# The Optimization of a Model for Predicting the Remaining Useful Life and Fault Diagnosis of Landing Gear

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Effects of Missing Data on the Prediction Model

#### 2.2. Challenges to Identifying Different Fault Causes through Vibration Signals

## 3. Methodology

#### 3.1. Basics of PHM Methodology

- First step: data acquisition;
- Second step: data preprocessing;
- Third step: feature extraction;
- Fourth step: feature selection;
- Last step: constructing a health assessment model, a fault diagnosis model, and an RUL model.

#### 3.2. Hyperparameter Optimization

#### 3.2.1. Optimization of the RUL Model

Algorithm 1: Random Search Optimization |

function RandomSearch(model, hyperparameter_space, iterations): |

best_score = 0 |

for i = 1 to iterations do: |

sampled_hyperparameters = SampleFrom(hyperparameter_space) |

current_model = TrainModel(model, sampled_hyperparameters) |

current_score = Evaluate(current_model) |

if current_score > best_score: |

best_score = current_score |

best_hyperparameters = sampled_hyperparameters |

return best_hyperparameters |

#### 3.2.2. Optimization of the Fault Diagnosis Model

Algorithm 2: Bayesian Optimization |

Function BayesianOptimization(model, hyperparameter_space, evaluations): |

Initialize an empty dataset D to store hyperparameters and their respective scores for i = 1 to evaluations do: surrogate_model = Fit(D) next_hyperparameters = SelectNext(surrogate_model, hyperparameter_space) new_score = Evaluate(model, next_hyperparameters) new_data_point = (next_hyperparameters, new_score) Add new_data_point to D best_hyperparameters = hyperparameters in D with the highest score return best_hyperparameters |

## 4. Results and Discussion

#### 4.1. Health Indicator Model

#### 4.2. RUL Prediction Model

#### 4.3. Optimization of the RUL Prediction Model

#### 4.4. Optimization of the Fault Diagnosis Model

#### 4.5. Applications of RUL Prediction Model and the Fault Diagnosis Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**Comparison of RUL prediction models for Airplane #1: (

**a**) before and (

**b**) after optimization.

**Figure 9.**Comparison of RUL prediction models for Airplane #2: (

**a**) before and (

**b**) after optimization.

**Figure 10.**Comparison of RUL prediction models for Airplane #3: (

**a**) before and (

**b**) after optimization.

**Figure 11.**Comparison of RUL prediction models for Airplane #4: (

**a**) before and (

**b**) after optimization.

**Figure 12.**Comparison of RUL prediction models for Airplane #5: (

**a**) before and (

**b**) after optimization.

**Figure 13.**Comparison of RUL prediction models for Airplane #6: (

**a**) before and (

**b**) after optimization.

**Figure 14.**Comparison of RUL prediction models for Airplane #7: (

**a**) before and (

**b**) after optimization.

Item No. | Malfunctioning Item | Number of Data |
---|---|---|

A | Rose wheel/tire | 224 |

B | Left main wheel/tire | 947 |

C | Right main wheel/tire | 899 |

D | Main wheel/tire | 57 |

E | Main wheel hub | 106 |

F | Nose wheel hub | 77 |

G | Nose landing gear drag brace | 25 |

H | Left landing gear drag brace | 25 |

I | Right landing gear drag brace | 41 |

J | Left landing gear brake | 69 |

K | Right landing gear brake | 61 |

L | Nose landing gear retract actuator | 22 |

M | Left landing gear retract actuator | 24 |

N | Right landing gear retract actuator | 27 |

O | Nose landing gear shock strut | 37 |

P | Left landing gear shock strut | 37 |

Q | Right landing gear shock strut | 40 |

R | Steering actuator | 27 |

S | Steering control valve | 5 |

T | Brake control valve | 26 |

Total | 2776 |

**Table 2.**Hyperparameters used in the AR algorithm [26].

No | Hyperparameter | Search Range | Description |
---|---|---|---|

1 | P | Integer (0–10) | Number of previous time steps. |

**Table 3.**Hyperparameters used in the ARIMA algorithm [26].

No | Hyperparameter | Search Range | Description |
---|---|---|---|

1 | P | Integer (0–10) | Number of previous time steps. |

2 | D | Integer (0–5) | Number of times differencing. |

3 | Q | Integer (0–10) | Number of previous forecast errors. |

No | Hyperparameter | Search Range | Description |
---|---|---|---|

1 | Kernel | Categorical (‘kernel_rbf’, ‘kernel_rq’, ‘kernel_expsine’, ‘kernel_matern’, ‘kernel_dotproduct’) | Kernel specifying the covariance function of the Gaussian process. |

