# Real-Time Identification of Time-Varying Cable Force Using an Improved Adaptive Extended Kalman Filter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equation of the Stay Cable Motion

#### 2.2. Cable Force Identification by IAEKF

#### 2.2.1. Update of the Error Covariance Matrices

#### 2.2.2. Update of the Fading-Factor Matrix

#### 2.3. Flowchart of the Proposed Method

## 3. Numerical Validation

^{2}; the mass per unit length is 32.93 kg/m; the static tension is ${T}_{0}=1470\mathrm{kN}$; the fundamental frequency at the initial tension is 0.903 Hz; and the damping ratio is assumed to be 1%.

## 4. Experimental Verification

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Alamdari, M.M.; Khoa, N.L.D.; Wang, Y.; Samali, B.; Zhu, X.Q. A multi-way data analysis approach for structural health monitoring of a cable-stayed bridge. Struct. Health Monit.
**2019**, 18, 35–48. [Google Scholar] [CrossRef] - Li, S.L.; Wei, S.Y.; Bao, Y.Q.; Li, H. Condition assessment of cables by pattern recognition of vehicle-induced cable tension ratio. Eng. Struct.
**2018**, 155, 1–15. [Google Scholar] [CrossRef] - Zhang, J.; Au, F.T.K.; Yang, D. Finite element model updating of long-span cable-stayed bridge by kriging surrogate model. Struct. Eng. Mech.
**2020**, 74, 157–173. [Google Scholar] - Zhang, X.; Peng, J.Y.; Cao, M.S.; Damjanovic, D.; Ostachowicz, W. Identification of instantaneous tension of bridge cables from dynamic responses: STRICT algorithm and applications. Mech. Syst. Signal Process.
**2020**, 142, 106729. [Google Scholar] [CrossRef] - Li, H.; Ou, J.P. The state of the art in structural health monitoring of cable-stayed bridges. J. Civ. Struct. Health Monit.
**2016**, 6, 43–67. [Google Scholar] [CrossRef] - Xu, M.Q.; Li, J.; Wang, S.Q.; Yang, N.; Hao, H. Damage detection of wind turbine blades by Bayesian multivariate cointegration. Ocean Eng.
**2022**, 258, 111603. [Google Scholar] - Wang, S.Q.; Xu, M.Q. Modal strain energy-based structural damage identification: A review and comparative study. Struct. Eng. Int.
**2019**, 29, 234–248. [Google Scholar] [CrossRef] - Xu, M.Q.; Guo, J.; Wang, S.Q.; Li, J.; Hao, H. Structural damage identification with limited modal measurements and ultra-sparse Bayesian regression. Struct. Control Health Monit.
**2021**, 28, e2729. [Google Scholar] [CrossRef] - Zarbaf, S.E.H.A.M.Z.; Norouzi, M.; Allemang, R.; Hunt, V.; Helmicki, A.; Venkatesh, C. Vibration-based cable condition assessment: A novel application of neural networks. Eng. Struct.
**2018**, 177, 291–305. [Google Scholar] [CrossRef] - Ren, J.Y.; Zhang, B.; Zhu, X.Q.; Li, S.H. Damaged cable identification in cable-stayed bridge from bridge deck strain measurements using support vector machine. Adv. Struct. Eng.
**2022**, 25, 754–771. [Google Scholar] [CrossRef] - Hou, J.L.; Li, C.; Jankowski, L.; Shi, Y.K.; Su, L.; Yu, S.; Geng, T.S. Damage identification of suspender cables by adding virtual supports with the substructure isolation method. Struct. Control Health Monit.
**2020**, 28, e2677. [Google Scholar] [CrossRef] - Bao, Y.Q.; Shi, Z.Q.; Beck, J.L.; Li, H.; Hou, T.Y. Identification of time-varying cable tension forces based on adaptive sparse time-frequency analysis of cable vibrations. Struct. Control Health Monit.
**2016**, 24, e1889. [Google Scholar] [CrossRef] - Bao, Y.Q.; Guo, Y.B.; Li, H. A machine learning-based approach for adaptive sparse time frequency analysis used in structural health monitoring. Struct. Health Monit.
**2020**, 19, 1963–1975. [Google Scholar] [CrossRef] - Liu, J.L.; Zheng, J.Y.; Wei, X.J.; Ren, W.X.; Laory, I. A combined method for instantaneous frequency identification in low frequency structures. Eng. Struct.
**2019**, 194, 370–383. [Google Scholar] [CrossRef] - Liu, J.L.; Ren, W.X.; Wang, Z.C.; Hu, Y.D. Instantaneous frequency identification based on synchrosqueezing wavelet transformation. J. Vib. Shock
**2013**, 32, 37–42. (In Chinese) [Google Scholar] - Liu, J.L.; Zheng, J.Y.; Wei, X.J.; Liao, F.Y.; Luo, Y.P. A new instantaneous frequency extraction method for nonstationary response signals in civil engineering structures. J. Low Freq. Noise Vib. Act. Control
**2018**, 37, 146134841879053. [Google Scholar] [CrossRef] - Wang, C.; Ren, W.X.; Wang, Z.C.; Zhu, H.P. Instantaneous frequency identification of time-varying structures by continuous wavelet transform. Eng. Struct.
