# Two-Dimensional, Two-Layer Quality Regression Model Based Batch Process Monitoring

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Batch Process Monitoring Based on Phase Mean PLS

^{2}statistics and square prediction error SPE statistics. Hotelling-T

^{2}statistics reflects the deviation degree of latent variables from the established model in amplitude and process data development trend. SPE describes the deviation degree of the measured value of the input variable from the latent variable space in the batch process [13].

^{2}and the control limit of the square prediction error SPE. The definitions of T

^{2}statistics and SPE statistics are as follows:

**W**

_{c}is the weight matrix. The detailed properties and calculations can be found in reference [8].

**F**distribution with the confidence level $\alpha $ and the degrees of freedom H and $I-H$, and H refers to the number of retained latent variables; ${g}_{c}{\mathit{\chi}}_{c,h,\alpha}^{2}$ means the ${\mathit{\chi}}^{2}$ distribution with the confidence level $\alpha $ and the proportional coefficient ${g}_{c}={s}_{c}/2{\mu}_{c}$; ${h}_{c}=2{{\mu}_{c}}^{2}/{s}_{c}$; ${\mu}_{c}$ refers to the mean value of SPE

_{c}; ${s}_{c}$ is the variance of SPE

_{c}.

^{2}statistics and online SPE statistics are calculated, and the calculation formula is as follows:

#### 2.2. Framework of Two-Dimensional, Two-Layer Regression Modeling Strategy

^{2}and SPE statistics when monitoring. Therefore, the statistical modeling and online monitoring of the batch process should not only consider a single mode but also carry out the quality analysis of multiple modes.

#### 2.3. Two-Dimensional, Two-Layer Regression Model

_{c}stands for the number of time-slices within phase c of mode m, m stands for the number of the historical modes, m = 1, 2, …, M, and c stands for the historical phases in each batch, c = 1, 2, …, C.

_{t}is the number of the time intervals within phase c. Then, the predictions based on the regression parameter of the whole phase, ${\overline{\mathsf{\alpha}}}_{t}$, are obtained:

^{2}statistics and online SPE statistics are calculated, and the calculation formula is as follows:

^{2}statistic of the current k-th moment, and $SP{E}_{t,k}$ is the SPE statistic of the current k-th moment; ${\tilde{\mathbf{z}}}_{t,k}$ is the residual vector at the k-th moment.

## 3. Illustration and Discussions

#### 3.1. Introduction of Injection Molding Process

#### 3.2. Normal Batch Monitoring

^{2}and SPE of the proposed method and the traditional method do not exceed the control limits in the injection phase and the packing-holding phase. In Figure 7, it can be seen that, in the plasticizing phase, T

^{2}and SPE of the proposed method do not exceed their respective control limits, while, for the traditional method, although T

^{2}does not exceed its control limit, there is an obvious period at the beginning of the phase during which the SPE value exceeds its control limit. In fact, the fluctuation of this batch is caused by the transition between the packing-holding phase and the plasticizing phase. During the transition, process characteristic change from the packing-holding phase characteristic to the plasticizing phase characteristic. This characteristic fluctuation is natural progress and does not affect the production, so this batch is normal. Because the traditional method which uses the phase mean model to represent the phase characteristic is more easily affected by the characteristic fluctuation away from the phase mean characteristic, this fluctuation is identified as a fault by the traditional method. However, this is not consistent with the conclusion from the real production. Because the proposed method comprehensively deals with the multi-phase and multi-mode problems by involving all quality-related phase and mode information in historical processes in the regression model, it is not easily affected by phase fluctuation, and it can offer the right monitoring result, in which this batch is normal during the plasticizing phase. In Figure 8, there is one point of SPE value of the proposed method exceeds the control limit. Generally, this point will not be considered as a fault. Therefore, it can be concluded from the monitoring results of the normal test batch that the proposed method is better than the traditional method because, by the two-dimensional, two-layer quality regression, all quality-related phase and mode information in historical processes have been extracted and utilized for quality prediction and monitoring.

#### 3.3. Abnormal Batch Monitoring

^{2}and SPE monitoring effects of the traditional method and the proposed method are shown in Figure 9, respectively. Compared with the traditional method, the amplitudes of the statistics of the proposed method are relatively larger. It can be concluded that the sensitivity of the proposed method is not affected, although it involves all quality-related phase and mode information in historical processes in the regression model, and it can identify the fault batch as fast as the traditional method, with an even larger alarm signal. So, the proposed method is better than the traditional method.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Operating Parameter | Set Value |
---|---|

Material | High density polyethylene (HDPE) |

Packing pressure | 25 Bar, 30 Bar, 35 Bar |

Packing time | 3 s |

Mold cooling water temperature | 25 °C |

Injection velocity | 24 mm/s |

Barrel temperature | 180 °C, 200 °C |

Cooling time | 15 s |

Number | Variable Description | Unit |
---|---|---|

1 | Screw speed | Mm/s |

2 | Plasticizing pressure | Bar |

3 | Nozzle temperature | °C |

4 | Cylinder pressure | Bar |

5 | SV2 valve opening | % |

6 | SV1 valve opening | % |

**Table 3.**Different operation modes caused by different packing pressure (PP) and different barrel temperature (BT).

Modes | PP/Bar | BT/°C |
---|---|---|

Mode 1 | 25 | 180 |

Mode 2 | 35 | 180 |

Mode 3 | 25 | 200 |

Mode 4 | 30 | 200 |

Mode 5 | 35 | 200 |

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**MDPI and ACS Style**

Zhao, L.; Huang, X.
Two-Dimensional, Two-Layer Quality Regression Model Based Batch Process Monitoring. *Processes* **2022**, *10*, 43.
https://doi.org/10.3390/pr10010043

**AMA Style**

Zhao L, Huang X.
Two-Dimensional, Two-Layer Quality Regression Model Based Batch Process Monitoring. *Processes*. 2022; 10(1):43.
https://doi.org/10.3390/pr10010043

**Chicago/Turabian Style**

Zhao, Luping, and Xin Huang.
2022. "Two-Dimensional, Two-Layer Quality Regression Model Based Batch Process Monitoring" *Processes* 10, no. 1: 43.
https://doi.org/10.3390/pr10010043