2 | Alpha | Integer ($1\times {10}^{-13}$ $1\times {10}^{-12}$, $1\times {10}^{-11}$, $1\times {10}^{-10}$, $1\times {10}^{-9}$, $1\times {10}^{-8}$,$1\times {10}^{-7}$) | Value added to the diagonal of the kernel matrix during fitting. |

3 | n_restarts_optimizer | Integer (0, 1, 2, 3) | Number of restarts of the optimizer for finding the kernel parameters that maximize the log-marginal likelihood. |

**Table 5.**Hyperparameters used in the XGBoost algorithm [21].

No | Hyperparameter | Search Range | Description |
---|---|---|---|

1 | n_estimators | Integer (100–10,000) | Number of boosting rounds. |

2 | max_depth | Integer (100–10,000) | Maximum tree depth for base learners. |

3 | learning_rate | Real number (0.01–0.5) | Boosting learning rate. |

4 | Booster | Categorical (‘gbtree’, ‘dart’) | Specify which booster to use: gbtree, gblinear, or dart. |

5 | Gamma | Real number (0–1000) | Minimum loss reduction required to make a further partition on a leaf node of the tree. |

6 | min_child_weight | Real number (1–100) | Minimum sum of instance weight (hessian) needed in a child. |

7 | Subsample | Real number (0.5–1) | Subsample ratio of the training instance. |

8 | colsample_bytree | Real number (0.5–1) | Subsample ratio of columns when constructing each tree. |

9 | reg_alpha | Real number (0–1) | L1 regularization term on weights. |

10 | reg_lambda | Real number (0–1) | L2 regularization term on weights. |

**Table 6.**Comparison of the results of the RUL prediction model with AR before and after optimization.

Airplane Number | Optimization | Actual HI of Latest Flight | Predicted HI of Latest Flight | RMSE | Error Range |
---|---|---|---|---|---|

Airplane #1 | Before | 0.379 | 0.391 | 0.013 | [−0.017, 0.026] |

After | 0.379 | 0.358 | 0.022 | [0.000, 0.047] | |

Airplane #2 | Before | 0.718 | 0.712 | 0.026 | [−0.081, 0.010] |

After | 0.718 | 0.685 | 0.030 | [−0.045, 0.048] | |

Airplane #3 | Before | 0.431 | 0.479 | 0.027 | [−0.048, 0.050] |

After | 0.431 | 0.436 | 0.050 | [−0.005, 0.071] | |

Airplane #4 | Before | 0.259 | 0.331 | 0.082 | [−0.133, 0.014] |

After | 0.259 | 0.251 | 0.040 | [−0.071, 0.012] | |

Airplane #5 | Before | 0.117 | 0.176 | 0.028 | [−0.062, 0.039] |

After | 0.117 | 0.077 | 0.056 | [−0.007, 0.081] | |

Airplane #6 | Before | 0.058 | 0.291 | 0.157 | [−0.233, −0.006] |

After | 0.058 | 0.040 | 0.022 | [−0.043, 0.043] | |

Airplane #7 | Before | 0.136 | 0.384 | 0.145 | [−0.249, −0.003] |

After | 0.136 | 0.155 | 0.040 | [−0.027, 0.063] | |

Average | Before | 0.0682 | |||

After | 0.0371 |

**Table 7.**Comparison of the results of the RUL prediction model with ARIMA before and after optimization.

Airplane Number | Optimization | Actual HI of Latest Flight | Predicted HI of Latest Flight | RMSE | Error Range |
---|---|---|---|---|---|

Airplane #1 | Before | 0.379 | 0.340 | 0.038 | [0.005, 0.065] |

After | 0.379 | 0.342 | 0.036 | [0.004, 0.064] | |

Airplane #2 | Before | 0.718 | 0.677 | 0.024 | [−0.050, 0.044] |

After | 0.718 | 0.677 | 0.024 | [−0.050, 0.044] | |

Airplane #3 | Before | 0.431 | 0.436 | 0.043 | [−0.022, 0.081] |

After | 0.431 | 0.435 | 0.044 | [−0.021, 0.081] | |

Airplane #4 | Before | 0.259 | 0.244 | 0.038 | [−0.066, 0.019] |

After | 0.259 | 0.243 | 0.037 | [−0.064, 0.020] | |

Airplane #5 | Before | 0.117 | 0.120 | 0.037 | [−0.008, 0.063] |

After | 0.117 | 0.120 | 0.037 | [−0.008, 0.063] | |

Airplane #6 | Before | 0.058 | −0.032 | 0.054 | [−0.037, 0.100] |

After | 0.058 | −0.032 | 0.054 | [−0.036, 0.100] | |

Airplane #7 | Before | 0.136 | 0.167 | 0.027 | [−0.044, 0.043] |

After | 0.136 | 0.151 | 0.025 | [−0.030, 0.045] | |

Average | Before | 0.0372 | |||

After | 0.0367 |

**Table 8.**Comparison of the results of the RUL prediction model with GPR before and after optimization.

Airplane Number | Optimization | Actual HI of Latest Flight | Predicted HI of Latest Flight | RMSE | Error Range |
---|---|---|---|---|---|

