Synchronized Bayesian state estimation in batch processes using a two-dimensional particle filter

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In chemical process control, estimation of the process states, e.g. concentration or properties of the reactant or resultant, in real time is a key issue. Batch processes are typically characterized by unequal batch lengths and unsynchronized batch trajectories, posing challenges for the state estimation. Due to ignorance of such challenges, many existing methods would produce poor estimates in real applications. We design a new state estimation approach employing Bayesian filtering with consideration of batch-to-batch dynamics. To characterize the dynamics across batches with different time profiles, a synchronized two-dimensional (2-D) state-space model is constructed that contains synchronously equal- and unequal-length situations. Based on this model, a novel formulation of the particle filter is derived where the particles evolve along both the time and the batch dimensions so as to approximate the synchronized 2-D optimal estimates. Also, the convergence concern is addressed from a practitioner’s viewpoint. To incorporate appropriate data from previous batches into the current estimation, an on-line synchronization method based on the dynamic time warping technique is developed using a new alignment performance measure together with a transfer alignment strategy. The performance of the proposal is evaluated by case study on a numerical example and a three-state batch reaction process.