Information-Theoretic Approach for Identifiability Assessment of Nonlinear Structural Finite-Element Models
Publication: Journal of Engineering Mechanics
Volume 145, Issue 7
Abstract
This paper presents an information-theoretic approach for identifiability assessment of model parameters in nonlinear finite-element (FE) model updating problems. Rooted in the Bayesian inference method, the proposed approach uses the Shannon information entropy as a measure of uncertainty in the model parameters. The difference in the entropy of a priori and a posteriori probability distribution functions of model parameters, which is referred to as the entropy gain, is used as a measure of information contained in each measurement channel about the model parameters. The entropy gain approach can be used for selection of estimation parameters, optimal sensor placement, and design of experiment. In this study, an approximate expression for the entropy gain is derived, and a three-step process is suggested for the identifiability assessment. The application of the proposed approach is demonstrated for a nonlinear structural system identification problem. Although the focus of this study is on nonlinear structural FE model identifiability, the provided approach can be used for identifiability assessment of other types of linear/nonlinear dynamic models.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
R. Astroza acknowledges the support received from the Chilean National Commission for Scientific and Technological Research (CONICYT), FONDECYT-Iniciacion Project No. 11160009, and the financial support received from the Universidad de los Andes, Chile, through the FAI (Fondo de Ayuda a la Investigación) research grant.
References
Ahmed, N. A., and D. V. Gokhale. 1989. “Entropy expressions and their estimators for multivariate distributions.” IEEE Trans. Inf. Theory 35 (3): 688–692. https://doi.org/10.1109/18.30996.
Argyris, C., C. Papadimitriou, and P. Panetsos. 2016. “Bayesian optimal sensor placement for modal identification of civil infrastructures.” J. Smart Cities 2 (2): 69–86. https://doi.org/10.18063/JSC.2016.02.001.
Astroza, R., H. Ebrahimian, and J. P. Conte. 2015. “Material parameter identification in distributed plasticity FE models of frame-type structures using nonlinear stochastic filtering.” J. Eng. Mech. 141 (5): 04014149. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000851.
Astroza, R., H. Ebrahimian, and J. P. Conte. 2017. “Batch and recursive Bayesian estimation methods for nonlinear structural system identification.” In Risk and reliability analysis: Theory and applications, 341–64. Cham, Switzerland: Springer.
Balan, T. A., F. C. Filippou, and E. P. Popov. 1997. “Constitutive model for 3D cyclic analysis of concrete structures.” J. Eng. Mech. 123 (2): 143–153. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:2(143).
Bayard, D. S., F. Y. Hadaegh, and D. R. Meldrum. 1988. “Optimal experiment design for identification of large space structures.” Automatica 24 (3): 357–364. https://doi.org/10.1016/0005-1098(88)90076-3.
Bellman, R., and K. J. Astrom. 1970. “On structural identifiability.” Math. Biosci. 7 (3–4): 329–339. https://doi.org/10.1016/0025-5564(70)90132-X.
Bowden, R. 1973. “The theory of parametric identification.” Econometrica 41 (6): 1069–1074. https://doi.org/10.2307/1914036.
Breitung, K. 1989. “Asymptotic approximations for probability integrals.” Probab. Eng. Mech. 4 (4): 187–190. https://doi.org/10.1016/0266-8920(89)90024-6.
CESMD (Center for Engineering Strong Motion Data). 2014. “Center for engineering strong motion data, CESMD: A cooperative effort.” Accessed September 2014. http://strongmotioncenter.org/.
Chatzis, M. N., E. N. Chatzi, and A. W. Smyth. 2015. “On the observability and identifiability of nonlinear structural and mechanical systems.” Struct. Control Health Monit. 22 (3): 574–593. https://doi.org/10.1002/stc.1690.
Chopra, A. K. 2012. Dynamics of structures: Theory and applications to earthquake engineering. 4th ed. Englewood Cliffs, NJ: Prentice-Hall.
Chow, H. M., H. F. Lam, T. Yin, and S. K. Au. 2011. “Optimal sensor configuration of a typical transmission tower for the purpose of structural model updating.” Struct. Control Health Monit. 18 (3): 305–320. https://doi.org/10.1002/stc.372.
Conte, J. P., P. K. Vijalapura, and M. Meghella. 2003. “Consistent finite-element response sensitivity analysis.” J. Eng. Mech. 129 (12): 1380–1393. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:12(1380).
