Homomorphically encrypted neural network modeling and predictive control for data-privacy-preserving control systems

Traditional model predictive control (MPC) relies on accurate mechanistic models, which are often costly or infeasible to obtain for complex, nonlinear systems. Machine learning-based MPC (ML-MPC) addresses this challenge by learning predictive models from data. However, its deployment in real-world systems is often limited by data privacy concerns. To address this challenge, we propose a homomorphic encryption (HE)-based framework that enables fully privacy-preserving ML-MPC. We develop a homomorphic encrypted recurrent neural network (HE-RNN) that supports training and inference on ciphertext inputs, with nonlinear activations approximated by Chebyshev polynomials to ensure HE compatibility. We further analyze ciphertext-induced errors and establish a generalization error bound for HE-RNN, providing theoretical guarantees. Integrating HE-RNN into MPC, we design a closed-loop control system that exchanges only encrypted signals between sensors, controllers, and actuators, with stability guarantees. Validation on a real-world photochemical reactor and a chemical process network in Aspen Plus Dynamics demonstrates the effectiveness of the proposed approach for encrypted modeling and control of nonlinear systems.