A Projective Weighted DTW Based Monitoring Approach for Multi-stage Processes with Unequal Durations

Multi-stage processes, such as batch and transition processes, often have unequal operation duration due to differing conditions, posing significant challenges to process monitoring. Although dynamic time warping (DTW) has been applied for offline synchronization, it cannot adequately align an evolving, incomplete online batch with completed historical batches due to inherent inconsistencies in their progression. Moreover, traditional methods generally overlook time-scale faults in the operational progress of the process, which undermines overall monitoring performance. To address these issues, a novel projective weighted DTW (PwDTW)-based method is proposed to monitor multi-stage processes with unequal durations. First, the asymmetric weighted DTW is adopted to offline align the original training dataset with different lengths, incorporating the Itakura parallelogram constraint to restrict the region of the warping path. Then, the PwDTW with an open-ended strategy is proposed to handle the online asynchronization problem by assessing the progress and similarity of the ongoing trajectory against each training trajectory. Further, the k-nearest neighbor (KNN) is used to identify the most similar subsequences of the training dataset with the online trajectory. Leveraging these subsequences, two monitoring indices are designed to monitor the process in not only amplitude scale but also time scale. The two indices reflect both the strength and speed of the process. Finally, a benchmark Tennessee Eastman process and a practical semiconductor manufacturing case are introduced to prove the effectiveness of the proposed method.

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