# Identification of Time-Varying External Force Using Group Sparse Regularization and Redundant Dictionary

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{1}-norm regularization, the proposed method can further improve the identified accuracy of unknown external force and greatly enhance the computational efficiency for the force identification problem.

## 1. Introduction

_{2}-norm of force vector with the Tikhonov matrix [10,13]. Many studies have been conducted on the application of the Tikhonov regularization in force identification problems [18]. However, the Tikhonov regularization method has some unavoidable limitations and disadvantages. For example, the approximate solution directly obtained from the Tikhonov regularization is too smooth, and the identified result may lack some details that the desired real solution might possess [19].

_{1}-norm regularization with different dictionaries for both impact force identification and harmonic force identification problems, and the identified accuracy can be improved by comparing it with the traditional Tikhonov regularization method. However, the sparsity of solution obtained from the standard l

_{1}-norm regularization is considered at the independent atom level, the complexity in the process of this force identification method is high, and the solving process is time-consuming. Moreover, for time-varying external force identification, local identification results may be unsatisfactory.

_{1}-norm regularization, the group sparse regularization method considers the sparsity of solutions in some group structures, which can reduce the complexity in the process of solving problems. In a special force identification problem, i.e., impact force identification, Qiao et al. [34] employs the prior information that the duration time of the actual impact force is quite short, and reasonable group sparsity of impact force in the time domain is considered, so group sparse regularization is reasonably introduced for impact force identification. Because the time-varying external force is usually not sparse in the time domain, the above prior information cannot be directly employed. How to introduce the group sparse regularization into the time-varying external force identification problem is an important work. If the group structures of atoms are not reasonable enough, the group sparse regularization method may not improve the identified accuracy of time-varying force. For example, the method in Ref. [35] directly uses the group number to determine the group structures of atoms; by comparing the identified results in Ref. [26], the identified accuracy of time-varying force component cannot be improved.

_{1}-norm regularization. Furthermore, some experiments are carried out on a simply supported beam to validate the effectiveness of the proposed method in Section 4. Finally, some conclusions are drawn in Section 5.

## 2. Theoretical Background

#### 2.1. Relationship between Structural Response and External Force

#### 2.2. Function Expansion Method for Force Identification Problem

#### 2.3. Sparse Regularization Techniques for Force Identification Problem

_{1}-norm regularization and group sparse regularization techniques are, respectively, introduced.

#### 2.3.1. Force Identification Problem Based on Standard l_{1}-Norm Regularization

_{1}-norm regularization [28] is a popular technique for obtaining sparse solutions to inverse problems. By combining Equation (7) with standard l

_{1}-norm regularization, the l

_{1}-norm of the coefficient vector is adopted to define the force identification problem. As a result, an unconstrained optimization problem can be defined as:

_{1}-norm of vector and $\lambda $ is a weight coefficient. When $\lambda \to \infty $, the solution of Equation (14) converges to the solution obtained from the least squares method.

_{1}-norm of vector $\alpha $ when the weight coefficient is given. Thus, the weight coefficient should first be determined for reasonable results, which will be introduced in the following section.

#### 2.3.2. Force Identification Problem Based on Group Sparse Regularization

_{1}-norm regularization only considers the standard sparsity of the solution, the underlying structure sparsity of the solution cannot be revealed. The structure sparsity of the solution is directly related to the underlying relevance of atoms in the given dictionary. Therefore, the underlying relevance of atoms should be analyzed at first, and the underlying structure of atoms, such as group structure, joint structure, or tree structure, can be reasonably determined. As a result, corresponding structured sparse regularization techniques can be introduced to the specific problem. That is to say, how to determine the underlying structure of atoms is an important issue for the structured sparse regularization methods.

#### 2.3.3. Solution Algorithm for Force Identification Problems

_{1}-norm regularization can be rewritten as:

_{1}-norm regularization, ${\left(\xb7\right)}_{{g}_{j}}$ is specified as ${\left(\xb7\right)}_{j}$, which is the j-th element of the vector, and $p=L$. $\epsilon $ is the tolerance error and is equal to 1 × 10

^{−6}.

