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http://dx.doi.org/10.12989/sss.2019.24.2.209

An improved Kalman filter for joint estimation of structural states and unknown loadings  

He, Jia (College of Civil Engineering, Hunan University)
Zhang, Xiaoxiong (College of Civil Engineering, Hunan University)
Dai, Naxin (The School of Civil Engineering, The University of South China)
Publication Information
Smart Structures and Systems / v.24, no.2, 2019 , pp. 209-221 More about this Journal
Abstract
The classical Kalman filter (KF) provides a practical and efficient way for state estimation. It is, however, not applicable when the external excitations applied to the structures are unknown. Moreover, it is known the classical KF is only suitable for linear systems and can't handle the nonlinear cases. The aim of this paper is to extend the classical KF approach to circumvent the aforementioned limitations for the joint estimation of structural states and the unknown inputs. On the basis of the scheme of the classical KF, analytical recursive solution of an improved KF approach is derived and presented. A revised form of observation equation is obtained basing on a projection matrix. The structural states and the unknown inputs are then simultaneously estimated with limited measurements in linear or nonlinear systems. The efficiency and accuracy of the proposed approach is verified via a five-story shear building, a simply supported beam, and three sorts of nonlinear hysteretic structures. The shaking table tests of a five-story building structure are also employed for the validation of the robustness of the proposed approach. Numerical and experimental results show that the proposed approach can not only satisfactorily estimate structural states, but also identify unknown loadings with acceptable accuracy for both linear and nonlinear systems.
Keywords
Kalman filter; state estimation; load identification; limited measurements; nonlinear hysteretic structures;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 Bernal, D. (2013), "Kalman filter damage detection in the presence of changing process and measurement noise", Mech. Syst. Signal Pr., 39(1-2), 361-371. DOI: 10.1016/j.ymssp.2013.02.012.   DOI
2 Ching, J. and Beck, J.L. (2007), "Real-time reliability estimation for serviceability limit states in structures with uncertain dynamic excitation and incomplete output data", Probab. Eng. Mech., 22(1), 50-62. DOI: 10.1016/j.probengmech.2006.05.006.   DOI
3 Eftekhar Azam, S., Ghisi, A. and Mariani, S. (2012a), "Parallelized sigma-point Kalman filtering for structural dynamics", Comput. Struct., 92-93, 193-205. DOI:10.1016/j.compstruc.2011.11.004.   DOI
4 Eftekhar Azam, S., Bagherinia, M. and Mariani, S. (2012b), "Stochastic system identification via particle and sigma-point Kalman filtering", Scientia Iranica, 19(4), 982-991. DOI: 10.1016/j.scient.2012.06.007.   DOI
5 Gao, F. and Lu, Y. (2006), "A Kalman-filter based time-domain analysis for structural damage diagnosis with noisy signals," J. Sound Vib., 297(3-5), 916-930. DOI: 10.1016/j.jsv.2006.05.007.   DOI
6 Eftekhar Azam, S., Chatzi, E. and Papadimitriou, C. (2015a), "A dual Kalman filter approach for state estimation via output-only acceleration measurements", Mech. Syst. Signal Pr., 60-61, 866-886. DOI:10.1016/j.ymssp.2015.02.001.   DOI
7 Eftekhar Azam, S., Chatzi, E., Papadimitriou, C. and Smyth, A. (2015b), "Experimental validation of the Kalman-type filters for online and real-time state and input estimation", J. Vib. Sound, DOI: 10.1177/1077546315617672.
8 Eftekhar Azam, S., Mariani, S. and Attari, N.K.A. (2017), "Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters", Nonlinear Dynam., 89(2), 1489-1511. DOI: 10.1007/s11071-017-3530-1.   DOI
9 Gillijns, S. and De Moor, B. (2007a), "Unbiased minimumvariance input and state estimation for linear discrete-time systems", Automatica, 43, 111-116. DOI: 10.1016/j.automatica.2006.08.002.   DOI
10 Gillijns, S. and De Moor, B. (2007b), "Unbiased minimumvariance input and state estimation for linear discrete-time systems with direct feedthrough", Automatica, 43, 934-937. DOI: 10.1016/j.automatica.2006.11.016.   DOI
11 He, J., Xu, Y.L., Zhang, C.D. and Zhang, X.H. (2015), "Optimum control system for earthquake-excited building structures with minimal number of actuators and sensors", Smart Struct. Syst., 16(6), 981-1002. DOI: 10.12989/sss.2015.16.6.981.   DOI
