# A Two-Stage Approach for Damage Diagnosis of Structures Based on a Fully Distributed Strain Mode under Multigain Feedback Control

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Two-Stage Damage Diagnosis Method Using Strain-Based, Closed-Loop Systems under Multigain Feedback Control

#### 2.1. Eigenstructure Assignment Using Strain Output Feedback

#### 2.2. Construction of Closed-Loop Systems

**G**

_{1},

**G**

_{2}, ……, and

**G**

_{n}can be obtained.

#### 2.3. MSEBI Method for Damage Localization

#### 2.4. Hybrid ANN-PSO Algorithm for Damage Quantification

_{j}is the output signal of j-th node in the output layer, while the input

_{q}is the input signal from the q-th node in the previous layer. The train parameters (weights and biases) are crucial in the ANN models, which determine recognition errors between the real and expected outputs and decide the success of the established network. However, due to the use of back propagation algorithm based on gradient descent, the neural network may find local minima rather than globe optima when the network is complex. To address this problem, the PSO algorithm is employed to find the optimized weights and biases in order to prevent the value from being trapped into local optimal.

#### 2.5. Flowchart of the Proposed Damage Diagnosis Approach

**G**i of each subsystem;

## 3. Numerical Simulation

#### 3.1. Brief Description of a Structural Example

#### 3.2. Assignment of the Multiple Closed-Loop System

#### 3.3. Damage Cases

#### 3.4. Damage Diagnosis Results

#### 3.4.1. Damage Localization Results

#### 3.4.2. Damage Quantification Results

_{1}= 1.5 and c

_{2}= 1.5, respectively, the weight value is 0.3. Then, the proposed hybrid ANN-PSO method is carried out, and the stop criteria of iteration are set as follows: the maximum iteration process number is 200 or the maximum tolerance limit of objective fitness function is smaller than 10

^{−5}, and the results are shown in Figure 10.

#### 3.4.3. Comparative Discussion with One-Stage Damage Diagnosis Using Sensitivity Matrix

#### 3.4.4. Comparative Discussion with the ANN-Only Algorithm for Damage Quantification

## 4. Concluding Remarks

- (i)
- A multigain closed-loop system is established to improve the sensitivity of strain mode shapes for structural damage in different spans, with which the MSEBI method and hybrid ANN-PSO algorithm are proposed to locate and quantify the small damage of the multispan structure, and the performance of the proposed method is validated through a numerical example of a two-span beam structure;
- (ii)
- The MCL system performs more effectively than the OL and SCL systems for detecting local damage, while the MCL system requires fewer actuators than the SCL system, making it more economical and practical for structure testing;
- (iii)
- Compared with the one-stage, sensitivity-based damage detection approach, the two-stage method has a better effect and accuracy, which can help avoid misjudgment for undamaged elements and realize fast damage detection;
- (iv)
- The hybrid PSO-ANN algorithm has a better detection effect, while the ANN-only algorithm may easily become trapped in local minima and become less accurate;
- (v)
- The proposed closed-loop damage diagnosis method is carried out in real time online, which is difficult to implement in practice; thus, further research on method implementation based on virtual output feedback should be investigated in the future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 9.**Damage localization results based on the MSEBI method: (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3; (

**d**) Case 4; (

**e**) Case 5; (

**f**) Case 6; (

**g**) Case 7.

**Figure 10.**Damage quantification results based on the hybrid ANN-PSO algorithm: (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3; (

**d**) Case 4; (

**e**) Case 5; (

**f**) Case 6; (

**g**) Case 7.

**Figure 11.**Comparison results between the one-stage and two-stage damage detection methods: (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3; (

**d**) Case 4; (

**e**) Case 5; (

**f**) Case 6; (

**g**) Case 7.

**Figure 12.**Comparison results between the ANN-only and ANN-PSO algorithms: (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3; (

**d**) Case 4; (

**e**) Case 5; (

**f**) Case 6; (

**g**) Case 7.

**Figure 13.**Regression of the training process for the simulated example—single damage: (

**a**) ANN-PSO; (

**b**) ANN-only.

Parameter | Young’s Modulus | Sectional Area | Moment of Inertia | Density | Damping Ratio |
---|---|---|---|---|---|

Value | 3.25 × 10^{10} MPa | 0.4 m^{2} | 6.69 × 10^{−2} m^{4} | 2370 kg/m^{3} | 0.01 |

System Type | Open-Loop Actuators Nodes | Closed-Loop Actuators Nodes | Amplification Coefficient | Optimal Results |
---|---|---|---|---|

OL | 9 | - | - | - |

SCL | 9 | 5, 12, 20, 27 | ${a}_{1}={10}^{13}$ ${a}_{2}={10}^{14}$ ${a}_{3}={10}^{12}$ | $\theta =\left\{\begin{array}{ccc}1.47& 1.11& 0.92\end{array}\right\}$ ${d}_{1}=\left\{\begin{array}{cccc}0.26& -0.32& 0.10& -0.84\end{array}\right\}$ ${d}_{2}=\left\{\begin{array}{cccc}-0.30& 0.85& 0.69& -0.61\end{array}\right\}$ ${d}_{3}=\left\{\begin{array}{cccc}0.62& -0.45& 0.28& 0.11\end{array}\right\}$ |

