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.