# Estimation of Feeding Composition of Industrial Process Based on Data Reconciliation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Data Reconciliation

#### 2.2. M-Estimator

- •
- $\rho $ is is continuous;
- •
- $\rho (\xi )=\rho (-\xi )$;
- •
- $\rho (\xi )\ge 0$ and $\rho $ is integrable
- •
- $\rho ({\xi}_{1})\le \rho ({\xi}_{2})$, for $\left|{\xi}_{i}\right|<\left|{\xi}_{j}\right|$;
- •
- $\rho (0)=0$.

- •
- IF is limited;
- •
- IF is continuous or piecewise continuous;
- •
- $\mathrm{IF}(-\xi )=-\mathrm{IF}(\xi )$;
- •
- IF is nearly linear near the origin ($\mathrm{IF}(\xi )\approx k\cdot \xi ,k\ne 0$, for small $\xi $), but this characteristic is not necessary;
- •
- the rejection point of IF (the point where IF is zero) is finite to suppress large deviations.

## 3. Data Reconciliation Based on a Novel Robust Estimator

#### 3.1. A Novel Robust Estimator

#### 3.2. The Tuning Parameter of the Novel Robust Estimator

#### 3.3. Linear Case

#### 3.3.1. There Are Two Gross Errors in the Measurement Variables

#### 3.3.2. There Are Three Gross Errors in the Measurement Variables

#### 3.4. Nonlinear Case

## 4. Feeding Composition Estimation Based on Iterative Data Reconciliation

#### 4.1. Industrial Process Description

#### 4.2. Iterative Robust Hierarchical Data Reconciliation and Composition Estimation Framework

## 5. Results from Real Industrial Data

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Kuehn, D.R.; Davidson, H. Computer control II. Mathematics of control. Chem. Eng. Prog.
**1961**, 57, 44–47. [Google Scholar] - Narasimhan, S.; Jordache, C. Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data; Elsevier: Amsterdam, The Netherlands, 1999. [Google Scholar]
- Prata, D.M.; Lima, E.L.; Pinto, J.C. In-Line Monitoring of Bulk Polypropylene Reactors Based on Data Reconciliation Procedures. Macromol. Symp.
**2008**, 271, 26–37. [Google Scholar] [CrossRef] - Prata, D.M.; Lima, E.L.; Pinto, J.C. Nonlinear Dynamic Data Reconciliation in Real Time in Actual Processes. Comput. Aided Chem. Eng.
**2009**, 27, 47–54. [Google Scholar] - Abu-El-Zeet, Z.H.; Roberts, P.D.; Becerra, V.M. Enhancing model predictive control using dynamic data reconciliation. Alche J.
**2002**, 48, 324–333. [Google Scholar] [CrossRef] - Sun, B.; Yang, C.; Wang, Y.; Gui, W.; Craig, I.; Olivier, L. A comprehensive hybrid first principles/machine learning modeling framework for complex industrial processes. J. Process Control
**2020**, 86, 30–43. [Google Scholar] [CrossRef] - Faber, R.; Arellano-Garcia, H.; Li, P.; Wozny, G. An optimization framework for parameter estimation of large-scale systems. Chem. Eng. Process.
**2007**, 46, 1085–1095. [Google Scholar] [CrossRef] - Schladt, M.; Hu, B. Soft Sensor Based on Nonlinear Steady-State Data Reconciliation in the Process Industry. Chem. Eng. Process.
**2007**, 46, 1107–1115. [Google Scholar] [CrossRef] - Su, Q.; Bommireddy, Y.; Shah, Y.; Ganesh, S.; Moreno, M.; Liu, J.; Gonzalez, M.; Yasdanpanah, N.; O’Connor, T.; Reklaitis, G.V.; et al. Data reconciliation in the Quality-byDesign (QbD) implementation of pharmaceutical continuous tablet manufacturing. Int. J. Pharm.
**2019**, 563, 259–272. [Google Scholar] [CrossRef] - Chiari, M.; Bussani, G.; Grotolli, M.G.; Pierucci, S. On-line Data Reconciliation and Optimisation: Refinery Applications. Comput. Chem. Eng.
**1997**, 21, s1185–s1190. [Google Scholar] [CrossRef] - Lee, M.H.; Lee, S.J.; Han, C.; Chang, K.S.; Kim, S.H.; Sang, H.Y. Hierarchical on-line data reconciliation and optimization for an industrial utility plant. Comput. Chem. Eng.
**1998**, 22, s247–s254. [Google Scholar] [CrossRef] - Almásy, G.A.; Sztanó, T. Checking and correction of measurements on the basis of linear system model. Probl. Control Inf. Theory
**1975**, 4, 57–69. [Google Scholar] - Mah, R.S.H.; Tamhane, A.C. Detection of gross errors in process data. Aiche J.
**2010**, 28, 828–830. [Google Scholar] [CrossRef] - Mah, R.S.; Stanley, G.M.; Downing, D.M. Reconciliation and rectification of process flow and inventory data. Ind. Eng. Chem. Process Des. Dev.
**1976**, 15, 175–183. [Google Scholar] [CrossRef] - Narasimhan, S.; Mah, R.S.H. Generalized likelihood ratio method for gross error identification. Aiche J.
