# Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Work

_{2}and CO

_{2}[28].

## 3. Methods

#### 3.1. Combustion Parameters

#### 3.2. Sensor Signals Window

#### 3.3. Discrete Wavelet Transform and Feature Extraction

#### 3.4. Extreme Gradient Boosting (XGBoost) Regression and Feature Importance (FI)

## 4. Results

#### 4.1. PFP Regression and FI

#### 4.2. MFB50 Regression and FI

#### 4.3. Summary

**Table 3.**Summary of regression results for all selected combustion parameters of introduced approach and comparison method.

DWT + XGBoost | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Engine 1 | Engine 2 | |||||||||||

Validation | Test | Validation | Test | |||||||||

Target | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} |

PFP | $3.98$ | $3.00$ | $0.98$ | $5.69$ | $4.35$ | $0.96$ | $3.87$ | $2.97$ | $0.87$ | $3.7$ | $2.89$ | $0.82$ |

MFB10 | $1.38$ | $1.09$ | $0.83$ | $1.43$ | $1.14$ | $0.77$ | $0.84$ | $0.66$ | $0.53$ | $0.82$ | $0.65$ | $0.48$ |

MFB50 | $0.79$ | $0.55$ | $0.97$ | $0.89$ | $0.63$ | $0.95$ | $0.71$ | $0.54$ | $0.80$ | $0.66$ | $0.52$ | $0.74$ |

MFB90 | $1.42$ | $1.05$ | $0.96$ | $1.43$ | $1.09$ | $0.79$ | $1.63$ | $1.20$ | $0.91$ | $1.42$ | $1.10$ | $0.74$ |

Time/Frequency + XGBoost | ||||||||||||

Engine 1 | Engine 2 | |||||||||||

Validation | Test | Validation | Test | |||||||||

Target | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} |

PFP | $6.25$ | $4.77$ | $0.96$ | $7.73$ | $6.06$ | $0.92$ | $5.24$ | $4.12$ | $0.76$ | $4.99$ | $3.95$ | $0.67$ |

MFB10 | $1.70$ | $1.34$ | $0.73$ | $1.90$ | $1.50$ | $0.59$ | $0.97$ | $0.76$ | $0.39$ | $0.96$ | $0.76$ | $0.30$ |

MFB50 | $1.78$ | $1.24$ | $0.86$ | $2.03$ | $1.44$ | $0.71$ | $1.01$ | $0.77$ | $0.60$ | $0.90$ | $0.71$ | $0.51$ |

MFB90 | $2.60$ | $1.88$ | $0.86$ | $3.01$ | $2.17$ | $0.69$ | $1.91$ | $1.47$ | $0.63$ | $1.82$ | $1.40$ | $0.56$ |

## 5. Discussion

#### 5.1. KS Position

#### 5.2. Towards a Theoretic Explanation of FI

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

AC | Approximation coefficients |

CA | Crank angle |

CWT | Continuous wavelet transfrom |

DC | Detailed coefficients |

DWT | Discrete wavelet transform |

ECU | Engine control unit |

FI | Feature importance |

IMEP | Indicated mean effective pressure |

KS | Knock sensor |

MAE | Mean absolute error |

MDI | Mean decrease in impurity |

MFB10 | Mass fraction burned 10% |

MFB50 | Mass fraction burned 50% |

MFB90 | Mass fraction burned 90% |

OPs | Operating points |

PFP | Peak firing pressure |

PS | Pressure sensor |

RMSE | Root mean square error |

RMS | Root mean square |

${R}^{2}$ | Coefficient of determination |

SCE | Single cylinder engine |

SHAP | Shapley additive explanations |

XGBoost | Extreme gradient boosting |

## References

- Maurya, R.K. Reciprocating Engine Combustion Diagnostics; Springer Nature: Cham, Switzerland, 2019. [Google Scholar]
- Pla, B.; De La Morena, J.; Bares, P.; Jiménez, I. Adaptive in-cylinder pressure model for spark ignition engine control. Fuel
**2021**, 299, 120870. [Google Scholar] [CrossRef] - Siano, D.; Bozza, F.; D’Agostino, D.; Panza, M.A. The Use of Vibrational Signals for On-Board Knock Diagnostics Supported by In-Cylinder Pressure Analyses; Technical Report; SAE International: Warrendale, PA, USA, 2014. [Google Scholar] [CrossRef]
- Chauvin, J.; Grondin, O.; Nguyen, E.; Guillemin, F. Real-time combustion parameters estimation for HCCI-diesel engine based on knock sensor measurement. IFAC Proc. Vol.
