# Degradation Tendency Measurement of Aircraft Engines Based on FEEMD Permutation Entropy and Regularized Extreme Learning Machine Using Multi-Sensor Data

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Fast Ensemble Empirical Mode Decomposition

- (1)
- Initialize the number of replicated times M, the amplitude of added white noises, and m = 1.
- (2)
- Add a random Gaussian white noise sequence ${n}_{m}(t)$ into the original time series $x(t)$ to generate a noise-added signal ${X}_{m}(t)$,$${X}_{m}(t)=x(t)+{n}_{m}(t).$$
- (3)
- Decompose the noise-added signal ${X}_{m}(t)$ into a series of intrinsic mode functions (IMFs) and a residue using the EMD method,$${X}_{m}(t)={\displaystyle \sum _{i=1}^{n}{c}_{i,m}(t)+}{r}_{n,m}(t),$$
- (4)
- If m < M, then repeat step (2) to step (3) with m = m + 1, and add different white noise sequences each time.
- (5)
- Calculate the ensemble mean of the M trials for each IMF and the residue as the final results,$${c}_{i}(t)=\frac{1}{M}{\displaystyle \sum _{m=1}^{M}{c}_{i,m}}(t),\text{\hspace{1em}}i=1,2,\dots ,n,\text{}m=1,2,\dots ,M,$$$${r}_{n}(t)=\frac{1}{M}{\displaystyle \sum _{m=1}^{M}{r}_{n,m}(t)},\text{\hspace{1em}}m=1,2,\dots ,M,$$

#### 2.2. Extreme Learning Machine

- (1)
- For a given training set $D=\left\{\left({x}_{i},{\mathrm{T}}_{i}\right)|{x}_{i}\in {R}^{n},{T}_{i}\in {R}^{m},i=1,2,\dots ,N\right\}$, set the activation function containing L hidden layer nodes as $g(x)$.
- (2)
- The network output of ELM can be expressed as$${f}_{L}(x)={\displaystyle \sum _{i=1}^{L}{\mathsf{\beta}}_{i}}g\left({\mathsf{\omega}}_{i}{x}_{i}+{b}_{i}\right)={y}_{i},$$

- (3)
- The objective function of ELM can be formulated as$$E={\displaystyle \sum _{i=1}^{N}\Vert {y}_{i}-{T}_{i}\Vert}.$$

- (4)
- The output weight matrix $\tilde{\mathsf{\beta}}$ can be obtained by the following formula:$$\tilde{\mathsf{\beta}}={H}^{-1}T,$$

#### 2.3. Regularized Extreme Learning Machine

## 3. Proposed Degradation Tendency Measurement Method

#### 3.1. Construction of Synthesized Degradation Index Using Multi-Sensor Data

#### 3.2. Reconstruction of Intrinsic Mode Functions Based on Permutation Entropy Theory

#### 3.3. Procedure of the Proposed Method

## 4. Experimental Results and Discussion

#### 4.1. Model Performance Evaluation

#### 4.2. SDI Series Construction

#### 4.3. SDI Series Decomposition and IMF Reconstruction

#### 4.4. Degradation Tendency Measurement of Aircraft Engines

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The structure of an extreme learning machine (ELM) network [35].

**Figure 3.**The simplified schematic illustration of the aircraft engine model [37].

**Figure 4.**The constructed synthesized degradation index (SDI) series based on the multi-dimensional sensor data.

**Figure 7.**Refactored IMFS (RIMFs) from fast ensemble empirical mode decomposition (FEEMD) decomposition results.

**Figure 10.**Analysis of degradation tendency measuring results in terms of mean absolute percentage error (MAPE), mean absolute error (MAE), and coefficient of determination (R

^{2}).

Mode | Setting Parameter 1 | Setting Parameter 2 | Setting Parameter 3 |
---|---|---|---|

1 | 0 | 0 | 100 |

2 | 10 | 0.25 | 20 |

3 | 20 | 0.7 | 0 |

4 | 25 | 0.62 | 80 |

5 | 35 | 0.84 | 60 |

6 | 42 | 0.84 | 40 |

**Table 2.**Description of the selected sensor monitoring parameters for the construction of the synthesized degradation index (SDI).

Index | Symbol | Description | Units |
---|---|---|---|

1 | T24 | Total temperature at low-pressure compressor outlet | °R |

2 | T30 | Total temperature at high-pressure compressor outlet | °R |

3 | T50 | Total temperature at low-pressure turbine outlet | °R |

4 | P30 | Total pressure at high-pressure compressor outlet | psia |

5 | Ps30 | Static pressure at high-pressure compressor outlet | psia |

6 | Phi | Ratio of fuel flow to Ps30 | pps/psia |

7 | BPR | Bypass ratio | - |

**Table 3.**Permutation entropy (PE) values of intrinsic mode functions (IMFs) decomposed by fast ensemble empirical mode decomposition (FEEMD) algorithm.

IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | IMF7 | IMF8 | r_{0} | |
---|---|---|---|---|---|---|---|---|---|

PE | 0.902 | 0.687 | 0.475 | 0.371 | 0.277 | 0.178 | 0.161 | 0.161 | 0 |

RIMFs | IMFs Contained | H_{p} |
---|---|---|

RIMF1 | IMF1 | [0.702, 0.902] |

RIMF2 | IMF2 | [0.487, 0.687] |

RIMF3 | IMF3, IMF4, IMF5 | [0.275, 0.475] |

RIMF4 | IMF6, IMF7, IMF8, r_{0} | [0, 0.178] |

**Table 5.**Forecasting accuracy comparison of the eight models. RELM—regularized extreme learning machine; MAPE—mean absolute percentage error; MAE—mean absolute error; R

^{2}—coefficient of determination.

Models | Forecasting Accuracy | ||
---|---|---|---|

MAPE (%) | MAE | R^{2} | |

FEEMD-PE/RELM | 3.552 | 0.029 | 0.979 |

EMD-PE/RELM | 5.134 | 0.043 | 0.976 |

FEEMD/RELM | 9.847 | 0.075 | 0.971 |

RELM | 12.577 | 0.093 | 0.781 |

ELM | 13.556 | 0.099 | 0.721 |

SVR | 17.279 | 0.125 | 0.498 |

ARIMA | 20.965 | 0.169 | 0.503 |

BPNN | 21.322 | 0.165 | 0.475 |

RELM | ELM | SVR | ARIMA | BPNN | |
---|---|---|---|---|---|

t (s) | 2.31 | 2.14 | 15.37 | 10.22 | 9.93 |

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**MDPI and ACS Style**

Jiang, W.; Xu, Y.; Shan, Y.; Liu, H.
Degradation Tendency Measurement of Aircraft Engines Based on FEEMD Permutation Entropy and Regularized Extreme Learning Machine Using Multi-Sensor Data. *Energies* **2018**, *11*, 3301.
https://doi.org/10.3390/en11123301

**AMA Style**

Jiang W, Xu Y, Shan Y, Liu H.
Degradation Tendency Measurement of Aircraft Engines Based on FEEMD Permutation Entropy and Regularized Extreme Learning Machine Using Multi-Sensor Data. *Energies*. 2018; 11(12):3301.
https://doi.org/10.3390/en11123301

**Chicago/Turabian Style**

Jiang, Wei, Yanhe Xu, Yahui Shan, and Han Liu.
2018. "Degradation Tendency Measurement of Aircraft Engines Based on FEEMD Permutation Entropy and Regularized Extreme Learning Machine Using Multi-Sensor Data" *Energies* 11, no. 12: 3301.
https://doi.org/10.3390/en11123301