Data-Driven Reconstruction of Dynamical Forces and Responses through Bayesian Expectation-Maximization and Modal Reduction Methods
Publication: Journal of Engineering Mechanics
Volume 149, Issue 6
Abstract
This paper presents a new Bayesian approach for estimating input and state in linear structures using response-only measurements. The proposed approach benefits from a modally reduced state-space model, which circumvents the dimensionality of dynamical responses in complex structures through a low-dimensional subspace. It also substitutes unknown physical forces with a set of equivalent modal forces, which is beneficial when the magnitude and location of input forces are unknown. In this work, these forces are characterized through the conventional random walk model and a class of stationary Gaussian processes. Subsequently, an augmented state-space model is constructed to describe modal states and input loads. Based on this model, a Bayesian expectation-maximization (BEM) methodology is developed to identify the input, state, and noise parameters. This noise identification perspective activates uncertainty quantification in joint input-state estimation problems and enables quantifying the degree of confidence in the estimated quantities. When the proposed method is tested using numerical and experimental examples, accurate estimations and reasonable uncertainty bounds are acquired for the dynamical state and input forces. Although the literature reports the superiority of the Gaussian process latent force model over the random walk model without using a unified noise calibration strategy, this study, to our best knowledge, is the first effort to compare and interpret the results on a consistent basis where the noise and input characteristics are all identified from the data through BEM.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This study has received financial support from the Hong Kong Research Grants Council under Project Nos. 16212918 and 16211019. The authors would like to acknowledge the IASC-ASCE SHM Task Group for making the data sets publicly available.
References
Astroza, R., H. Ebrahimian, and J. P. Conte. 2019. “Performance comparison of Kalman-based filters for nonlinear structural finite element model updating.” J. Sound Vib. 438 (Jan): 520–542. https://doi.org/10.1016/j.jsv.2018.09.023.
Aucejo, M., O. De Smet, and J.-F. Deü. 2019. “Practical issues on the applicability of Kalman filtering for reconstructing mechanical sources in structural dynamics.” J. Sound Vib. 442 (Mar): 45–70. https://doi.org/10.1016/j.jsv.2018.10.060.
Bernal, D. 2011. “The zero-order hold in time domain identification: An unnecessary operating premise.” Struct. Control Health Monit. 18 (5): 510–518. https://doi.org/10.1002/stc.383.
Chatzi, E. N., and A. W. Smyth. 2009. “The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing.” Struct. Control Health Monit. 16 (1): 99–123. https://doi.org/10.1002/stc.290.
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.
Ching, J., J. L. Beck, K. A. Porter, and R. Shaikhutdinov. 2006. “Bayesian state estimation method for nonlinear systems and its application to recorded seismic response.” J. Eng. Mech. 132 (4): 396–410. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:4(396).
Dertimanis, V. K., E. N. Chatzi, S. E. Azam, and C. Papadimitriou. 2019. “Input-state-parameter estimation of structural systems from limited output information.” Mech. Syst. Signal Process. 126 (Jul): 711–746. https://doi.org/10.1016/j.ymssp.2019.02.040.
Dyke, S., D. Bernal, J. Beck, and C. Ventura. 2003. “Experimental phase II of the structural health monitoring benchmark problem.” In Proc., 16th ASCE Engineering Mechanics Conf., 1–7. Reston, VA: ASCE.
Eftekhar Azam, S., E. Chatzi, and C. Papadimitriou. 2015. “A dual Kalman filter approach for state estimation via output-only acceleration measurements.” Mech. Syst. Signal Process. 60–61 (Aug): 866–886. https://doi.org/10.1016/j.ymssp.2015.02.001.
Johnson, E. A., H. F. Lam, L. S. Katafygiotis, and J. L. Beck. 2004. “Phase I IASC-ASCE structural health monitoring benchmark problem using simulated data.” J. Eng. Mech. 130 (1): 3–15. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:1(3).
Kalman, R. E. 1960. “A new approach to linear filtering and prediction problems.” J. Basic Eng. 82 (1): 35–45. https://doi.org/10.1115/1.3662552.