**2013**, 52, 17–25. [Google Scholar] [CrossRef] - Wang, C.; Zhang, J.; Zhu, H.P. A combined method for time-varying parameter identification based on variational mode decomposition and generalized Morse wavelet. Int. J. Struct. Stab. Dyn.
**2020**, 10, 2050077. [Google Scholar] [CrossRef] - Hou, S.T.; Dong, B.; Fan, J.H.; Wu, G.; Wang, H.C.; Han, Y.T.; Zhao, X.J. Variational mode decomposition based time-varying force identification of stay cables. Appl. Sci.
**2021**, 11, 1254. [Google Scholar] [CrossRef] - Xue, S.L.; Shen, R.L. Real time cable force identification by short time sparse time domain algorithm with half wave. Measurement
**2020**, 152, 107355. [Google Scholar] [CrossRef] - Yang, Y.C.; Li, S.L.; Nagarajaiah, S.; Li, H.; Zhou, P. Real-time output-only identification of time varying cable tension from accelerations via complexity pursuit. J. Struct. Eng.
**2016**, 142, 1–10. [Google Scholar] [CrossRef] - Li, H.; Zhang, F.J.; Jin, Y.Z. Real-time identification of time-varying tension in stay cables by monitoring cable transversal acceleration. Struct. Control Health Monit.
**2014**, 21, 1100–1117. [Google Scholar] [CrossRef] - Zhang, F.J.; Li, H.; Mao, C.X. Identification of time-variant stay cable tension force using a wavelet method in combination with extended Kalman filter. J. Civ. Eng. Manag.
**2013**, 30, 1–5. (In Chinese) [Google Scholar] - Lei, Y.; Yang, N. Simultaneous identification of structural time-varying physical parameters and unknown excitations using partial measurements. Eng. Struct.
**2020**, 214, 110672. [Google Scholar] [CrossRef] - Yang, N.; Li, J.; Lei, Y.; Hao, H. Identification of time-varying nonlinear structural physical parameters by integrated WMA and UKF/UKF-UI. Nonlinear Dyn.
**2021**, 106, 681–706. [Google Scholar] [CrossRef] - Hoshiya, M.; Saito, E. Structural identification by extended Kalman filter. J. Eng. Mech. (ASCE)
**1984**, 110, 1757–1771. [Google Scholar] [CrossRef] - Lei, Y.; Chen, F.; Zhou, H. An algorithm based on two-step Kalman filter for intelligent structural damage detection. Struct. Control Health Monit.
**2014**, 22, 694–706. [Google Scholar] [CrossRef] - Papadimitriou, C.; Fritzen, C.P.; Kraemer, P.; Ntotsios, E. Fatigue predictions in entire body of metallic structures from a limited number of vibration sensors using Kalman filtering. Struct. Control Health Monit.
**2011**, 18, 554–557. [Google Scholar] [CrossRef] - Yuen, K.V.; Kuok, S.C. Online updating and uncertainty quantification using nonstationary output-only measurement. Mech. Syst. Signal Process.
**2016**, 66–67, 62–77. [Google Scholar] [CrossRef] - Yang, J.N.; Lin, S.L.; Huang, H.W.; Zhou, L. An adaptive extended Kalman filter for structural damage identification. Struct. Control Health Monit.
**2006**, 13, 849–867. [Google Scholar] [CrossRef] - Huang, Q.; Xu, Y.L.; Liu, H.J. An efficient algorithm for simultaneous identification of time-varying structural parameters and unknown excitations of a building structure. Eng. Struct.
**2015**, 98, 29–37. [Google Scholar] [CrossRef] - Yang, Y.H.; Nagayama, T.; Xue, K. Structure system estimation under seismic excitation with an adaptive extended Kalman filter. J. Sound Vib.
**2020**, 489, 115690. [Google Scholar] [CrossRef] - Mu, H.Q.; Kuok, S.C.; Yuen, K.V. Stable robust extended Kalman filter. J. Aerosp. Eng.
**2016**, 30, B4016010. [Google Scholar] [CrossRef] - Eftekhar Azam, S.; Mariani, S. Online damage detection in structural systems via dynamic inverse analysis: A recursive Bayesian approach. Eng. Struct.
**2018**, 159, 28–45. [Google Scholar] [CrossRef] - Shafei, A.; Shafei, H. A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment. Multibody Syst. Dyn.
**2016**, 38, 21–42. [Google Scholar] [CrossRef] - Askari, M.; Yu, Y.; Zhang, C.W.; Samali, B.; Gu, X.Y. Real-time tracking of structural stiffness reduction with unknown inputs, using self-adaptive recursive least-square and curvature-change techniques. Int. J. Struct. Stab. Dyn.
**2019**, 19, 1950123. [Google Scholar] [CrossRef] - Yang, N.; Lei, Y.; Li, J.; Hao, H.; Huang, J.S. A substructural and wavelet multiresolution approach for identifying time-varying physical parameters by partial measurements. J. Sound Vib.
**2022**, 523, 116737. [Google Scholar] [CrossRef] - Zhou, D.H.; Frank, P.M. Strong tracking filtering of nonlinear time-varying stochastic systems with coloured noise: Application to parameter estimation and empirical robustness analysis. Int. J. Control
**1996**, 65, 295–307. [Google Scholar] [CrossRef]

