.-C.: Online updating and uncertainty quantification using nonstationary output-only measurement. Mech. Syst. Signal Process. 66–67, 62–77 (2016). doi: 10.1016/j.ymssp.2015.05.019 CrossRefGoogle Scholar
  88. 88.
    Lim, J.: Particle filtering for nonlinear dynamic state systems with unknown noise statistics. Nonlinear Dyn. 78(2), 1369–1388 (2014). doi: 10.1007/s11071-014-1523-x CrossRefGoogle Scholar
  89. 89.
    Yang, Y., Gao, W.: An optimal adaptive Kalman filter. J. Geod. 80(4), 177–183 (2006)CrossRefMATHGoogle Scholar
  90. 90.
    Boutayeb, M., Rafaralahy, H., Darouach, M.: Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems. IEEE Trans. Autom. Control 42(4), 581–586 (1997). doi: 10.1109/9.566674 MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    Zang, C., Imregun, M.: Structural damage detection using artificial neural networks and measured FRF data reduced via principal component projection. J. Sound Vib. 242(5), 813–827 (2001)CrossRefGoogle Scholar
  92. 92.
    Sahin, M., Shenoi, R.: Quantification and localisation of damage in beam-like structures by using artificial neural networks with experimental validation. Eng. Struct. 25(14), 1785–1802 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of ThessalyVólosGreece
  2. 2.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  3. 3.Structural Engineering DepartmentBuilding and Housing Research Center (BHRC)TehranIran

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