## 3. Numerical Simulations

^{3}. The length is 1.52 m for horizontal and vertical components. Rayleigh damping is adopted, and the first two damping ratios are assumed to be 1%.

_{s}= 640 Hz, and the sampling time is T = 1 s. As a result, the number of sampling points is N = 640.

#### 3.1. Comparative Study between Two Force Identification Methods

_{1}-norm regularization and group sparse regularization are, respectively, shown in Figure 3 and Figure 4.

_{1}-norm regularization. The RPEs of the standard l

_{1}-norm regularization and group sparse regularization are 18.4109% and 15.7292%, respectively. Because the atom coefficients ${\alpha}_{2j}$ and ${\alpha}_{2j+1}$ have the relationship shown in Equation (15), the local identification accuracy of the time-varying external force can be further improved. These results indicate that the way to define group structures in this study is reasonable, which can improve.

_{1}-norm regularization and group sparse regularization techniques is 65.58 s and 7.63 s, respectively. That is to say, the group sparse regularization can also effectively enhance the calculated efficiency.

_{1}-norm regularization.

#### 3.2. Influence of Different Noise Levels and Measurement Points

_{1}-norm regularization. It is reasonable to consider the potential group structures of atoms in the given dictionary, so the proposed method can not only further improve the identification accuracy of the external force, but also significantly enhance the calculated efficiency. This way, defining objective function can effectively accord with the prior knowledge of atom relevance in the given dictionary.

## 4. Experimental Verifications

#### 4.1. Experimental Setup

^{11}N/m

^{2}. Mode testing and signal acquisition were, respectively conducted in this study, so 11 accelerometers were evenly placed on the beam, and the distance of each two adjacent accelerometers was 0.12 m.

_{v}

_{1}= 1.0531 × 10

^{7}N·m

^{−1}and k

_{v}

_{2}= 2.0495 × 10

^{7}N·m

^{−1}, respectively. Meanwhile, the rotational spring coefficients were set as k

_{r}

_{1}= 7.5103 × 10

^{1}N·m/rad and k

_{r}

_{2}= 6.2786 × 10

^{1}N·m/rad, respectively.

^{−3}, 1.8644 × 10

^{−3}, and 3.8858 × 10

^{−3}, were directly used to calculate the transfer matrix. Moreover, the comparisons on the first three mode shapes are plotted in Figure 11. For intuitive comparison, each modal shape vector is scaled by its 2-norm.

#### 4.2. Verification of Proposed Method

_{1}-norm regularization and group sparse regularization were selected by the L-curve criterion, as, respectively, shown in Figure 12 and Figure 13. As a result, the weight coefficients were determined, and the identified external force obtained from the proposed method was compared with that obtained from the standard l

_{1}-norm regularization, as shown in Figure 14.

_{1}-norm regularization and group sparse regularization are 31.56% and 21.73%, respectively. Because the updated FEM was adopted for force identification, the errors in the transfer matrix were inevitable. In this situation, by adding the regularization constraint, defined based on the group structures, the local identification accuracy obtained by the proposed method is greatly improved. It indicates the group structures of atoms are reasonable enough when updated FEM is used. Meanwhile, the computation time for the standard l

_{1}-norm regularization and the proposed method is 16.35 s and 3.59 s, respectively.

