A Matrix Block-Based Physics-Informed Probabilistic Quality-Relevant Monitoring Model

Physical laws widely exist in many industrial processes, which can be used to enhance model reliability and avoid physically inconsistent results. In this study, a probabilistic linear latent variable model is developed with readily accessible physical knowledge (PI-PQM) for quality-relevant monitoring applications. A block-matrix-based structure is adopted in the proposed method to satisfy the properties of different variables. Within this framework, physical principles (i.e., boundedness and monotonicity) are imposed on the selected variables as inequation constraints. In particular, both constraints are formulated with a skew normal distribution, which can be recast as a combination of normal and truncated normal distributions. After that, the variational inference (VI) algorithm is utilized for estimating the latent variables and model parameters of the proposed method. It is noted that the effect of the physical constraints is analyzed by comparing the latent variables and coefficient matrices of constrained and unconstrained variables. The performance of the proposed method is illustrated using two numerical simulations and a real industrial application. Experimental results verify that physical information helps the proposed method achieve accurate convergence to the real structure of the collected data. Besides, the monitoring sensitivity for both process- and quality-related anomalies is also effectively improved after adding physical constraints.