A Data-Driven Tube Model Predictive Control Framework with Recursive Feasibility and Stability Guarantees

The use of deep neural networks as fast approximators of optimal control laws is explored in nonlinear Model Predictive Control, primarily for their ability to reduce online computational complexity. Despite their practical advantages, integrating such learned components into robust Model Predictive Control frameworks raises several questions regarding stability, constraint satisfaction, and robustness under uncertainty. In this work, we develop a learning-based tube-based MPC scheme for discrete-time nonlinear systems subject to additive disturbances. The proposed approach uses imitation learning to construct a neural approximation of the nominal control policy, which is combined with a linear feedback mechanism to ensure closed-loop stability. To account for approximation errors and external disturbances, we reformulate the tube dynamics by explicitly incorporating neural network errors. We derive probabilistic bounds on neural network error using statistical learning theory and systematically propagate them through the closed-loop error dynamics. This enables the construction of robust positively invariant sets that ensure recursive feasibility and constraint satisfaction. The framework is developed in the context of data-driven control architectures and is motivated by applications in structural health monitoring, where reliable interaction between physical systems and their digital counterparts is important. Numerical experiments on representative test cases are performed to verify the theoretical guarantees.