Shapley-Based Multi-Objective Distributional Control of Cascaded Refining Systems With Deep Temporal Modeling

Accurately controlling cascaded refining processes is usually difficult in pulp and papermaking systems when using model predictive control (MPC), as the MPC framework requires the simultaneous regulation of deterministic targets and the reshaping output probability density functions under nonlinear and large time-delay conditions. Deep learning can facilitate advanced data-driven modeling and can be assimilated into MPC, but is usually limited by the model interpretability and the complex requirements of multi-objective distributional control. This study proposes a deep neural network-based MPC framework that integrates a predictive model with interpretable optimization strategies to overcome the aforementioned limitations. To enable precise prediction, we first develop a temporal autoencoder network that simultaneously forecasts the Schopper-Riegler beating degree and parameterizes the fiber length distribution, as well as to effectively capture the system’s nonlinear characteristics. Furthermore, to tackle multi-objective challenges, we propose an optimization strategy that integrates a predictive cost-driven adaptive weighting mechanism and an adaptive gradient descent algorithm to dynamically balance objectives and accelerate convergence to optimal control. Also, to address interpretability and optimal control issues, we formulate a dynamic Shapley value analysis to adaptively refine the control optimization feasible region by capturing time-varying feature interactions. Finally, the efficiency of the proposed model is validated through a case study on this system, demonstrating that significant improvements can be achieved in terms of control performance, response accuracy, and interpretability.