Fault-Tolerant Soft Sensor Modeling Based on a Two-Dimensional Group Distributionally Robust Optimization Framework

In industrial automation and intelligence, fault tolerance mechanisms have always been an attractive topic. To develop soft sensors with fault tolerance for different types of faults and unforeseen new faults, this article proposes a two-dimensional group distributionally robust optimization (2D-GDRO) framework for fault-tolerant soft sensor modeling. We propose to describe the potential distributions of new fault conditions with an uncertainty set and optimize the soft sensor model by minimizing the worst-case risk over the uncertainty set. Considering the restricted representation range of the uncertainty set constructed directly from a mixture distribution of a limited number of existing fault conditions in the training set, a two-dimensional uncertainty set is designed at the group dimension and the sample dimension. To efficiently train a fault-tolerant soft sensor within the 2D-GDRO framework, we introduce a triple-interleaved optimization algorithm. This algorithm integrates mini-batch stochastic gradient descent, exponentiated gradient ascent, and group-wise SoftMax techniques. Finally, the fault tolerance of the 2D-GDRO framework based soft sensor is verified using the Tennessee-Eastman process and the real three-phase flow facility. The experimental results show that 2D-GDRO outperforms other training frameworks in average soft sensing accuracy under new fault conditions. Note to Practitioners—This article addresses the need for fault-tolerant soft sensors in industrial automation, where systems must continue to function reliably despite various types of faults and unforeseen faults. The solution proposed is a 2D-GDRO framework, which aims to make soft sensors more resilient to such faults by preparing them for worst-case scenarios during the training phase. This framework enhances the fault tolerance of soft sensors by designing an uncertainty set at two dimensions: the group dimension and the sample dimension, which allows soft sensors to better generalize beyond known fault conditions. The 2D-GDRO framework uses an efficient triple-interleaved optimization algorithm. While the 2D-GDRO framework significantly improves fault tolerance, its performance depends on the diversity of fault conditions represented in the training data. In cases where fault scenarios are highly unpredictable, additional refinement may be required. The fault-tolerant soft sensor based on 2D-GDRO can be applied across various industries, such as chemical manufacturing, oil and gas, and energy production, ensuring that the predictive monitoring remain reliable even in the presence of unknown new faults.