## 5. Conclusions

- The proposed method is an effective technique for indirectly measuring time-varying external force from acceleration responses in a few measurement points; both identified external forces and reconstructed responses are in good agreement with the measured values.
- Compared with the standard l
_{1}-norm regularization, the proposed method can further improve the force identification results in both numerical simulations and experimental verifications. - The relevance of atoms in the redundant dictionary can obviously reduce the complexity in the process of force identification. When the same algorithm is used, the proposed method can enhance computational efficiency by comparing with the standard l
_{1}-norm regularization.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Song, J.; Kang, W.-H.; Lee, Y.-J.; Chun, J. Structural System Reliability: Overview of Theories and Applications to Optimization. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civ. Eng.
**2021**, 7, 03121001. [Google Scholar] [CrossRef] - Jerez, D.J.; Jensen, H.A.; Beer, M. Reliability-Based Design Optimization of Structural Systems under Stochastic Excitation: An Overview. Mech. Syst. Signal Process.
**2022**, 166, 108397. [Google Scholar] [CrossRef] - Zhu, X.Q.; Law, S.S. Recent Developments in Inverse Problems of Vehicle–bridge Interaction Dynamics. J. Civ. Struct. Health Monit.
**2016**, 6, 107–128. [Google Scholar] [CrossRef] - Shokravi, H.; Shokravi, H.; Bakhary, N.; Heidarrezaei, M.; Rahimian Koloor, S.S.; Petrů, M. Vehicle-Assisted Techniques for Health Monitoring of Bridges. Sensors
**2020**, 20, 3460. [Google Scholar] [CrossRef] - Sujon, M.; Dai, F. Application of Weigh-in-Motion Technologies for Pavement and Bridge Response Monitoring: State-of-the-Art Review. Autom. Constr.
**2021**, 130, 103844. [Google Scholar] [CrossRef] - Nie, Z.; Shen, Z.; Li, J.; Hao, H.; Lin, Y.; Ma, H. Output-Only Complete Mode Shape Identification of Bridges Using a Limited Number of Sensors. Mech. Syst. Signal Process.
**2022**, 178, 109246. [Google Scholar] [CrossRef] - Kamal, M.; Rahman, M.M. Advances in Fatigue Life Modeling: A Review. Renew. Sustain. Energy Rev.
**2018**, 82, 940–949. [Google Scholar] [CrossRef] - Kalayci, C.B.; Karagoz, S.; Karakas, Ö. Soft Computing Methods for Fatigue Life Estimation: A Review of the Current State and Future Trends. Fatigue Fract. Eng. Mater. Struct.
**2020**, 43, 2763–2785. [Google Scholar] [CrossRef] - Liu, J.; Li, B. A Novel Strategy for Response and Force Reconstruction under Impact Excitation. J. Mech. Sci. Technol.
**2018**, 32, 3581–3596. [Google Scholar] [CrossRef] - Sanchez, J.; Benaroya, H. Review of Force Reconstruction Techniques. J. Sound Vib.
**2014**, 333, 2999–3018. [Google Scholar] [CrossRef] - Liu, H.; Luo, Z.; Yu, L. A Semi-Convex Function for Both Constant and Time-Varying Moving Force Identification. Mech. Syst. Signal Process.
**2021**, 146, 107062. [Google Scholar] [CrossRef] - Jacquelin, E.; Bennani, A.; Hamelin, P. Force Reconstruction: Analysis and Regularization of a Deconvolution Problem. J. Sound Vib.
**2003**, 265, 81–107. [Google Scholar] [CrossRef] - Inoue, H.; Harrigan, J.J.; Reid, S.R. Review of Inverse Analysis for Indirect Measurement of Impact Force. Appl. Mech. Rev.
**2001**, 54, 503–524. [Google Scholar] [CrossRef] - Li, J.; Hao, H. Substructural Interface Force Identification with Limited Vibration Measurements. J. Civ. Struct. Health Monit.
**2016**, 6, 395–410. [Google Scholar] [CrossRef] - Sanchez, J.; Benaroya, H. Asymptotic Approximation Method of Force Reconstruction: Proof of Concept. Mech. Syst. Signal Process.
**2017**, 92, 39–63. [Google Scholar] [CrossRef] - Yan, G.; Sun, H.; Büyüköztürk, O. Impact Load Identification for Composite Structures Using Bayesian Regularization and Unscented Kalman Filter. Struct. Control Health Monit.
**2017**, 24, e1910. [Google Scholar] [CrossRef] - Chen, Z.; Chan, T.H.T. A Truncated Generalized Singular Value Decomposition Algorithm for Moving Force Identification with Ill-Posed Problems. J. Sound Vib.
