Fault detection of batch processes based on multivariate functional kernel principal component analysis

• Traditional KPCA method is combined with the approach of functional data analysis.
• The proposed MFKPCA method is used for batch process monitoring.
• Three complementary statistics (T2, SPE and MSE) are used for fault detection.

In some batch processes, the variables' trajectories often show obviously functional nature and can be considered as smooth functions rather than just vectors, then the collected three-way data array essentially can be transformed into a two-way function matrix. For this purpose, the approach of functional data analysis (FDA) is introduced to express the variables' functions in this article, which can highlight the subtle differences in the shape of variables' trajectories between normal batches and faulty ones, and can also easily handle the problems of irregular three-way data array, such as unequal batch length, different sampling rates, missing data, etc. Based on the function matrix, a novel monitoring method called multivariate functional kernel principal component analysis (MFKPCA) is proposed for fault detection of batch processes, which directly eigen-decomposes the two-way function matrix in the function space and uses the kernel trick to handle the nonlinear correlations. Different from the traditional PCA method, MFKPCA designs three complementary control charts for batch process monitoring based on three statistics including Hotelling's T2, squared prediction error (SPE) and mean squared error (MSE). Finally, some simulations and an industrial semiconductor etch process are used to illustrate the validity of the proposed monitoring method.