# Real-Time Prognostics of Engineered Systems under Time Varying External Conditions Based on the COX PHM and VARX Hybrid Approach

## Abstract

**:**

## 1. Introduction

#### 1.1. Failure Prognostics

#### 1.2. Applications of Vector Autoregressive (VAR) Models to the Failure Prognostics

#### 1.3. Survival Analysis Combined with Sensor Signal Forecasting Techniques

## 2. Methodology

#### 2.1. Research Framework

#### 2.2. Cox Proportional Hazards Model with Time-Varying Covariates

#### 2.3. Similarity Matching of Generated Reliability Indices

#### 2.4. Vector Autoregressive Models with Exogenous Variables (VARX)

#### 2.5. Conditional Granger Causality

#### 2.6. Grey Model with Fourier Series Calibration (FGM)

## 3. Case Studies

#### 3.1. Data Sets Employed in the Research

#### 3.2. Data Preprocessing

#### 3.3. Implementation of the Hybrid Approach

#### 3.3.1. Implementation of the Cox PHM with Time-Varying Covariates

#### 3.3.2. Implementation of the Pairwise Conditional Granger Causality (CGC) Tests

#### 3.4. Implementation of the VARX Model

#### 3.5. Forecast of the External Conditions Using the FGM

#### 3.6. Prediction of RULs for the Test Set Utilising the Models Based on the Train Set

## 4. Results

#### 4.1. Cox PHM Fitting Results Based on the Training Set

#### 4.2. Results of Similarity Matching of Reliability Indices

#### 4.3. VARX Fitting Results

#### 4.4. Fitting and Prediction of the FGM

#### 4.5. Results on the RUL Prediction of Turbofan Units in the Test Set

## 5. Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Inner structure of the turbofan employed in the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS) data sets [78].

**Figure 5.**Similarity matching between the incomplete reliability index of the 37th unit in the test set and its five most similar reliability indices of the units in the run-to-failure training set.The similarity decreases with the figure order. (

**a**) Incomplete reliability index of the 37th unit in the test set matched with the complete reliability index of the 45th unit in the training set. (

**b**) Incomplete reliability index of the 37th unit in the test set matched with the complete reliability index of the 28th unit in the training set. (

**c**) Incomplete reliability index of the 37th unit in the test set matched with the complete reliability index of the 98th unit in the training set. (

**d**) Incomplete reliability index of the 37th unit in the test set matched with the complete reliability index of the 93rd unit in the training set. (

**e**) Incomplete reliability index of the 37th unit in the test set matched with the complete reliability index of the 61st unit in the training set.

**Figure 6.**Box plot of the RMSE and the NRMSE of the in-sample fitting of the VARX models based on training set units.

**Figure 7.**In-sample fitting of the first operational conditions of the unit 37 in FD001 test sets based on the FGM(1, 1) and ARIMA/ARMA calibration.

**Figure 8.**In-sample fitting of the second operational conditions of the unit 37 in FD001 test sets based on the FGM(1, 1) and ARIMA/ARMA calibration.

**Figure 9.**Prediction of the first operational conditions of the unit 37 in FD001 test sets based on the FGM(1, 1) and ARIMA/ARMA calibration.

**Figure 10.**Prediction of the second operational conditions of the unit 37 in FD001 test sets based on the FGM(1, 1) and ARIMA/ARMA calibration.

**Figure 11.**Predictions of the incomplete reliability index of the test set unit 37th considering its five most similar reliability indices of the training set units. The online prediction of the reliability index for the unit 37th in the test set is updated by future values of the external operational conditions which are predicted by means of the FGM model.

**Figure 12.**Predictions of the reliability function of the test set unit 37th considering its five most similar reliability indices of the training set units.

**Table 1.**Fitting results of the Cox PHM with time-varying covariates based on the run-to-failure records of training set units.

Variable | Hazard Ratio | Lower 95% | Upper 95% | z | p Value | Partial Log-Likelihood | Penaliser |
---|---|---|---|---|---|---|---|

PC 1 | 3.59 | 2.67 | 4.83 | 8.43 | <0.005 | −328.05 | 0.01 |

PC 2 | 1.61 | 0.87 | 2.98 | 1.50 | 0.13 | ||

PC 3 | 0.71 | 0.26 | 1.94 | −0.66 | 0.51 |

**Table 2.**Comparisons between the Root Mean Square Error (RMSE) calculated by the proposed approach and other papers.

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**MDPI and ACS Style**

Zhu, H.
Real-Time Prognostics of Engineered Systems under Time Varying External Conditions Based on the COX PHM and VARX Hybrid Approach. *Sensors* **2021**, *21*, 1712.
https://doi.org/10.3390/s21051712

**AMA Style**

Zhu H.
Real-Time Prognostics of Engineered Systems under Time Varying External Conditions Based on the COX PHM and VARX Hybrid Approach. *Sensors*. 2021; 21(5):1712.
https://doi.org/10.3390/s21051712

**Chicago/Turabian Style**

Zhu, Hongmin.
2021. "Real-Time Prognostics of Engineered Systems under Time Varying External Conditions Based on the COX PHM and VARX Hybrid Approach" *Sensors* 21, no. 5: 1712.
https://doi.org/10.3390/s21051712