**1987**, 33, 1514–1521. [Google Scholar] [CrossRef] - Tong, H.; Crowe, C.M. Detection of gross erros in data reconciliation by principal component analysis. Aiche J.
**2010**, 41, 1712–1722. [Google Scholar] [CrossRef] - Yu, M.; Hong-Ye, S.U.; Jian, C.H.U. A support vector regression approach for recursive simultaneous data reconciliation and gross error detection in nonlinear dynamical systems. Acta Autom. Sin.
**2009**, 35, 707–716. [Google Scholar] - Zhang, Z.; Chen, J. Simultaneous data reconciliation and gross error detection for dynamic systems using particle filter and measurement test. Comput. Chem. Eng.
**2014**, 69, 66–74. [Google Scholar] [CrossRef] - Yuan, Y.; Khatibisepehr, S.; Huang, B.; Li, Z. Bayesian method for simultaneous gross error detection and data reconciliation. Aiche J.
**2015**, 61, 3232–3248. [Google Scholar] [CrossRef] - Tjoa, I.B.; Biegler, L.T. Simultaneous strategies for data reconciliation and gross error detection of nonlinear systems. Comput. Chem. Eng.
**1991**, 15, 679–690. [Google Scholar] [CrossRef] - Johnson, L.P.M.; Kramer, M.A. Maximum likelihood data rectification: Steady-state systems. Alche J.
**1995**, 41, 2415–2426. [Google Scholar] [CrossRef] - Arora, N.; Biegler, L.T. Redescending estimators for data reconciliation and parameter estimation. Comput. Chem. Eng.
**2001**, 25, 1585–1599. [Google Scholar] [CrossRef] - Wang, D.; Romagnoli, J.A. A Framework for Robust Data Reconciliation Based on a Generalized Objective Function. Ind. Eng. Chem. Res.
**2003**, 42, 3075–3084. [Google Scholar] [CrossRef] - Özyurt, D.B.; Pike, R.W. Theory and practice of simultaneous data reconciliation and gross error detection for chemical processes. Comput. Chem. Eng.
**2004**, 28, 381–402. [Google Scholar] [CrossRef] - Ragot, J.; Chadli, M.; Maquin, D. Mass balance equilibration: A robust approach using contaminated distribution. Aiche J.
**2005**, 51, 1569–1575. [Google Scholar] [CrossRef] - Prata, D.M.; Schwaab, M.; Lima, E.L.; Pinto, J.C. Simultaneous robust data reconciliation and gross error detection through particle swarm optimization for an industrial polypropylene reactor. Chem. Eng. Sci.
**2010**, 65, 4943–4954. [Google Scholar] [CrossRef] - Zhang, Z.; Shao, Z.; Chen, X.; Wang, K.; Qian, J. Quasi-weighted least squares estimator for data reconciliation. Comput. Chem. Eng.
**2010**, 34, 154–162. [Google Scholar] [CrossRef] - Llanos, C.E.; Sanchéz, M.C.; Maronna, R.A. Robust estimators for data reconciliation. Ind. Eng. Chem. Res.
**2015**, 54, 5096–5105. [Google Scholar] [CrossRef] - Alighardashi, H.; Magbool Jan, N.; Huang, B. Expectation Maximization Approach for Simultaneous Gross Error Detection and Data Reconciliation Using Gaussian Mixture Distribution. Ind. Eng. Chem. Res.
**2017**, 56, 14530–14544. [Google Scholar] [CrossRef] - Xie, S.; Yang, C.; Yuan, X.; Wang, X.; Xie, Y. A novel robust data reconciliation method for industrial processes. Control Eng. Pract.
**2019**, 83, 203–212. [Google Scholar] [CrossRef] - de Menezes, D.Q.F.; Prata, D.M.; Secchi, A.R.; Pinto, J.C. A review on robust M-estimators for regression analysis. Comput. Chem. Eng.
**2021**, 147, 107254. [Google Scholar] [CrossRef] - Rey, W.J.J. Introduction to Robust and Quasi-Robust Statistical Methods; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Fair, R.C. On the robust estimation of econometric models. Ann. Econ. Soc. Meas.
**1974**, 3, 667. [Google Scholar] - Jin, S.Y.; Li, X.W.; Huang, Z.J.; Meng, L. A new target function for robust data reconciliation. Ind. Eng. Chem. Res.
**2012**, 51, 10220–10224. [Google Scholar] [CrossRef] - Hoaglin, D.C.; Mosteller, F.; Tukey, J.W. Understanding Robust and Exploratory Data Analysis; Wiley: New York, NY, USA, 1983; Volume 3. [Google Scholar]
- Albuquerque, J.S.; Biegler, L.T. Data reconciliation and gross-error detection for dynamic systems. Aiche J.
**1996**, 42, 2841–2856. [Google Scholar] [CrossRef] - Shevlyakov, G.; Morgenthaler, S.; Shurygin, A. Redescending m-estimators. J. Stat. Plann. Inference
**2008**, 138, 2906–2917. [Google Scholar] [CrossRef] [Green Version] - Zhou, X.J.; Yang, C.H.; Gui, W.H. State transition algorithm. J. Ind. Manag. Optim.
**2012**, 8, 1039–1056. [Google Scholar] [CrossRef] [Green Version]