**2008**, 41, 8501–8507. [Google Scholar] [CrossRef] [Green Version] - Aliramezani, M.; Koch, C.R.; Shahbakhti, M. Modeling, diagnostics, optimization, and control of internal combustion engines via modern machine learning techniques: A review and future directions. Prog. Energy Combust. Sci.
**2022**, 88, 100967. [Google Scholar] [CrossRef] - Lounici, M.S.; Loubar, K.; Balistrou, M.; Tazerout, M. Investigation on heat transfer evaluation for a more efficient two-zone combustion model in the case of natural gas SI engines. Appl. Therm. Eng.
**2011**, 31, 319–328. [Google Scholar] [CrossRef] [Green Version] - Posch, S.; Pirker, G.; Kefalas, A.; Wimmer, A. Development of a Virtual Sensor to Predict Cylinder Pressure Signal based on Knock Sensor Signal; Technical report; SAE Technical International: Warrendale, PA, USA, 2022. [Google Scholar]
- Wang, Q.; Sun, T.; Lyu, Z.; Gao, D. A Virtual In-Cylinder Pressure Sensor Based on EKF and Frequency-Amplitude-Modulation Fourier-Series Method. Sensors
**2019**, 19, 3122. [Google Scholar] [CrossRef] [Green Version] - Businaro, A.; Cavina, N.; Corti, E.; Mancini, G.; Moro, D.; Ponti, F.; Ravaglioli, V. Accelerometer Based Methodology for Combustion Parameters Estimation. Energy Procedia
**2015**, 81, 950–959. [Google Scholar] [CrossRef] [Green Version] - Han, R.; Bohn, C.; Bauer, G. Recursive engine in-cylinder pressure estimation using Kalman filter and structural vibration signal. IFAC-PapersOnLine
**2018**, 51, 700–705. [Google Scholar] [CrossRef] - Siano, D.; Valentino, G.; Bozza, F.; Iacobacci, A.; Marchitto, L. A Non-Linear Regression Technique to Estimate from Vibrational Engine Data the Instantaneous In-Cylinder Pressure Peak and Related Angular Position; Technical Report; SAE International: Warrendale, PA, USA, 2016. [Google Scholar] [CrossRef]
- Norouzi, A.; Heidarifar, H.; Shahbakhti, M.; Koch, C.R.; Borhan, H. Model Predictive Control of Internal Combustion Engines: A Review and Future Directions. Energies
**2021**, 14, 6251. [Google Scholar] [CrossRef] - Taglialatela, F.; Lavorgna, M.; Mancaruso, E.; Vaglieco, B. Determination of combustion parameters using engine crankshaft speed. Mech. Syst. Signal Process.
**2013**, 38, 628–633. [Google Scholar] [CrossRef] - Johnsson, R. Cylinder pressure reconstruction based on complex radial basis function networks from vibration and speed signals. Mech. Syst. Signal Process.
**2006**, 20, 1923–1940. [Google Scholar] [CrossRef] - Bennett, C.; Dunne, J.; Trimby, S.; Richardson, D. Engine cylinder pressure reconstruction using crank kinematics and recurrently-trained neural networks. Mech. Syst. Signal Process.
**2017**, 85, 126–145. [Google Scholar] [CrossRef] - Jia, L.; Naber, J.D.; Blough, J.R. Review of sensing methodologies for estimation of combustion metrics. J. Combust.
**2016**, 2016, 8593523. [Google Scholar] [CrossRef] [Green Version] - Siano, D.; D’Agostino, D. Knock Detection in SI Engines by Using the Discrete Wavelet Transform of the Engine Block Vibrational Signals. Energy Procedia
**2015**, 81, 673–688. [Google Scholar] [CrossRef] [Green Version] - Delvecchio, S.; Bonfiglio, P.; Pompoli, F. Vibro-acoustic condition monitoring of Internal Combustion Engines: A critical review of existing techniques. Mech. Syst. Signal Process.
**2018**, 99, 661–683. [Google Scholar] [CrossRef] - Jang, Y.I.; Sim, J.Y.; Yang, J.R.; Kwon, N.K. The optimal selection of mother wavelet function and decomposition level for denoising of dcg signal. Sensors
**2021**, 21, 1851. [Google Scholar] [CrossRef] [PubMed] - Alqahtani, M.; Gumaei, A.; Mathkour, H.; Maher Ben Ismail, M. A genetic-based extreme gradient boosting model for detecting intrusions in wireless sensor networks. Sensors
**2019**, 19, 4383. [Google Scholar] [CrossRef] [Green Version] - Chakraborty, D.; Elzarka, H. Early detection of faults in HVAC systems using an XGBoost model with a dynamic threshold. Energy Build.
**2019**, 185, 326–344. [Google Scholar] [CrossRef] - Flores, V.; Keith, B. Gradient boosted trees predictive models for surface roughness in high-speed milling in the steel and aluminum metalworking industry. Complexity
**2019**, 2019, 1536716. [Google Scholar] [CrossRef] [Green Version] - Leon-Medina, J.X.; Anaya, M.; Parés, N.; Tibaduiza, D.A.; Pozo, F. Structural damage classification in a Jacket-type wind-turbine foundation using principal component analysis and extreme gradient boosting. Sensors
**2021**, 21, 2748. [Google Scholar] [CrossRef] - Rao, H.; Shi, X.; Rodrigue, A.K.; Feng, J.; Xia, Y.; Elhoseny, M.; Yuan, X.; Gu, L. Feature selection based on artificial bee colony and gradient boosting decision tree. Appl. Soft Comput.