**2017**, 401, 297–310. [Google Scholar] [CrossRef] - Liu, R.; Dobriban, E.; Hou, Z.; Qian, K. Dynamic Load Identification for Mechanical Systems: A Review. Arch. Comput. Methods Eng.
**2022**, 29, 831–863. [Google Scholar] [CrossRef] - Ren, C.; Wang, N.; Liu, Q.; Liu, C. Dynamic Force Identification Problem Based on a Novel Improved Tikhonov Regularization Method. Math. Probl. Eng.
**2019**, 2019, 1–13. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Shepard, W.S. An Improved Method for the Reconstruction of a Distributed Force Acting on a Vibrating Structure. J. Sound Vib.
**2006**, 291, 369–387. [Google Scholar] [CrossRef] - Sun, R.; Chen, G.; He, H.; Zhang, B. The Impact Force Identification of Composite Stiffened Panels under Material Uncertainty. Finite Elem. Anal. Des.
**2014**, 81, 38–47. [Google Scholar] [CrossRef] - Qiao, B.; Chen, X.; Xue, X.; Luo, X.; Liu, R. The Application of Cubic B-Spline Collocation Method in Impact Force Identification. Mech. Syst. Signal Process.
**2015**, 64–65, 413–427. [Google Scholar] [CrossRef] - Aucejo, M.; De Smet, O. An Iterated Multiplicative Regularization for Force Reconstruction Problems. J. Sound Vib.
**2018**, 437, 16–28. [Google Scholar] [CrossRef] - Aucejo, M.; De Smet, O. A Novel Algorithm for Solving Multiplicative Mixed-Norm Regularization Problems. Mech. Syst. Signal Process.
**2020**, 144, 106887. [Google Scholar] [CrossRef] - Pan, C.; Ye, X.; Zhou, J.; Sun, Z. Matrix Regularization-Based Method for Large-Scale Inverse Problem of Force Identification. Mech. Syst. Signal Process.
**2020**, 140, 106698. [Google Scholar] [CrossRef] - Liu, H.; Yu, L.; Luo, Z.; Pan, C. Compressed Sensing for Moving Force Identification Using Redundant Dictionaries. Mech. Syst. Signal Process.
**2020**, 138, 106535. [Google Scholar] [CrossRef] - Qiao, B.; Zhang, X.; Wang, C.; Zhang, H.; Chen, X. Sparse Regularization for Force Identification Using Dictionaries. J. Sound Vib.
**2016**, 368, 71–86. [Google Scholar] [CrossRef] - Bao, Y.; Li, H.; Chen, Z.; Zhang, F.; Guo, A. Sparse L
_{1}Optimization-Based Identification Approach for the Distribution of Moving Heavy Vehicle Loads on Cable-Stayed Bridges. Struct. Control Health Monit.**2016**, 23, 144–155. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Wang, L.; Cao, H.; Xie, Y. An Improved Iterative Tikhonov Regularization Method for Solving the Dynamic Load Identification Problem. Int. J. Comput. Methods Eng. Sci. Mech.
**2015**, 16, 292–300. [Google Scholar] [CrossRef] - Song, X.; Zhang, Y.; Liang, D. Load Identification for a Cantilever Beam Based on Fiber Bragg Grating Sensors. Sensors
**2017**, 17, 1733. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Qiu, B.; Lu, Y.; Qu, X.; Li, X. Experimental Research on a Hybrid Algorithm for Localisation and Reconstruction of the Impact Force Applied to a Rectangular Steel Plate Structure. Sensors
**2022**, 22, 8123. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; Li, K. Sparse Identification of Time-Space Coupled Distributed Dynamic Load. Mech. Syst. Signal Process.
**2021**, 148, 107177. [Google Scholar] [CrossRef] - Qiao, B.; Mao, Z.; Liu, J.; Zhao, Z.; Chen, X. Group Sparse Regularization for Impact Force Identification in Time Domain. J. Sound Vib.
**2019**, 445, 44–63. [Google Scholar] [CrossRef] - Zhang, L.; Liang, Y.; Yu, L. Moving Force Identification Based on Group Lasso and Compressed Sensing. Int. J. Struct. Stab. Dyn.
**2022**, 22, 1–28. [Google Scholar] [CrossRef] - Feng, W.; Li, Q.; Lu, Q.; Li, C.; Wang, B. Group Relevance Vector Machine for Sparse Force Localization and Reconstruction. Mech. Syst. Signal Process.
**2021**, 161, 107900. [Google Scholar] [CrossRef] - Zou, J.; Li, H.; Liu, G. Split Bregman Algorithm for Structured Sparse Reconstruction. IEEE Access
**2018**, 6, 21560–21569. [Google Scholar] [CrossRef] - Liu, H.; Yu, L.; Luo, Z.; Chen, Z. Multi-Strategy Structural Damage Detection Based on Included Angle of Vectors and Sparse Regularization. Struct. Eng. Mech.
**2020**, 75, 415–424. [Google Scholar] - Hansen, P.C. Regularization Tools Version 4.0 for Matlab 7.3. Numer. Algorithms
**2007**, 46, 189–194. [Google Scholar] [CrossRef]