**Figure 8.**The framework of iterative robust hierarchical data reconciliation and composition estimation.

M-Estimator | Tuning Parameter | |
---|---|---|

1 | Welsch | $\{\begin{array}{l}{c}_{w}=2.3828{E}_{ff}=90\%\\ {c}_{w}=2.9846{E}_{ff}=95\%\\ {c}_{w}=3.9077{E}_{ff}=98\%\\ {c}_{w}=4.7343{E}_{ff}=99\%\end{array}$ |

2 | Xie | $\{\begin{array}{l}{c}_{x}=1.6705{E}_{ff}=90\%\\ {c}_{x}=1.9597{E}_{ff}=95\%\\ {c}_{x}=2.3409{E}_{ff}=98\%\\ {c}_{x}=2.6359{E}_{ff}=99\%\end{array}$ |

3 | Proposed | $\{\begin{array}{l}{c}_{p}=1.3082{E}_{ff}=90\%\\ {c}_{p}=1.5424{E}_{ff}=95\%\\ {c}_{p}=2.0942{E}_{ff}=98\%\\ {c}_{p}=2.6060{E}_{ff}=99\%\end{array}$ |

Stream | True | Original Meas. | Meas. with Gross Error | Proposed | Xie | Welsch | Cauchy | Fair |
---|---|---|---|---|---|---|---|---|

${x}_{1}$ | 5 | 4.995 | 4.995 | 5.0111 | 5.0152 | 5.0346 | 5.0519 | 5.1777 |

${x}_{2}$ | 15 | 14.91 | 16.91 | 15.0420 | 15.0540 | 15.1202 | 15.1970 | 15.6413 |

${x}_{3}$ | 15 | 15.01 | 15.01 | 15.0420 | 15.0540 | 15.1202 | 15.1970 | 15.6413 |

${x}_{4}$ | 5 | 5.002 | 5.002 | 4.9987 | 4.9985 | 5.0053 | 5.0230 | 5.1795 |

${x}_{5}$ | 10 | 9.98 | 10.98 | 10.0433 | 10.0555 | 10.1149 | 10.1740 | 10.4618 |

${x}_{6}$ | 5 | 5.019 | 5.019 | 5.0322 | 5.0403 | 5.0803 | 5.1221 | 5.2841 |

${x}_{7}$ | 5 | 5.014 | 5.014 | 5.0111 | 5.0152 | 5.0346 | 5.0519 | 5.1777 |

SSE | -- | -- | -- | 0.0067 | 0.0110 | 0.0510 | 0.1287 | 1.2117 |

TER | -- | -- | -- | 0.9424 | 0.9263 | 0.8457 | 0.7606 | 0.2953 |

RER | -- | -- | -- | 0.9101 | 0.8838 | 0.7501 | 0.6008 | −0.2626 |

Stream | True | Original Meas. | Meas. with Gross Error | Proposed | Xie | Welsch | Cauchy | Fair |
---|---|---|---|---|---|---|---|---|