12 Lei, Y., Luo, S. and Su, Y. (2016), "Data fusion based improved Kalman filter with unknown inputs and without collocated acceleration measurements", Smart Struct. Syst., 18(3), 375-387. DOI: 10.12989/sss.2016.18.3.375.   DOI
13 Hernandez, E.M. (2011), "A natural observer for optimal state estimation in second order linear structural systems", Mech. Syst. Signal Pr., 25(8), 2938-2947. DOI:10.1016/j.ymssp.2011.06.003.   DOI
14 Hernandez, E.M., Bernal, B. and Caracoglia, L. (2013), "On-line monitoring of wind-induced stresses and fatigue damage in instrumented structures", Struct. Control. Health Monit., 20(10), 1291-1302. DOI: 10.1002/stc.1536.   DOI
15 Hsieh, C.S. (2009), "Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs", Automatica, 45(9), 2149-2153. DOI: 10.1016/j.automatica.2009.05.004.   DOI
16 Hu, R.P., Xu, Y.L., Lu, X., Zhang, C.D., Zhang, Q.L. and Ding, J.M. (2017), "Integrated multi-type sensor placement and response reconstruction method for high-rise buildings under unknown seismic loading", Struct. Des. Tall Spec. Build., 27(6), e1453, DOI: 10.1002/tal.1453.   DOI
17 Kim, D.W. and Park, C.S. (2017), "Application of Kalman filter for estimating a process disturbance in a building space", Sustainability, 9, 1868. DOI: 10.3390/su9101868.   DOI
18 Liu, L., Zhu, J., Su, Y. and Lei, Y. (2016), "Improved Kalman filter with unknown inputs based on data fusion of partial acceleration and displacement measurements", Smart Struct. Syst., 17(6), 903-915. DOI: 10.12989/sss.2016.17.6.903.   DOI
19 Lourens, E., Papadimitriou, C., Gillijns, S., Reynders, E., De Roeck, G. and Lombaert, G. (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 Pr., 29, 310-327. DOI: 10.1016/j.ymssp.2012.01.011.   DOI
20 Lourens, E., Reynders, E., DeRoeck, G., Degrande, G. and Lombaert, G. (2012b), "An augmented Kalman filter for force identification in structural dynamics", Mech. Syst. Signal Pr., 27, 446-460. DOI: 10.1016/j.ymssp.2011.09.025.   DOI
21 Naets, F., Cuadrado, J. and Desmet, W. (2015), "Stable force identification in structural dynamics using Kalman filtering and dummy-measurements", Mech. Syst. Signal Pr., 50-51, 235-248. DOI: 10.1016/j.ymssp.2014.05.042.   DOI
22 Pan, S.W., Su, H.Y., Wang, H. and Chu, J. (2011), "The study of joint input and state estimation with Kalman filtering", Trans. Inst. Meas. Control, 33(8), 901-918. DOI: 10.1177/0142331210361551.   DOI
23 Papadimitriou, C., Fritzen, C., Kraemer, P. and Ntotsios, E. (2011), "Fatigue predictions in entire body of metallic structures from a limited number of vibration sensors using Kalman filtering", Struct. Control. Health Monit., 18, 554-573. DOI: 10.1002/stc.395.   DOI
24 Ren, P. and Zhou, Z. (2017), "Strain estimation of truss structures based on augmented Kalman filtering and modal expansion", Adv. Mech. Eng., 9(11), 1-10. DOI: 10.1177/1687814017735788
25 Smyth, A. and Wu, M. (2007), "Multi-rate Kalman filtering for the data fusion of displacement and acceleration response measurements in dynamic system monitoring", Mech. Syst. Signal Pr., 21(2), 706-723. DOI: 10.1016/j.ymssp.2006.03.005.   DOI
26 Vicario, F., Phan, M.Q., Betti, R. and Longman, R.W. (2015), "Output-only observer/Kalman filter identification (O3KID)", Struct. Control. Health Monit., 22(5), 847-872. DOI: 10.1002/stc.1719.   DOI
27 Welch, G. and Bishop, G. (1995), "An Introduction to the Kalman Filter", Technical Report, University of North Carolina at Chapel Hill Chapel Hill, NC, USA, DOI: 10.1145/800233.807054.
28 Zhi, L.H., Fang, M.X. and Li, Q.S. (2017), "Estimation of wind loads on a tall building by an inverse method", Struct. Control. Health Monit., 24, e1908, DOI: 10.1002/stc.198   DOI
29 Wu, A.L., Loh, C.H., Yang, J.N., Weng, J.H., Chen, C.H. and Ueng, T.S. (2009), "Input force identification: Application to soil-pile interaction", Struct. Control Vib.. Control, 16, 223-240, DOI: 10.1002/stc.308.
30 Zhi, L.H., Li, Q.S. and Fang, M.X. (2016), "Identification of wind loads and estimation of structural responses of super-tall buildings by an inverse method", Comput.-Aided Civil Infrastruct. Eng., 31, 966-982. DOI: 10.1111/mice.12241.   DOI
31 Zhu, S., Zhang, X.H., Xu, Y.L. and Zhan, S. (2013), "Multi-type sensor placement for multi-scale response reconstruction", Adv. Struct. Eng., 16(10), 1779-1797. DOI: 10.1260/1369-4332.16.10.1779.   DOI
32 Zhang, C.D. and Xu, Y.L. (2017), "Structural damage identification via response reconstruction under unknown excitation", Struct. Control. Health Monit., 24(8), e1953, DOI: 10.1002/stc.1953.   DOI