Projective Weighted Dynamic Time Warping-Based Monitoring Method for Multi-Stage Processes with Unequal Durations

Academic Background

In modern manufacturing industries, online monitoring of multi-stage processes (such as batch and transition processes) is crucial for improving product quality and reducing failure risks. However, due to varying operating conditions, the durations of these processes are often unequal, posing significant challenges to monitoring. Traditional Dynamic Time Warping (DTW) methods can be used for offline synchronization but fail to effectively align ongoing batches with completed historical batches during online data processing due to inherent differences in their progression. Additionally, traditional methods typically overlook time-scale faults in operational processes, which undermines overall monitoring performance. To address these issues, this paper proposes a novel Projective Weighted Dynamic Time Warping (PWDTW) method for monitoring multi-stage processes with unequal durations.

This study aims to solve the practical problem of online monitoring of multi-stage processes from both amplitude and time perspectives. Traditional methods generally focus only on offline synchronization and monitoring of signal amplitudes, making it difficult to effectively identify issues such as processes progressing too fast or too slow. By introducing the PWDTW method, this paper not only achieves offline synchronization but also evaluates the progress of new samples online and designs two monitoring indices to detect amplitude and time-scale anomalies.

Source of the Paper

This paper is co-authored by Ying Zheng, Peiming Wang, Yang Wang, and David Shan-Hill Wong. Ying Zheng and Peiming Wang are affiliated with the School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Yang Wang is with the Department of Data Science, City University of Hong Kong, and David Shan-Hill Wong is with the Department of Chemical Engineering, National Tsing Hua University. The paper was published in IEEE Transactions on Automation Science and Engineering and formally released in 2025.

Research Process and Results

1. Offline Alignment

In the offline phase, researchers first collected a series of multi-stage process data under Normal Operating Conditions (NOC). These datasets contain trajectories of varying lengths. To align these data, the researchers employed the Asymmetric Weighted Dynamic Time Warping (AWDTW) method, combined with the Itakura Parallelogram Constraint, to limit the region of the warping path. By aligning historical data, synchronized datasets of the same length were generated.

Specific Steps: - Data Collection and Reference Set Selection: A reference set with a length close to the average value was selected from the NOC datasets. - AWDTW Alignment: AWDTW was used to align each dataset with the reference set, generating aligned datasets. - Itakura Constraint Optimization: By setting the Itakura parallelogram constraint factor, the region of the warping path was limited to avoid over-warping.

2. Online Monitoring

In the online phase, researchers proposed the Projective Weighted Dynamic Time Warping (PWDTW) method, combined with an open-ended strategy, to handle online asynchronization issues. This method evaluates the progress of the online trajectory against historical trajectories to identify the historical subsequence most similar to the current online data. Subsequently, the k-Nearest Neighbor (kNN) algorithm was used to identify the training subsequence most similar to the online trajectory, and two monitoring indices were designed to detect amplitude and time-scale anomalies.

Specific Steps: - Online Data Collection: Collect online process test data and compute its PWDTW distance with the aligned datasets. - kNN Clustering: Use the kNN algorithm to find the historical subsequence most similar to the online data. - Monitoring Index Design: Design two monitoring indices for detecting amplitude and time-scale anomalies.

3. Monitoring Indices

This paper designed two monitoring indices: the amplitude-scale change rate index (δk) and the time-scale counting index (c̄). The δk index is used to detect sudden changes in process amplitude, while the c̄ index evaluates whether the process progress speed is abnormal. Through these two indices, researchers can simultaneously monitor the strength and speed of the process, enabling more comprehensive monitoring.

4. Case Studies

To validate the method’s effectiveness, it was applied to the Tennessee Eastman Process (TEP) and a real-world semiconductor manufacturing process. In the TEP, researchers simulated four typical faults and demonstrated through experiments that the PWDTW method could effectively detect these faults. In the semiconductor manufacturing process, the paper showed how the PWDTW method detected long-term trends and time-scale anomalies.

Specific Results: - TEP Case: The PWDTW method successfully detected amplitude-scale faults (such as over-target and under-target) and time-scale faults (such as operations being too fast or too slow). - Semiconductor Manufacturing Case: The method successfully identified long-term trends and time-scale anomalies in the semiconductor manufacturing process, validating its feasibility in industrial applications.

Conclusions and Significance

The PWDTW method proposed in this paper demonstrates significant advantages in online monitoring of multi-stage processes. By combining Asymmetric Weighted DTW and the Itakura constraint, this paper not only achieves efficient alignment of historical data but also ensures precise matching of online data through an open-ended strategy. Additionally, the design of two monitoring indices enables researchers to comprehensively evaluate amplitude and time-scale anomalies.

Significance of the Research: - Scientific Value: This paper proposes a new online monitoring method capable of effectively handling multi-stage processes with unequal durations, addressing the limitations of traditional methods in online applications. - Practical Value: The successful application of this method in the TEP and semiconductor manufacturing processes demonstrates its broad applicability in industries such as energy production and pharmaceuticals.

Research Highlights

1. Comprehensive Synchronization Method: This paper achieves not only offline synchronization but also develops an online synchronization strategy for real-time monitoring of new data progress.
2. Novel Online Asynchronization Strategy: An open-ended strategy based on PWDTW is proposed to online identify the historical subsequence most matching the current progress.
3. Dual-Scale Monitoring Indices: Two monitoring indices are designed to detect amplitude and time-scale anomalies, providing a more comprehensive evaluation of the process.

Other Valuable Information

This work was supported by the Hubei Province International Science and Technology Cooperation Project (2024EHA033) and the Interdisciplinary Research Program of Huazhong University of Science and Technology (2024JCYJ031). Additionally, the PWDTW method has broad applicability and can be adapted for monitoring systems with varying durations in other industries.

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Through this research, the PWDTW method has been theoretically validated and demonstrated powerful monitoring capabilities in real-world industrial applications, providing a new solution for online monitoring of multi-stage processes.