**2019**, 74, 634–642. [Google Scholar] [CrossRef] - Sun, R.; Wang, G.; Zhang, W.; Hsu, L.T.; Ochieng, W.Y. A gradient boosting decision tree based GPS signal reception classification algorithm. Appl. Soft Comput.
**2020**, 86, 105942. [Google Scholar] [CrossRef] - Xuan, P.; Sun, C.; Zhang, T.; Ye, Y.; Shen, T.; Dong, Y. Gradient boosting decision tree-based method for predicting interactions between target genes and drugs. Front. Genet.
**2019**, 10, 459. [Google Scholar] [CrossRef] [PubMed] - Nishat Toma, R.; Kim, J.M. Bearing fault classification of induction motors using discrete wavelet transform and ensemble machine learning algorithms. Appl. Sci.
**2020**, 10, 5251. [Google Scholar] [CrossRef] - Zelenka, J.; Kammel, G.; Wimmer, A.; Bärow, E.; Huschenbett, M. Analysis of a prechamber ignited HPDI gas combustion concept. In SAE Technical Papers; SAE International: Warrendale, PA, USA, 2020. [Google Scholar] [CrossRef]
- Kirsten, M. Detektion Klopfender Verbrennung in Diesel/Erdgas-Dual-Fuel-Motoren. Ph.D. Thesis, Graz University of Technology, Graz, Austria, 2016. [Google Scholar]
- Pischinger, R.; Klell, M.; Sams, T. Thermodynamik der Verbrennungskraftmaschine; Springer: Vienna, Austria, 2009. [Google Scholar]
- Pipitone, E. A comparison between combustion phase indicators for optimal spark timing. J. Eng. Gas Turbines Power
**2008**, 130, 052808. [Google Scholar] [CrossRef] - Eriksson, L.; Thomasson, A. Cylinder state estimation from measured cylinder pressure traces-a survey. IFAC-PapersOnLine
**2017**, 50, 11029–11039. [Google Scholar] [CrossRef] - Hosseinzadeh, M. Robust control applications in biomedical engineering: Control of depth of hypnosis. In Control Applications for Biomedical Engineering Systems; Elsevier: Amsterdam, The Netherlands, 2020; pp. 89–125. [Google Scholar]
- Kefalas, A.; Ofner, A.B.; Pirker, G.; Posch, S.; Geiger, B.C.; Wimmer, A. Detection of knocking combustion using the continuous wavelet transformation and a convolutional neural network. Energies
**2021**, 14, 439. [Google Scholar] [CrossRef] - Addison, P.S. The Illustrated Wavelet Transform Handbook Introductory Theory and Applications in Science, Engineering, Medicine and Finance; lOP Publishing Ltd.: Bristol, UK, 2002. [Google Scholar]
- Saeed, A.; Ragai, H.F. Implementation of fast discrete wavelet transform for vibration analysis on an FPGA. In Proceedings of the 2012 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP), Poznan, Poland, 18–20 July 2012; pp. 1–5. [Google Scholar] [CrossRef]
- Barandas, M.; Folgado, D.; Fernandes, L.; Santos, S.; Abreu, M.; Bota, P.; Liu, H.; Schultz, T.; Gamboa, H. Tsfel: Time series feature extraction library. SoftwareX
**2020**, 11, 100456. [Google Scholar] [CrossRef] - Chakraborty, D.; Elzarka, H. Advanced machine learning techniques for building performance simulation: A comparative analysis. J. Build. Perform. Simul.
**2019**, 12, 193–207. [Google Scholar] [CrossRef] - Chen, T.; Guestrin, C. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd ACM Sigkdd International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016; pp. 785–794. [Google Scholar]
- Ilay Adler, A.; Painsky, A. Feature Importance in Gradient Boosting Trees with Cross-Validation Feature Selection. arXiv
**2021**, arXiv:2109.05468. [Google Scholar] - Breiman, L.; Friedman, J.H.; Olshen, R.A.; Stone, C.J. Classification and Regression Trees; Routledge: London, UK, 2017. [Google Scholar]
- Lundberg, S.M.; Erion, G.; Chen, H.; DeGrave, A.; Prutkin, J.M.; Nair, B.; Katz, R.; Himmelfarb, J.; Bansal, N.; Lee, S.I. From local explanations to global understanding with explainable AI for trees. Nat. Mach. Intell.
**2020**, 2, 56–67. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of experimental setup with marked positions for in-cylinder PS (cyan) and KS (orange).

