Ronghuan Li

The state‐space models are often used to describe the characteristics of many physical or chemical processes, but unknown system parameters bring difficulties to the solution of the state equation. This paper addresses the challenge of accurate parameter and state estimation for input‐nonlinear state‐space models with random missing data. To tackle this issue, an expectation‐maximization‐based Kalman smoothing (EM‐KS) algorithm is derived by transforming the state‐space models into the form of a pseudo‐linear equation. The unknown intermediate states are obtained through KS. In the pseudo‐linear equation, all unknown parameters are treated as a vector and estimated under the framework of the EM algorithm. By treating the missing data as latent variables, the proposed algorithm ingeniously avoids the missing data points and can obtain effective parameter estimation even in the case of a high data missing ratio. Simulation results demonstrate the effectiveness of the proposed method, while comparison with the existing approach further demonstrates its reliability.

Deep probabilistic learning networks have been applied in industrial soft sensors. However, they face significant challenges in latent variable inference, deep learning backend implementation, and labeled data scarcity. The first challenge arises when covariates directly infer the latent variable, potentially leading to inaccuracies. The second stems from discrepancies between the theoretical probabilistic distribution and practical instance-based deep learning backends. The third is commonly encountered in soft sensor applications, where unlabeled data often go unused, reducing accuracy. To address these challenges, this work proposes a novel semi-supervised probabilistic learning network (SS-PLN) for soft sensor modeling with partially labeled data. The first issue is addressed by formulating an optimization problem as the model’s learning objective. This optimization is efficiently solved by investigating the input of inference network through analyzing the optimal solution’s structure. For the second issue, mean and covariance equations are used to represent probabilistic distributions, ensuring effective deep learning backends. The third issue is addressed by integrating supervised and unsupervised probabilistic learning networks to form the SS-PLN model, thereby maximizing the use of unlabeled data and enhancing the soft sensing performance. The feasibility and effectiveness of the proposed SS-PLN are validated through comparisons with recent semi-supervised learning methods, using data from two industrial processes. Note to Practitioners—Soft sensors are extensively used to predict key variables in industrial processes, especially when direct measurement is challenging or infeasible. While deep learning models have shown promise for this purpose, several challenges remain. Directly inferring latent variables from covariates often results in inaccuracies due to the complexity of their relationships. Additionally, discrepancies between theoretical probabilistic distributions and practical instance-based implementations complicate soft sensor design, particularly under partially labeled data conditions. This paper proposes a novel semi-supervised probabilistic learning network (SS-PLN) to address these challenges. The SS-PLN leverages both labeled and unlabeled data to enhance prediction accuracy. By formulating a learnable optimization problem, the accuracy of inferring latent variable is improved. The mean and covariance expressions are utilized to represent the probabilistic distributions, facilitating numerical implementation. A supervised and an unsupervised probabilistic networks are integrated to construct SS-PLN, enabling effective modeling of partially labeled data. Experimental results from a sugar crystallization process and a steel-making process demonstrate the effective and suitable soft sensing performance of SS-PLN.