# Distributed Piezoelectric Sensor System for Damage Identification in Structures Subjected to Temperature Changes

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Principal Component Analysis

#### 2.1.1. PCA Modeling

#### 2.1.2. Normalization: Group Scaling

#### 2.1.3. Projection of New Data onto the PCA Model

#### 2.2. Machine Learning

#### 2.2.1. Nearest Neighbor Pattern Classification

- Fine k-NN: a nearest neighbor classifier that makes finely-detailed distinctions between classes with the number of neighbors set to one.
- Medium k-NN: a nearest neighbor classifier with fewer distinctions than a fine k-NN with the number of neighbors set to 10.
- Coarse k-NN: a nearest neighbor between classes, with the number of neighbors set to 100.
- Cosine k-NN: a nearest neighbor classifier that uses the cosine distance metric. The cosine distance between two vectors u and v is defined as:$$\begin{array}{c}\hfill 1-{\displaystyle \frac{u\xb7v}{\left|u\right|\xb7\left|v\right|}},\end{array}$$
- Cubic k-NN: a nearest neighbor classifier that uses the cubic distance metric. The cubic distance between two n-dimensional vectors u and v is defined as:$$\begin{array}{c}\hfill \sqrt[3]{\sum _{i=1}^{n}{|{u}_{i}-{v}_{i}|}^{3}}.\end{array}$$
- Weighted k-NN: a nearest neighbor classifier that uses distance weighting. The weighted Euclidean distance between two n-dimensional vectors u and v is defined as:$$\begin{array}{c}\hfill \sqrt{\sum _{i=1}^{n}{w}_{i}{({x}_{i}-{y}_{i})}^{2}},\end{array}$$

#### 2.2.2. Decision Trees

- Compared with other machine learning methods, trees are simple and easy to understand.
- Decision trees use different methods and can be combined to obtain a single prediction.
- The combination of different trees usually produces better results.
- Because of its simplicity, more elaborated methods can produce better results in classification and regression tasks.

#### 2.2.3. Support Vector Machines

## 3. Damage Classification Methodology

#### Data Acquisition System

## 4. Experimental Setup and Results

- (i)
- an aluminum plate with four piezoelectric transducers; and
- (ii)
- a composite plate of carbon fiber polymer with six piezoelectric transducers.

#### 4.1. First Specimen: Aluminum Plate

- ${T}_{1}={10}^{\circ};$
- ${T}_{2}={20}^{\circ};$
- ${T}_{3}={30}^{\circ};$
- ${T}_{4}={40}^{\circ};$ and
- ${T}_{5}={45}^{\circ}$.

- no damage (healthy or pristine structure);
- Damage 1;
- Damage 2; and
- Damage 3.

#### 4.2. Second Specimen: Carbon Fiber Plate

- ${T}_{1}={0}^{\circ};$
- ${T}_{2}={10}^{\circ};$
- ${T}_{3}={20}^{\circ};$
- ${T}_{4}={30}^{\circ};$
- ${T}_{5}={40}^{\circ};$ and
- ${T}_{6}={45}^{\circ}$.

- no damage (healthy or pristine structure);
- Damage 1;
- Damage 2; and
- Damage 3.

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 15.**First principal component versus second principal component in the aluminum plate described in Section 4.1.

**Figure 16.**Confusion matrix using: (

**a**) subspace k-NN; (

**b**) weighted k-NN; (

**c**) fine k-NN; and (

**d**) fine Gaussian SVM.

**Figure 17.**Confusion matrix using: (

**a**) rusboosted trees; (

**b**) boosted trees; (

**c**) coarse k-NN; and (

**d**) coarse Gaussian SVM.

**Figure 21.**First principal component versus second principal component in the carbon fiber plate described in Section 4.2.

**Figure 23.**Confusion matrix machines with good behavior. (

**a**) Subspace k-NN; (

**b**) weighted k-NN; (

**c**) bagged Trees; (

**d**) cubic SVM.

**Figure 24.**Confusion matrix machines with bad behavior. (

**a**) Rusboosted trees; (

**b**) boosted trees; (

**c**) coarse k-NN; (

**d**) coarse Gaussian SVM.

**Table 1.**Percentage of correct decisions for the healthy structure and the structure with Damage 1, 2 and 3, for the twenty different machine learning strategies (aluminum plate).