Metric | Haar | Db4 | Sym4 | Coif6 |
---|---|---|---|---|

MAE | 3.00 | 3.97 | 3.94 | 4.31 |

RMSE | 3.98 | 5.12 | 5.11 | 5.57 |

${R}^{2}$ | 0.98 | 0.97 | 0.97 | 0.97 |

Description | Name | Equations |
---|---|---|

1. Index of minimum | arg_min | ${\varphi}_{min}=argmin\left(c\right)$ |

2. Index of maximum | arg_max | ${\varphi}_{max}=argmax\left(c\right)$ |

3. Variance | var | ${\sigma}^{2}={\sum}_{i=1}^{n}(\frac{{({c}_{i}-\overline{c})}^{2}}{n-1}$) |

4. Maximum | max | ${c}_{max}=max\left(c\right)$ |

5. Minimum | min | ${c}_{min}=min\left(c\right)$ |

6. Maximum gradient | max_grad | $\frac{\partial c}{\partial \varphi}{|}_{max}=max\left(\frac{\partial c}{\partial \varphi}\right)$ |

7. Mean difference | mean_diff | $\overline{d}=\overline{{c}_{i+1}-{c}_{i}}$ |

8. Root mean square | rms | ${c}_{rms}=\frac{\sqrt{{\sum}_{i=1}^{n}{{c}_{i}}^{2}}}{n}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kefalas, A.; Ofner, A.B.; Pirker, G.; Posch, S.; Geiger, B.C.; Wimmer, A.
Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting. *Sensors* **2022**, *22*, 4235.
https://doi.org/10.3390/s22114235

**AMA Style**

Kefalas A, Ofner AB, Pirker G, Posch S, Geiger BC, Wimmer A.
Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting. *Sensors*. 2022; 22(11):4235.
https://doi.org/10.3390/s22114235

**Chicago/Turabian Style**

Kefalas, Achilles, Andreas B. Ofner, Gerhard Pirker, Stefan Posch, Bernhard C. Geiger, and Andreas Wimmer.
2022. "Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting" *Sensors* 22, no. 11: 4235.
https://doi.org/10.3390/s22114235