MCL | 9 | 5, 12 | ${a}_{1}={10}^{15}$ ${a}_{2}={10}^{13}$ ${a}_{3}={10}^{11}$ | $\theta =\left\{\begin{array}{ccc}1.39& 1.09& 0.94\end{array}\right\}$ ${d}_{1}=\left\{\begin{array}{cc}-0.16& 0.99\end{array}\right\}$ ${d}_{2}=\left\{\begin{array}{cc}0.76& 0.05\end{array}\right\}$ ${d}_{3}=\left\{\begin{array}{cc}0.21& -0.96\end{array}\right\}$ |

23 | 20, 27 | ${a}_{1}={10}^{14}$ ${a}_{2}={10}^{15}$ ${a}_{3}={10}^{14}$ | $\theta =\left\{\begin{array}{ccc}1.26& 1.01& 1.00\end{array}\right\}$ ${d}_{1}=\left\{\begin{array}{cc}0.91& -0.03\end{array}\right\}$ ${d}_{2}=\left\{\begin{array}{cc}0.60& 0.72\end{array}\right\}$ ${d}_{3}=\left\{\begin{array}{cc}-0.16& 0.83\end{array}\right\}$ |

Damage Case | Damaged Elements | Case Type | Damaged Degrees (%) |
---|---|---|---|

1 | 1 | Single | 5 |

2 | 8 | Single | 10 |

3 | 15 | Single | 5 |

4 | 8, 15 | Double | 10, 5 |

5 | 8, 23 | Double | 10, 10 |

6 | 7, 8 | Double | 10, 10 |

7 | 8, 15, 23 | Triple | 10, 5, 10 |

Damage Case | Actual Loctation | OL-Identified Loctation | SCL-Identified Loctation | MCL-Identified Loctation |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 8 | 8 | 8 | 8 |

3 | 15 | 15 | 15 | 15 |

4 | 8, 15 | 8, 15 | 8, 15 | 8, 15 |

5 | 8, 23 | 8, 23 | 8, 23 | 8, 23 |

6 | 7, 8 | 7, 8 | 7, 8 | 7, 8 |

7 | 8, 15, 23 | 8, 15, 23 | 8, 15, 23 | 8, 15, 23 |

Damage Case | Actual Damaged Elements | Actual Damaged Degree (%) | OL | SCL | MCL | |||
---|---|---|---|---|---|---|---|---|

Identified Results (%) | Error Norm | Identified Results (%) | Error Norm | Identified Results (%) | Error Norm | |||

1 | 1 | 5 | 7.15 | 2.15 | 6.22 | 1.22 | 6.15 | 1.15 |

2 | 8 | 10 | 11.23 | 1.23 | 10.96 | 0.96 | 10.81 | 0.81 |

3 | 15 | 5 | 6.47 | 1.47 | 6.19 | 1.19 | 5.75 | 0.75 |

4 | 8 | 10 | 11.65 | 1.96 | 11.10 | 1.37 | 10.95 | 1.35 |

15 | 5 | 6.06 | 5.81 | 5.95 | ||||

5 | 8 | 10 | 9.23 | 1.69 | 9.11 | 1.16 | 9.53 | 1.08 |

23 | 10 | 11.50 | 10.75 | 10.97 | ||||

6 | 7 | 10 | 9.04 | 2.31 | 9.40 | 1.38 | 9.45 | 1.29 |

8 | 10 | 12.10 | 11.24 | 11.16 | ||||

7 | 8 | 10 | 9.67 | 2.38 | 9.34 | 1.98 | 9.48 | 1.40 |

15 | 5 | 6.61 | 6.55 | 5.86 | ||||

23 | 10 | 11.72 | 11.04 | 10.96 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, Z.; Dong, K.; Fang, Z.; Liu, Y.
A Two-Stage Approach for Damage Diagnosis of Structures Based on a Fully Distributed Strain Mode under Multigain Feedback Control. *Sustainability* **2022**, *14*, 10019.
https://doi.org/10.3390/su141610019

**AMA Style**

Zhou Z, Dong K, Fang Z, Liu Y.
A Two-Stage Approach for Damage Diagnosis of Structures Based on a Fully Distributed Strain Mode under Multigain Feedback Control. *Sustainability*. 2022; 14(16):10019.
https://doi.org/10.3390/su141610019

**Chicago/Turabian Style**

Zhou, Zheng, Kaizhi Dong, Ziwei Fang, and Yang Liu.
2022. "A Two-Stage Approach for Damage Diagnosis of Structures Based on a Fully Distributed Strain Mode under Multigain Feedback Control" *Sustainability* 14, no. 16: 10019.
https://doi.org/10.3390/su141610019