Airplane #1 | Before | 0.379 | 0.704 | 0.163 | [−0.325, −0.002] |

After | 0.379 | 0.314 | 0.063 | [0.024, 0.094] | |

Airplane #2 | Before | 0.718 | 0.771 | 0.068 | [−0.136, −0.043] |

After | 0.718 | 0.707 | 0.027 | [−0.080, 0.014] | |

Airplane #3 | Before | 0.431 | 0.474 | 0.027 | [−0.043, 0.050] |

After | 0.431 | 0.446 | 0.037 | [−0.030, 0.071] | |

Airplane #4 | Before | 0.259 | 0.238 | 0.039 | [−0.073, 0.027] |

After | 0.259 | 0.243 | 0.032 | [−0.060, 0.020] | |

Airplane #5 | Before | 0.117 | −0.107 | 0.206 | [0.043, 0.268] |

After | 0.117 | 0.173 | 0.031 | [−0.058, 0.049] | |

Airplane #6 | Before | 0.058 | 0.171 | 0.051 | [−0.113, −0.005] |

After | 0.058 | 0.009 | 0.037 | [−0.027, 0.071] | |

Airplane #7 | Before | 0.136 | 0.264 | 0.068 | [−0.131, 0.029] |

After | 0.058 | 0.009 | 0.037 | [−0.027, 0.071] | |

Average | Before | 0.0888 | |||

After | 0.0398 |

Item | Malfunctioning Item | Accuracy | Precision Rate | Recall Rate | F1 Score |
---|---|---|---|---|---|

A | Nose wheel/tire | 99.96% | 100.00% | 96.88% | 98.41% |

B | Left main wheel/tire | 99.73% | 100.00% | 95.56% | 97.73% |

C | Right main wheel/tire | 99.80% | 100.00% | 96.44% | 98.19% |

D | Main wheel/tire | 99.97% | 100.00% | 91.23% | 95.41% |

E | Main wheel hub | 99.95% | 100.00% | 92.45% | 96.08% |

F | Nose wheel hub | 99.97% | 100.00% | 93.51% | 96.64% |

G | Nose landing gear drag brace | 99.99% | 100.00% | 96.00% | 97.96% |

H | Left landing gear drag brace | 100.0% | 100.00% | 100.00% | 100.00% |

I | Right landing gear drag brace | 100.0% | 100.00% | 100.00% | 100.00% |

J | Left landing gear brake | 99.98% | 100.00% | 95.45% | 97.67% |

K | Right landing gear brake | 99.98% | 100.00% | 95.08% | 97.48% |

L | Nose landing gear retract actuator | 100.00% | 100.00% | 100.00% | 100.00% |

M | Left landing gear retract actuator | 100.00% | 100.00% | 100.00% | 100.00% |

N | Right landing gear retract actuator | 99.99% | 100.00% | 96.30% | 98.11% |

O | Nose landing gear shock strut | 99.99% | 100.00% | 97.30% | 98.63% |

P | Left landing gear shock strut | 100.00% | 100.00% | 100.00% | 100.00% |

Q | Right landing gear shock strut | 99.99% | 100.00% | 97.50% | 98.73% |

R | Steering actuator | 100.00% | 100.00% | 100.00% | 100.00% |

S | Steering control valve | 100.00% | 100.00% | 100.00% | 100.00% |

T | Brake control valve | 99.99% | 100.00% | 92.31% | 96.00% |

Average | 99.96% | 100.00% | 96.80% | 98.35% |

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## Share and Cite

**MDPI and ACS Style**

Chang, Y.-J.; Hsu, H.-K.; Hsu, T.-H.; Chen, T.-T.; Hwang, P.-W.
The Optimization of a Model for Predicting the Remaining Useful Life and Fault Diagnosis of Landing Gear. *Aerospace* **2023**, *10*, 963.
https://doi.org/10.3390/aerospace10110963

**AMA Style**

Chang Y-J, Hsu H-K, Hsu T-H, Chen T-T, Hwang P-W.
The Optimization of a Model for Predicting the Remaining Useful Life and Fault Diagnosis of Landing Gear. *Aerospace*. 2023; 10(11):963.
https://doi.org/10.3390/aerospace10110963

**Chicago/Turabian Style**

Chang, Yuan-Jen, He-Kai Hsu, Tzu-Hsuan Hsu, Tsung-Ti Chen, and Po-Wen Hwang.
2023. "The Optimization of a Model for Predicting the Remaining Useful Life and Fault Diagnosis of Landing Gear" *Aerospace* 10, no. 11: 963.
https://doi.org/10.3390/aerospace10110963