${x}_{1}$ | 5 | 4.995 | 4.995 | 5.0080 | 5.0164 | 5.0574 | 5.0323 | 4.3533 |

${x}_{2}$ | 15 | 14.91 | 16.91 | 15.0390 | 15.0552 | 15.1425 | 15.1802 | 15.1833 |

${x}_{3}$ | 15 | 15.01 | 15.01 | 15.0390 | 15.0552 | 15.1425 | 15.1802 | 15.1833 |

${x}_{4}$ | 5 | 5.002 | 5.002 | 4.9991 | 4.9984 | 5.0037 | 5.0244 | 5.3066 |

${x}_{5}$ | 10 | 9.98 | 10.98 | 10.0399 | 10.0568 | 10.1388 | 10.1558 | 9.8767 |

${x}_{6}$ | 5 | 5.019 | 5.019 | 5.0320 | 5.0404 | 5.0814 | 5.1235 | 5.5233 |

${x}_{7}$ | 5 | 5.014 | 3.514 | 5.0080 | 5.0164 | 5.0574 | 5.0323 | 4.3533 |

SSE | -- | -- | -- | 0.0058 | 0.0115 | 0.0731 | 0.1071 | 1.2866 |

TER | -- | -- | -- | 0.9743 | 0.9642 | 0.9111 | 0.8954 | 0.3474 |

RER | -- | -- | -- | 0.9641 | 0.9470 | 0.8621 | 0.8446 | 0.1268 |

Stream | Standard Deviation | True | Meas. | Proposed | Xie | Welsch | Cauchy | Fair |
---|---|---|---|---|---|---|---|---|

${x}_{1}$ | 0.5 | 4.5124 | 4.5360 | 4.5378 | 4.5280 | 4.5379 | 4.5796 | 4.4727 |

${x}_{2}$ | 0.6 | 5.5819 | 5.9070 | 5.5754 | 5.5655 | 5.5331 | 5.5360 | 5.6514 |

${x}_{3}$ | 0.2 | 1.9260 | 1.8074 | 1.9221 | 1.9223 | 1.9200 | 1.9153 | 1.9321 |

${x}_{4}$ | 0.2 | 1.4560 | 1.4653 | 1.4653 | 1.4842 | 1.4924 | 1.5096 | 1.4655 |

${x}_{5}$ | 0.5 | 4.8545 | 4.5491 | 4.8156 | 4.8010 | 4.8083 | 4.7882 | 4.7870 |

${u}_{1}$ | -- | 11.070 | -- | 11.0988 | 11.1178 | 11.1939 | 11.2079 | 10.9076 |

${u}_{2}$ | -- | 0.6147 | -- | 0.6143 | 0.6160 | 0.6187 | 0.6168 | 0.6104 |

${u}_{3}$ | -- | 2.0504 | -- | 2.0469 | 2.0349 | 2.0317 | 2.0345 | 2.0372 |

SSE | -- | -- | -- | 0.0031 | 0.0067 | 0.0222 | 0.0333 | 0.0377 |

TER | -- | -- | -- | 0.8950 | 0.8192 | 0.7751 | 0.6631 | 0.7994 |

RER | -- | -- | -- | 0.8805 | 0.8008 | 0.7324 | 0.5930 | 0.7693 |

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**MDPI and ACS Style**

Luan, Y.; Jiang, M.; Feng, Z.; Sun, B.
Estimation of Feeding Composition of Industrial Process Based on Data Reconciliation. *Entropy* **2021**, *23*, 473.
https://doi.org/10.3390/e23040473

**AMA Style**

Luan Y, Jiang M, Feng Z, Sun B.
Estimation of Feeding Composition of Industrial Process Based on Data Reconciliation. *Entropy*. 2021; 23(4):473.
https://doi.org/10.3390/e23040473

**Chicago/Turabian Style**

Luan, Yusi, Mengxuan Jiang, Zhenxiang Feng, and Bei Sun.
2021. "Estimation of Feeding Composition of Industrial Process Based on Data Reconciliation" *Entropy* 23, no. 4: 473.
https://doi.org/10.3390/e23040473