**Figure 2.**Identified cable force by using the traditional EKF with empirical fading factor and noise error covariance matrices.

**Figure 3.**Identified cable force by the proposed method when RMS = 2% and the mutation amplitude is 10%.

**Figure 4.**Identified cable force by the proposed method when RMS = 2%. (

**a**) The mutation amplitude is 5%; (

**b**) The mutation amplitude is 15%.

**Figure 5.**Identified cable force by the proposed method when RMS = 6% and the mutation amplitude is 10%.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, N.; Li, J.; Xu, M.; Wang, S.
Real-Time Identification of Time-Varying Cable Force Using an Improved Adaptive Extended Kalman Filter. *Sensors* **2022**, *22*, 4212.
https://doi.org/10.3390/s22114212

**AMA Style**

Yang N, Li J, Xu M, Wang S.
Real-Time Identification of Time-Varying Cable Force Using an Improved Adaptive Extended Kalman Filter. *Sensors*. 2022; 22(11):4212.
https://doi.org/10.3390/s22114212

**Chicago/Turabian Style**

Yang, Ning, Jun Li, Mingqiang Xu, and Shuqing Wang.
2022. "Real-Time Identification of Time-Varying Cable Force Using an Improved Adaptive Extended Kalman Filter" *Sensors* 22, no. 11: 4212.
https://doi.org/10.3390/s22114212