**1. Introduction**

Batch process products play an increasingly important role in modern human life. In order to meet the ever-changing market demand of modern society, the safe and reliable operation of batch processes and continuous and stable product quality have gradually become the focus of attention in the processing industry [1,2]. The characteristics of batch operation processes are more complex than that of continuous industrial processes and have more abundant statistical characteristics. In order to enhance the safety of the batch production process and its control system, it is urgen<sup>t</sup> to establish a suitable processmonitoring system to monitor the production process.

Currently, data-driven methods [3–5] of extracting information from process data and modeling monitoring have become a hotspot in process-monitoring research. With the advancement of sensor technology, almost all industrial objects are equipped with different types of sensing devices. This results in a large amount of data being obtained in an industrial process. The data-driven methods extract information hidden in data by analyzing and mining collected industrial data, which may help reveal the operation mode of the industrial process and trace the fault reasons. In recent years, data-driven methods are continuously developed and perfected, batch process monitoring and fault diagnosis technologies based on data-driven methods have increasingly become research hotspots of people, and theories of the batch process monitoring and fault diagnosis technologies are continuously and deeply developed.

Multivariate statistical analysis methods do not require the acquisition of process mechanism knowledge; they only require the use of historical data to build models. These

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**Citation:** Zhao, L.; Huang, X.; Yu, H.Quality-Analysis-Based Process Monitoring for Multi-Phase Multi-Mode Batch Processes. *Processes* **2021**, *9*, 1321. https:// doi.org/10.3390/pr9081321

Academic Editors: Marco S. Reis and Furong Gao

Received: 31 May 2021 Accepted: 27 July 2021 Published: 29 July 2021

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**Copyright:** © 2021 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/).

methods can effectively extract key information in data, eliminate redundancy and remarkably reduce data dimensionality so that the process running state can be directly displayed in a two-dimensional statistical monitoring graph. Before the 1990s, researchers have generally simply treated batch processes as special continuous processes of limited duration, with no theoretical system of research specifically directed to batch process monitoring. Due to the essential difference of the characteristics of the batch process and the continuous process, a satisfactory effect is difficult to obtain in the batch process.

Aiming at the three-dimensional data characteristics of the batch process, a trilinear decomposition model can be established to directly investigate the three-dimensional data structure [6]. The data are stored and analyzed by using a trilinear decomposition model, and structural information of the data can be retained. In summary, there are six different two-dimensional matrix unfolding modes [7], which mainly reflect the different arrangemen<sup>t</sup> modes inside the data.

Nomikos [8–10] proposed multi-way principal components analysis (MPCA) and multi-way partial least squares (MPLS) methods, innovatively extending the successful application of multivariate statistical analysis methods to batch processes. Different internationally academic institutions and teams, including Wold professor [11] of Umea University, English Martin professor [12] of Newcastle University, proposed their own methodology, which facilitates the study of batch process monitoring. Corresponding models for monitoring have been established based on the model under normal conditions. When influenced by abnormal disturbances, the process variable correlations are changed, thereby deviating from the laws and characteristics under normal conditions. Corresponding multivariate statistics are calculated and compared with the monitoring control limits defined in advance, and the occurrence of abnormal working conditions can be detected.

In the batch process, the multi-phase nature is another important nature. In recent years, many scholars have conducted considerable research into process monitoring and quality analysis of batch processes [13–15]. Most researches were carried out by establishing different models to obtain different characteristics and dividing a cycle of a batch into phases, due to the cognition that the correlation of variables in the same phase is similar, and the correlation of variables in different phases is very different. Some scholars study the characteristics of the phases, e.g., the problem of transitions between adjacent phases [16] and the problem of non-uniform durations [17]. In addition, the scholars suggested that phases contribute to the final quality together, and individual phase models should be connected in some way during the modeling process. Therefore, a recursive quality regression method aiming at the multi-phase characteristics of the batch process was proposed [18], where the regression on the process variables in the current phase and the residual quality obtained in the previous phase was carried out to extract important quality information between the phase.

In addition, due to the influence of various factors, there are multi-mode characteristics in the batch process. In the whole operation process, process changes in batch direction lead to different process states and different process characteristics. In this way, monitoring and quality prediction for only one process state may lead to inaccurate analysis and monitoring results. In order to solve this problem, some scholars proposed to build an integrated model that can include both the common model and the specific model [19]. However, these methods barely evaluate the changes along the batch direction, in which models are in general updated arbitrarily, decreasing the efficiency of the monitoring system, as well as increasing the chance of introducing disturbances into the process model. Some scholars have proposed a specific modeling method for a specific process state [20]. However, the process variation along the batch direction may be too slow to be divided into several states. In addition, in the batch production process, when a new mode is generated, the corresponding model is built in the mode library and saved in the mode library. However, the relationship between these modes is not analyzed and judged. As new modes are generated one after another, all new modes must be saved, which makes the mode library

larger and larger. Therefore, a quality prediction method based on the relationship between modes is proposed to extract information from historical modes [21].

In recent years, monitoring of multiple characteristics of batch processes has also been the direction of many scholars. A process-monitoring method based on multi-mode Fisher discriminant analysis to solve the problem of multi-mode monitoring of batch process was proposed [22], which overcomes the limitation of the single operation mode assumption. Taking the whole batch trajectory as the research object, based on the dynamic time warping method, the obtained data are automatically classified from the perspective of data distribution to reflect the differences in batch direction. For the batch process with multi-phase characteristics, a two-phase PLS regression model based on phase analysis and different statistical analyses was proposed [23]. At the first level, multiple PLS models are used to monitor a single point in time. At the second level, the final quality is predicted. Through these two different levels of models, real-time monitoring and accurate quality prediction are organically combined. Due to the calibration and modeling problems caused by operation switching (or moving to different phases), a new evolutionary PLS method is proposed, which can be used to predict intermediate quality measurement and to detect process faults avoiding false positives [24].

In this work, both multi-phase quality analysis and multi-mode quality analysis are conducted at the same time to develop a comprehensive process-monitoring strategy based on the quality prediction of batch processes. The multi-phase and multi-mode batch process concerned here involves variety in two directions. One is the multi-phase direction, the other is the multi-mode direction, and the processing methods of the two directions are different due to different process characteristics. In the multi-phase direction, the phase residual recursive model is unitized to connect the contributions of the successive phases on the final quality together, while in the multi-mode direction, the relationship between the current mode and the historical mode is analyzed and extracted to obtain more quality-related information for quality prediction. Firstly, the time-slice modeling method and the goodness-of-fit index are used to analyze the influence of different phases on the final quality and identify the critical-to-quality phases. Then, the phase mean model is introduced to analyze the phase characteristics and monitor the phase based on quality information. After that, single modes are analyzed, where the residual regression model of each phase is established with the quality variables of the current phase and the quality residual of the previous phase, and the current mode is predicted and monitored. In addition, for the quality prediction and monitoring of multiple modes, it is emphasized to extract the relationship between historical mode and new mode by between-mode modeling. This model contains more modal quality-related information and can better predict and monitor multiple modes. Finally, the strategy is applied to an injection molding process to illustrate the effectiveness of the strategy.

The rest of this paper includes four parts: the proposed method is presented in Section 2, including critical-to-quality phase identification, phase mean model, multi-phase residual recursive modeling for a single mode, between-mode modeling for multiple modes and model comparison and selection. In Section 3, the injection molding process is briefly introduced, and the method used is illustrated through an example to obtain the results and make a comparative analysis. At last, the conclusion is drawn.
