# Reciprocating Compressor Multi-Fault Classification Using Symbolic Dynamics and Complex Correlation Measure

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## Abstract

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## 1. Introduction

- Novel application in PHM of the Complex Correlation Measure (CCM) derived from the Poincaré plot of the vibration signal for extracting useful features for accurate classification of multi-fault in valves and roller bearings.
- Novel application of Symbolic Dynamics (SD) for classification of multi-fault of valves and roller bearings.
- Accurate classification of 13 different combined fault conditions (multi-fault scenario) of valves and roller bearing.
- Accurate classification of 17 fault conditions of valves in a reciprocating compressor.
- Comparison of three different set of features extracted from vibration signal for classifying the set of multi-fault previously mentioned.
- Comparison of two high performance Random Forest (RF) models applied to the problem of multi-fault classification of valves and roller bearings in a reciprocating compressor.

## 2. Poincaré Plot and Their Features

#### 2.1. Poincaré Plot

#### 2.2. Symbolic Dynamics

#### 2.3. Complex Correlation Measure

## 3. Random Forest

#### 3.1. Ensemble Bagged Trees

#### 3.2. Ensemble Subspace k-Nearest Neighbors

## 4. Experimental Test-Bed

#### 4.1. Reciprocating Compressor

#### 4.2. Dataset Vibration Signal Acquisition

## 5. Feature Extraction

#### 5.1. Symbolic Dynamics

#### 5.2. Complex Correlation Measure

#### 5.3. Statistical Features

## 6. Multi-Fault Classification

#### Parameters Selection

## 7. Results

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PHM | Prognostics and Health Management |

SVM | Support Vector Machines |

LSTM | Long Short-Term Memory model |

CCM | Complex Correlation Measure |

RF | Random Forest |

CART | Classification And Regression Tree |

OOB | Out of the Bag |

EDM | Electrical Discharge Machining |

SD | Symbolic Dynamics |

EBT | Ensemble Bagged Tree |

k-NN | k-Nearest Neighbors |

ESK | Ensemble Subspace k-NN |

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**Figure 2.**The Lorenz attractor is shown in terms of the phase-space trajectories as well as their intersection with a surface for obtaining the Poincaré map.

**Figure 3.**Poincaré map corresponding to the intersection of the phase space with the surface $\Sigma $.

**Figure 4.**Bi-dimensional representation of Lorenz model. (

**a**) Time evolution of $x\left(t\right)$. (

**b**) Time evolution of $y\left(t\right)$. (

**c**) Time evolution of $z\left(t\right)$. (

**d**) Poincaré plot for $x\left(t\right)$. The lag used for the Poincaré plot is $\tau =20$.

**Figure 5.**Poincaré plot showing a fragment of component $x\left(t\right)$ of Lorenz model including the set of symbols assigned.

**Figure 6.**Feature extraction using the symbolic dynamics calculated from the Poincaré plot. (

**a**) Poincaré plot from the Lorenz model constructed with the $x\left(t\right)$ component. The twelve angular regions are overlapped, and a single symbol is assigned to each of points included in the region (

**b**) Histogram calculated from the Poincaré plot for the Lorenz model. The histogram represents the set of features useful for fault classification.

**Figure 7.**Poincaré plot showing the triangles for calculating CCM. The Poincaré plot is considering a lag $\tau $.

**Figure 14.**The Poincaré plot is subdivided into 24 angular regions and each region is represented by a symbol.

**Figure 17.**CCM for a vibration signal of the healthy class (blue) and for the faulty class P2 (red). The calculation considers a variation of $\tau $ between 1 and 50.

Label | Valve, 2S–DV | Bearings, B1 |
---|---|---|

P1 | Healthy | Healthy |

P2 | Valve seat wear | Inner race crack |

P3 | Corrosion of valve plate | Inner race crack |

P4 | Fracture of valve plate | Inner race crack |

P5 | Broken Spring | Inner race crack |

P6 | Valve seat wear | Roller element crack |

P7 | Corrosion of valve plate | Roller element crack |

P8 | Fracture of valve plate | Roller element crack |

P9 | Broken Spring | Roller element crack |

P10 | Valve seat wear | Outer race crack |

P11 | Corrosion of valve plate | Outer race crack |

P12 | Fracture of valve plate | Outer race crack |

P13 | Broken Spring | Outer race crack |

**Table 2.**Percent accuracy for each of features types and models calculated for each vibration signal.

Features | Model | A1 | A2 | A3 |
---|---|---|---|---|

Statistical | EBT | 100 | 100 | 100 |

ESK | 100 | 100 | 100 | |

SD | EBT | 100 | 100 | 100 |

ESK | 100 | 100 | 100 | |

CCM | EBT | 99.4 | 96.8 | 94.2 |

ESK | 100 | 100 | 99.4 |

**Table 3.**Results for the EBT model using features extracted from signals recorded with the accelerometer A3.

Fault Condition | Sensitivity | Specificity | F1-Score |
---|---|---|---|

P1 | 0.92 | 1.00 | 0.96 |

P2 | 1.00 | 1.00 | 1.00 |

P3 | 1.00 | 1.00 | 1.00 |

P4 | 1.00 | 0.99 | 0.96 |

P5 | 1.00 | 0.99 | 0.96 |

P6 | 0.92 | 1.00 | 0.96 |

P7 | 0.92 | 0.99 | 0.92 |

P8 | 0.85 | 0.99 | 0.88 |

P9 | 1.00 | 0.99 | 0.96 |

P10 | 1.00 | 1.00 | 1.00 |

P11 | 1.00 | 1.00 | 1.00 |

P12 | 0.79 | 0.99 | 0.85 |

P13 | 0.90 | 0.98 | 0.82 |

**Table 4.**Percent accuracy for each of features types and models calculated over the dataset of valve faults in [20].

Features | Model | A1 | A2 | A3 | A4 |
---|---|---|---|---|---|

Statistical | EBT | 73.2 | 83.5 | 73.4 | 74.4 |

ESK | 52.6 | 61.7 | 57.2 | 49.0 | |

SD | EBT | 100 | 100 | 100 | 100 |

ESK | 100 | 100 | 100 | 100 | |

CCM | EBT | 91.7 | 98.7 | 95.3 | 97.9 |

ESK | 97.8 | 99.8 | 97.8 | 99.5 |

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**MDPI and ACS Style**

Cerrada, M.; Macancela, J.-C.; Cabrera, D.; Estupiñan, E.; Sánchez, R.-V.; Medina, R.
Reciprocating Compressor Multi-Fault Classification Using Symbolic Dynamics and Complex Correlation Measure. *Appl. Sci.* **2020**, *10*, 2512.
https://doi.org/10.3390/app10072512

**AMA Style**

Cerrada M, Macancela J-C, Cabrera D, Estupiñan E, Sánchez R-V, Medina R.
Reciprocating Compressor Multi-Fault Classification Using Symbolic Dynamics and Complex Correlation Measure. *Applied Sciences*. 2020; 10(7):2512.
https://doi.org/10.3390/app10072512

**Chicago/Turabian Style**

Cerrada, Mariela, Jean-Carlo Macancela, Diego Cabrera, Edgar Estupiñan, René-Vinicio Sánchez, and Ruben Medina.
2020. "Reciprocating Compressor Multi-Fault Classification Using Symbolic Dynamics and Complex Correlation Measure" *Applied Sciences* 10, no. 7: 2512.
https://doi.org/10.3390/app10072512