Kontoroupi, T., and A. W. Smyth. 2016. “Online noise identification for joint state and parameter estimation of nonlinear systems.” ASCE-ASME J. Risk Uncertainty Eng. Syst. Part A: Civ. Eng. 2 (3): B4015006. https://doi.org/10.1061/AJRUA6.0000839.
Kullaa, J. 2019. “Bayesian virtual sensing in structural dynamics.” Mech. Syst. Signal Process. 115 (Jan): 497–513. https://doi.org/10.1016/j.ymssp.2018.06.010.
Lei, Y., D. Xia, K. Erazo, and S. Nagarajaiah. 2019. “A novel unscented Kalman filter for recursive state-input-system identification of nonlinear systems.” Mech. Syst. Signal Process. 127 (Jul): 120–135. https://doi.org/10.1016/j.ymssp.2019.03.013.
Lourens, E., C. Papadimitriou, S. Gillijns, E. Reynders, G. De Roeck, and G. Lombaert. 2012a. “Joint input-response estimation for structural systems based on reduced-order models and vibration data from a limited number of sensors.” Mech. Syst. Signal Process. 29 (May): 310–327. https://doi.org/10.1016/j.ymssp.2012.01.011.
Lourens, E., E. Reynders, G. De Roeck, G. Degrande, and G. Lombaert. 2012b. “An augmented Kalman filter for force identification in structural dynamics.” Mech. Syst. Signal Process. 27 (1): 446–460. https://doi.org/10.1016/j.ymssp.2011.09.025.
Maes, K., M. N. Chatzis, R. Vandebril, and G. Lombaert. 2021. “Observability of modally reduced order models with unknown parameters.” Mech. Syst. Signal Process. 146 (Jan): 106993. https://doi.org/10.1016/j.ymssp.2020.106993.
Maes, K., A. Iliopoulos, W. Weijtjens, C. Devriendt, and G. Lombaert. 2016a. “Dynamic strain estimation for fatigue assessment of an offshore monopile wind turbine using filtering and modal expansion algorithms.” Mech. Syst. Signal Process. 76–77 (Aug): 592–611. https://doi.org/10.1016/j.ymssp.2016.01.004.
Maes, K., A. W. Smyth, G. De Roeck, and G. Lombaert. 2016b. “Joint input-state estimation in structural dynamics.” Mech. Syst. Signal Process. 70–71 (Mar): 445–466. https://doi.org/10.1016/j.ymssp.2015.07.025.
Mehra, R. 1970. “On the identification of variances and adaptive Kalman filtering.” IEEE Trans. Autom. Control 15 (2): 175–184. https://doi.org/10.1109/TAC.1970.1099422.
Mehra, R. 1972. “Approaches to adaptive filtering.” IEEE Trans. Autom. Control 17 (5): 693–698. https://doi.org/10.1109/TAC.1972.1100100.
Naets, F., J. Croes, and W. Desmet. 2015. “An online coupled state/input/parameter estimation approach for structural dynamics.” Comput. Methods Appl. Mech. Eng. 283 (Jan): 1167–1188. https://doi.org/10.1016/j.cma.2014.08.010.
Nayek, R., S. Chakraborty, and S. Narasimhan. 2019. “A Gaussian process latent force model for joint input-state estimation in linear structural systems.” Mech. Syst. Signal Process. 128 (Aug): 497–530. https://doi.org/10.1016/j.ymssp.2019.03.048.
Okubo, N., and K. Yamaguchi. 1995. “Prediction of dynamic strain distribution under operating condition by use of modal analysis.” In Proc., 13th Int. Modal Analysis Conf. Nashville, TN: Society for Experimental Mechanics.
Rasmussen, C. E., and C. K. Williams. 2005. Gaussian processes for machine learning, 4. Cambridge, MA: MIT Press. https://doi.org/10.7551/mitpress/3206.001.0001.
Rogers, T. J., K. Worden, and E. J. Cross. 2020. “On the application of Gaussian process latent force models for joint input-state-parameter estimation: With a view to Bayesian operational identification.” Mech. Syst. Signal Process. 140 (Jun): 106580. https://doi.org/10.1016/j.ymssp.2019.106580.
