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In this paper, the GTucker2 model is proposed for monitoring both even-length and uneven-length batch processes. The GTucker2 model has two prominent advantages. The first one is that it performs tensor decomposition on the three-way data array and thus avoids potential problems of information loss and “curse of dimensionality” induced by data unfolding. The second one is that it solves the uneven-length problem in a “natural” way without using batch trajectory synchronization, which prevents distorting data and fault patterns and guarantees higher modeling and monitoring precisions. An online batch process monitoring method is then developed by integrating GTucker2 with the moving data window technique. Three monitoring statistics named Q, R2, and T2 statistics are constructed for fault detection and diagnosis. The effectiveness and advantages of the GTucker2-based monitoring method are illustrated by two case studies in a benchmark fed-batch penicillin fermentation process.

A novel data projection algorithm named tensor global-local preserving projections (TGLPP) is proposed for three-way data arrays. TGLPP is able to preserve global and local structures of data simultaneously. TGLPP builds a unified framework for global structure preserving and local structure preserving. Tensor principal component analysis (TPCA) and tensor locality preserving projections (TLPP) are unified in the TGLPP framework. They are proved to be two limiting cases of TGLPP. A batch process monitoring method is then developed on the basis of TGLPP. The SPD statistic and R2 statistic are used for fault detection. The contribution plot is applied for fault identification. Two case studies are carried out to validate the efficiency of TGLPP-based monitoring method.

A novel method named tensor global–local structure analysis (TGLSA) is proposed for batch process monitoring. Different from principal component analysis (PCA) and locality preserving projections (LPP), TGLSA aims at preserving both global and local structures of data. Consequently, TGLSA has the ability to extract more meaningful information from data than PCA and LPP. Moreover, the tensor-based projection strategy makes TGLSA more applicable for the three-dimensional data than multiway-based methods, such as MPCA and MLPP. A TGLSA-based online monitoring approach is developed by combining TGLSA with a moving window technique. Two new statistics, i.e., SPD and R2 statistics, are constructed for fault detection and diagnosis. In particular, the R2 statistic is a novel monitoring statistic, which is proposed based on a support tensor domain description method. The effectiveness and advantages of the TGLSA-based monitoring approach are illustrated by a benchmark fed-batch penicillin fermentation process.

The flow regime transitions in an airlift reactor were investigated based on pressure fluctuation signals. Two time–frequency analysis methods, i.e., Wigner–Ville distribution and wavelet transform, were used to extract flow regime characteristics from pressure signals. The main frequency derived from the smoothed pseudo-Wigner–Ville distribution of the pressure signal was used to quantify flow regime transitions in the reactor. Two flow regime transition points were successfully detected from the evolution of main frequencies of pressure signals. In addition, the local dynamic characteristics of the pressure signal at different frequency bands were analyzed by use of the wavelet transform. A new flow regime identification method based on the wavelet entropy of the pressure signal was proposed. This method was confirmed to be reliable and efficient to detect flow regime transitions in the reactor.