**Figure 1.**Basic flowchart of force identification method based on sparse regularization and redundant dictionary.

**Figure 3.**Selection of near-optimal regularization parameter for standard l

_{1}-norm regularization.

**Figure 5.**Identified result, respectively, obtained from standard l

_{1}-norm regularization and group sparse regularization.

**Figure 6.**Comparative results between standard l

_{1}-norm regularization and group sparse regularization under different weight coefficients.

**Figure 7.**Experiment on beam structure for force identification. (

**a**) Experimental setup. (

**b**) Height of beam structure. (

**c**) Width of beam structure. (

**d**) Support of beam structure. (

**e**) Beam structure.

**Figure 12.**Selection of near-optimal regularization parameter for standard l

_{1}-norm regularization in experimental verifications.

**Figure 13.**Selection of near-optimal regularization parameter for group sparse regularization in experimental verifications.

**Figure 14.**Identified result, respectively, obtained from standard l

_{1}-norm regularization and group sparse regularization in experimental verifications.

**Figure 15.**Comparison of measurement and reconstructed acceleration responses at different measurement points. (

**a**) 1/2 span. (

**b**) 7/10 span.

Measurement Points | Noise Level | RPE | Weight Coefficient (λ) |
---|---|---|---|

y direction of nodes 3 and 7 | 1% | 10.3081% | 1.6207 × 10^{4} |

y direction of nodes 2 and 8 | 1% | 10.6728% | 1.0780 × 10^{4} |

y direction of nodes 3 and 7 | 5% | 15.7292% | 1.4873 × 10^{3} |

y direction of nodes 3, 7 and 8 | 5% | 17.8764% | 9.5138 × 10^{2} |

y direction of nodes 2 and 8 | 5% | 17.3825% | 8.8676 × 10^{2} |

y direction of nodes 3 and 7 | 10% | 24.1736% | 8.7850 × 10^{2} |

y direction of nodes 2 and 8 | 10% | 25.9328% | 6.5278 × 10^{2} |

Cases | Response Combinations | RPE | Weight Coefficient (λ) |
---|---|---|---|

Case 1 | 1/2 & 7/10 span | 21.7343% | 1.4462 × 10^{4} |

Case 1 | 3/10 & 7/10 span | 22.3612% | 1.5461 × 10^{4} |

Case 1 | 3/10 & 1/2 span | 17.5320% | 1.0173 × 10^{4} |

Case 2 | 1/2 & 7/10 span | 21.9778% | 5.3531 × 10^{3} |

Case 2 | 3/10 & 1/2 span | 19.8638% | 1.0930 × 10^{4} |

Case 3 | 3/10 & 1/2 span | 19.7285% | 5.5294 × 10^{3} |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 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**

Liu, H.; Ma, H.
Identification of Time-Varying External Force Using Group Sparse Regularization and Redundant Dictionary. *Sensors* **2023**, *23*, 151.
https://doi.org/10.3390/s23010151

**AMA Style**

Liu H, Ma H.
Identification of Time-Varying External Force Using Group Sparse Regularization and Redundant Dictionary. *Sensors*. 2023; 23(1):151.
https://doi.org/10.3390/s23010151

**Chicago/Turabian Style**

Liu, Huanlin, and Hongwei Ma.
2023. "Identification of Time-Varying External Force Using Group Sparse Regularization and Redundant Dictionary" *Sensors* 23, no. 1: 151.
https://doi.org/10.3390/s23010151