Cover, T. M., and J. A. Thomas. 2006. Elements of information theory. Hoboken, NJ: Wiley.
Cramer, H. 1946. Mathematical methods of statistics. Princeton, NJ: Princeton University Press.
Danai, K., S. A. Civjan, and M. M. Styckiewicz. 2013. “Sensor location selection for structures via identifiability analysis in the time-scale domain.” J. Sound Vib. 332 (24): 6296–6311. https://doi.org/10.1016/j.jsv.2013.06.015.
DiStefano, J. J., and C. Cobelli. 1980. “On parameter and structural identifiability: Nonunique observability/reconstructibility for identifiable systems, other ambiguities, and new definitions.” IEEE Trans. Autom. Control 25 (4): 830–833. https://doi.org/10.1109/TAC.1980.1102439.
Duijndam, A. J. W. 1988. “Bayesian estimation in seismic inversion. Part II: Uncertainty analysis.” Geophys. Prospect. 36 (8): 899–918. https://doi.org/10.1111/j.1365-2478.1988.tb02199.x.
Ebrahimian, H., R. Astroza, and J. P. Conte. 2015. “Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method.” Earthquake Eng. Struct. Dyn. 44 (10): 1495–1522. https://doi.org/10.1002/eqe.2532.
Ebrahimian, H., R. Astroza, J. P. Conte, and R. A. de Callafon. 2017. “Nonlinear finite element model updating for damage identification of civil structures using batch Bayesian estimation.” Mech. Syst. Signal Process. 84 (Part B): 194–222. https://doi.org/10.1016/j.ymssp.2016.02.002.
Filippou, F. C., E. P. Popov, and V. V. Bertero. 1983. Effects of bond deterioration on hysteretic behavior of reinforced concrete joints. Berkeley, CA: Earthquake Engineering Research Center, College of Engineering, UC Berkeley.
Goodwin, G. C., and R. L. Payne. 1977. Dynamic system identification: Experiment design and data analysis. New York: Academic Press.
Heredia-Zavoni, E., and L. Esteva. 1998. “Optimal instrumentation of uncertain structural systems subject to earthquake ground motions.” Earthquake Eng. Struct. Dyn. 27 (4): 343–362. https://doi.org/10.1002/(SICI)1096-9845(199804)27:4%3C343::AID-EQE726%3E3.0.CO;2-F.
International Code Council. 2011. 2012 international building code. Country Club Hills, IL: International Code Council.
Jacquez, J. A., and P. Greif. 1985. “Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design.” Math. Biosci. 77 (1–2): 201–227. https://doi.org/10.1016/0025-5564(85)90098-7.
Kammer, D. C. 1991. “Sensor placement for on-orbit modal identification and correlation of large space structures.” J. Guidance Control Dyn. 14 (2): 251–259. https://doi.org/10.2514/3.20635.
Katafygiotis, L. S., and J. L. Beck. 1998. “Updating models and their uncertainties. Part II: Model identifiability.” J. Eng. Mech. 124 (4): 463–467. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(463).
Katafygiotis, L. S., C. Papadimitriou, and H.-F. Lam. 1998. “A probabilistic approach to structural model updating.” Soil Dyn. Earthquake Eng. 17 (7–8): 495–507. https://doi.org/10.1016/S0267-7261(98)00008-6.
Kirkegaard, P. H., and R. Brincker. 1994. “On the optimal location of sensors for parametric identification of linear structural systems.” Mech. Syst. Signal Process. 8 (6): 639–647. https://doi.org/10.1006/mssp.1994.1045.
Kleiber, M., H. Antunez, T. D. Hien, and P. Kowalczyk. 1997. Parameter sensitivity in nonlinear mechanics: Theory and finite element computations. Chichester, UK: Wiley.
Mehra, R. K. 1974. “Optimal input signals for parameter estimation in dynamic systems: Survey and new results.” IEEE Trans. Autom. Control 19 (6): 753–768. https://doi.org/10.1109/TAC.1974.1100701.
Moon, T. K., and W. C. Stirling. 2000. Mathematical methods and algorithms for signal processing. Upper Saddle River, NJ: Prentice-Hall.
Nguyen, V. V., and E. F. Wood. 1982. “Review and unification of linear identifiability concepts.” SIAM Rev. 24 (1): 34–51. https://doi.org/10.1137/1024002.
OpenSees. 2014. “Open system for earthquake engineering simulation.” October 1, 2014. https://opensees.berkeley.edu/.