Sarkka, S. 2013. Bayesian filtering and smoothing. Cambridge, UK: Cambridge University Press.
Sarkka, S., A. Solin, and J. Hartikainen. 2013. “Spatiotemporal learning via infinite-dimensional Bayesian filtering and smoothing: A look at Gaussian process regression through Kalman filtering.” IEEE Signal Process Mag. 30 (4): 51–61. https://doi.org/10.1109/MSP.2013.2246292.
Sedehi, O., L. S. Katafygiotis, and C. Papadimitriou. 2020. “Hierarchical Bayesian operational modal analysis: Theory and computations.” Mech. Syst. Signal Process. 140 (Jun): 106663. https://doi.org/10.1016/j.ymssp.2020.106663.
Sedehi, O., C. Papadimitriou, D. Teymouri, and L. S. Katafygiotis. 2019. “Sequential Bayesian estimation of state and input in dynamical systems using output-only measurements.” Mech. Syst. Signal Process. 131 (Sep): 659–688. https://doi.org/10.1016/j.ymssp.2019.06.007.
Seo, S.-W., K.-J. Kim, and B.-K. Bae. 1998. “Estimation of operational strains from vibration measurements: An application to lead wires of chips on printed circuit board.” J. Sound Vib. 210 (5): 567–579. https://doi.org/10.1006/jsvi.1997.1327.
Song, M., R. Astroza, H. Ebrahimian, B. Moaveni, and C. Papadimitriou. 2020. “Adaptive Kalman filters for nonlinear finite element model updating.” Mech. Syst. Signal Process. 143 (Sep): 106837. https://doi.org/10.1016/j.ymssp.2020.106837.
Teymouri, D., O. Sedehi, L. S. Katafygiotis, and C. Papadimitriou. 2022. “A Bayesian expectation-maximization (BEM) methodology for joint input-state estimation and virtual sensing of structures.” Mech. Syst. Signal Process. 169 (Apr): 108602. https://doi.org/10.1016/j.ymssp.2021.108602.
Teymouri, D., O. Sedehi, L. S. Katafygiotis, and C. Papadimitriou. 2023. “Input-state-parameter-noise identification and virtual sensing in dynamical systems: A Bayesian expectation-maximization (BEM) perspective.” Mech. Syst. Signal Process. 185 (Feb): 109758. https://doi.org/10.1016/j.ymssp.2022.109758.
Yuen, K.-V. 2010. Bayesian methods for structural dynamics and civil engineering. Chichester, UK: Wiley.
Yuen, K.-V., and L. S. Katafygiotis. 2003. “Bayesian fast Fourier transform approach for modal updating using ambient data.” Adv. Struct. Eng. 6 (2): 81–95. https://doi.org/10.1260/136943303769013183.
Yuen, K.-V., and S. C. Kuok. 2016. “Online updating and uncertainty quantification using nonstationary output-only measurement.” Mech. Syst. Signal Process. 66–67 (Jan): 62–77. https://doi.org/10.1016/j.ymssp.2015.05.019.
Zou, J., E.-M. Lourens, and A. Cicirello. 2022. “Virtual sensing of subsoil strain response in monopile-based offshore wind turbines via Gaussian process latent force models.” Preprint, submitted July 13, 2022. https://arxiv.org/abs/2207.05901v1.
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© 2023 American Society of Civil Engineers.
History
Received: Oct 21, 2022
Accepted: Feb 3, 2023
Published online: Mar 29, 2023
Published in print: Jun 1, 2023
Discussion open until: Aug 29, 2023
ASCE Technical Topics:
- Analysis (by type)
- Bayesian analysis
- Continuum mechanics
- Dynamic response
- Dynamics (solid mechanics)
- Engineering fundamentals
- Engineering mechanics
- Gaussian process
- Mathematics
- Methodology (by type)
- Motion (dynamics)
- Numerical methods
- Probability
- Solid mechanics
- Stationary processes
- Statistical analysis (by type)
- Stochastic processes
- Structural dynamics
- Structural response
- Uncertainty principles
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