Machine Name | Healthy | Damage 1 | Damage 2 | Damage 3 |
---|---|---|---|---|

Medium Tree | 66% | 76% | 70% | 56% |

Simple Tree | 64% | 60% | 30% | 58% |

Complex Tree | 72% | 76% | 58% | 56% |

Linear SMV | 70% | 60% | 26% | 60% |

Quadratic SVM | 78% | 70% | 56% | 70% |

Cubic SVM | 86% | 68% | 66% | 72% |

Fine Gaussian SVM | 90% | 80% | 66% | 78% |

Medium Gaussian SVM | 76% | 80% | 56% | 74% |

Coarse Gaussian SVM | 94% | 64% | 14% | 38% |

Fine k-NN | 94% | 78% | 74% | 80% |

Medium k-NN | 80% | 62% | 64% | 74% |

Coarse k-NN | 94% | 42% | 2% | 24% |

Cosine k-NN | 84% | 58% | 78% | 72% |

Cubic k-NN | 80% | 64% | 62% | 76% |

Weighted k-NN | 94% | 66% | 68% | 80% |

Boosted Trees | 96% | 0% | 42% | 42% |

Bagged Trees | 84% | 70% | 66% | 78% |

Subspace Discriminant | 56% | 44% | 32% | 46% |

Subspace k-NN | 94% | 78% | 72% | 80% |

Rusboosted Trees | 98% | 0% | 42% | 0% |

**Table 2.**Percentage of correct decisions for the healthy structure and the structure with Damages 1, 2 and 3, for the twenty different machine learning strategies (composite plate).

Machine Name | Healthy | Damage 1 | Damage 2 | Damage 3 |
---|---|---|---|---|

Medium Tree | 55.00% | 63.33% | 60.83% | 52.50% |

Simple Tree | 40.00% | 60.00% | 63.33% | 42.50% |

Complex Tree | 57.50% | 64.17% | 75.83% | 65.83% |

Linear SVM | 41.67% | 59.17% | 45.00% | 47.50% |

Quadratic SVM | 65.83% | 73.33% | 85.00% | 75.50% |

Cubic SVM | 70.83% | 75.00% | 86.67% | 74.17% |

Fine Gaussian SVM | 59.17% | 64.17% | 83.33% | 78.33% |

Medium Gaussian SVM | 55.83% | 60.00% | 82.50% | 63.33% |

Coarse Gaussian SVM | 52.50% | 10.83% | 33.33% | 56.67% |

Fine k-NN | 63.33% | 61.67% | 80.00% | 70.00% |

Medium k-NN | 65.00% | 46.67% | 75.00% | 63.33% |

Coarse k-NN | 52.50% | 37.50% | 60.83% | 35.83% |

Cosine k-NN | 65.00% | 43.33% | 79.17% | 60.83% |

Cubic k-NN | 59.17% | 47.50% | 72.50% | 60.00% |

Weighted k-NN | 61.67% | 58.33% | 83.33% | 74.17% |

Boosted Trees | 16.67% | 62.50% | 60.83% | 71.67% |

Bagged Trees | 71.67% | 72.50% | 90.00% | 84.17% |

Subspace Discriminant | 33.33% | 45.83% | 45.00% | 55.83% |

Subspace k-NN | 70.83% | 72.50% | 89.17% | 82.50% |

Rusboosted Trees | 0.00% | 62.50% | 0.00% | 93.33% |

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## Share and Cite

**MDPI and ACS Style**

Vitola, J.; Pozo, F.; Tibaduiza, D.A.; Anaya, M. Distributed Piezoelectric Sensor System for Damage Identification in Structures Subjected to Temperature Changes. *Sensors* **2017**, *17*, 1252.
https://doi.org/10.3390/s17061252

**AMA Style**

Vitola J, Pozo F, Tibaduiza DA, Anaya M. Distributed Piezoelectric Sensor System for Damage Identification in Structures Subjected to Temperature Changes. *Sensors*. 2017; 17(6):1252.
https://doi.org/10.3390/s17061252

**Chicago/Turabian Style**

Vitola, Jaime, Francesc Pozo, Diego A. Tibaduiza, and Maribel Anaya. 2017. "Distributed Piezoelectric Sensor System for Damage Identification in Structures Subjected to Temperature Changes" *Sensors* 17, no. 6: 1252.
https://doi.org/10.3390/s17061252