Pant, S. 2018. “Information sensitivity functions to assess parameter information gain and identifiability of dynamical systems.” J. R. Soc. Interface 15 (142): 20170871. https://doi.org/10.1098/rsif.2017.0871.
Pant, S., and D. Lombardi. 2015. “An information-theoretic approach to assess practical identifiability of parametric dynamical systems.” Math. Biosci. 268: 66–79. https://doi.org/10.1016/j.mbs.2015.08.005.
Papadimitriou, C. 2004. “Optimal sensor placement methodology for parametric identification of structural systems.” J. Sound Vib. 278 (4–5): 923–947. https://doi.org/10.1016/j.jsv.2003.10.063.
Papadimitriou, C., J. L. Beck, and S. K. Au. 2000. “Entropy-based optimal sensor location for structural model updating.” J. Vib. Control 6 (5): 781–800. https://doi.org/10.1177/107754630000600508.
Papadimitriou, C., and G. Lombaert. 2012. “The effect of prediction error correlation on optimal sensor placement in structural dynamics.” Mech. Syst. Signal Process. 28: 105–127. https://doi.org/10.1016/j.ymssp.2011.05.019.
Raue, A., V. Becker, U. Klingmüller, and J. Timmer. 2010. “Identifiability and observability analysis for experimental design.” Chaos 20 (4): 045105. https://doi.org/10.1063/1.3528102.
Reid, J. G. 1977. “Structural identifiability in linear time-invariant systems.” IEEE Trans. Autom. Control 22 (2): 242–246. https://doi.org/10.1109/TAC.1977.1101474.
Rothenberg, T. J. 1971. “Identification in parametric models.” Econometrica 39 (3): 577–591. https://doi.org/10.2307/1913267.
Staley, R. M., and P. C. Yue. 1970. “On system parameter identifiability.” Inf. Sci. 2 (2): 127–138. https://doi.org/10.1016/S0020-0255(70)80046-9.
Tarantola, A. 2005. Inverse problem theory and methods for model parameter estimation. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Taucer, F. F., E. Spacone, and F. C. Filippou. 1991. A fiber beam-column element for seismic response analysis of reinforced concrete structures: UBC/EERC-91/17. Berkeley, CA: Earthquake Engineering Research Center, College of Engineering, UC Berkeley.
Teergele, J., and K. Danai. 2015. “Selection of outputs for distributed parameter systems by identifiability analysis in the time-scale domain.” Int. J. Syst. Sci. 46 (16): 2939–2954. https://doi.org/10.1080/00207721.2014.884251.
Tunali, E. T., and T. J. Tarn. 1987. “New results for identifiability of nonlinear systems.” IEEE Trans. Autom. Control 32 (2): 146–154. https://doi.org/10.1109/TAC.1987.1104544.
Udwadia, F. E. 1994. “Methodology for optimum sensor locations for parameter identification in dynamic systems.” J. Eng. Mech. 120 (2): 368–390. https://doi.org/10.1061/(ASCE)0733-9399(1994)120:2(368).
Van Trees, H. L. 2002. Optimum array processing, Part IV of detection, estimation, and modulation theory. New York: Wiley.
Xia, X., and C. H. Moog. 2003. “Identifiability of nonlinear systems with application to HIV/AIDS models.” IEEE Trans. Autom. Control 48 (2): 330–336. https://doi.org/10.1109/TAC.2002.808494.
Yuen, K. V., and S. C. Kuok. 2015. “Efficient Bayesian sensor placement algorithm for structural identification: A general approach for multi-type sensory systems.” Earthquake Eng. Struct. Dyn. 44 (5): 757–774. https://doi.org/10.1002/eqe.2486.
Zhang, Y., and A. Der Kiureghian. 1993. “Dynamic response sensitivity of inelastic structures.” Comput. Methods Appl. Mech. Eng. 108 (1–2): 23–36. https://doi.org/10.1016/0045-7825(93)90151-M.
Zona, A., M. Barbato, and J. P. Conte. 2006. “Finite element response sensitivity analysis of continuous steel-concrete composite girders.” Steel Compos. Struct. 6 (3): 183–202. https://doi.org/10.12989/scs.2006.6.3.183.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 145 • Issue 7 • July 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Jan 13, 2018
Accepted: Oct 3, 2018
Published online: Apr 22, 2019
Published in print: Jul 1, 2019
Discussion open until: Sep 22, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.