A Unifying View of Multivariate State Space Models for Soft Sensors in Industrial Processes

State-space formulations offer a flexible approach for developing soft sensors in industrial processes, leveraging both data information and domain knowledge of process dynamics. On one hand, the state vector introduces varying perspectives in modeling process dynamics. However, choosing the definition of a state vector that is appropriate for the data and problem at hand is not a simple task. In this study, we examine and bridge three hybrid models using the framework of state space equations. We explore three key aspects within this framework: problem formulation, state prediction, and parameter estimation by the Expectation-Maximization (EM) algorithm. We compare the three hybrid models and two recurrent neural networks (RNN) approaches on three real-world datasets from desulfuring, polymerization, and sulfur recovery processes. Results are analyzed from both the data perspective and the process perspective, aiming to enhance the understanding and implementation of soft sensors in dynamic settings, with potential implications for various industries relying on accurate and adaptable soft sensor technologies.