Adaptive Tube MPC: Beyond a Common Quadratically Stabilizing Feedback Gain

Abstract: This paper proposes an adaptive tube framework for model predictive control (MPC) of discrete-time linear time-invariant systems subject to parametric uncertainty and additive disturbances. In contrast to conventional tube-based MPC schemes that employ fixed tube geometry and constraint tightening designed for worst-case uncertainty, the proposed approach incorporates online parameter learning to progressively refine the parametric uncertainty set and update the parameter estimates. These updates are used to adapt the components of the MPC optimization problem, including the prediction model, feedback gain, terminal set, and tube cross-sections. As the uncertainty set contracts, the required amount of constraint tightening reduces and the tube shrinks accordingly, yielding less conservative control actions. Recursive feasibility, robust constraint satisfaction, and closed-loop stability are formally established. Furthermore, the framework does not require the existence of a common quadratically stabilizing linear feedback gain for the entire parametric uncertainty set, thereby relaxing a standard assumption in existing tube-based MPC formulations. Numerical examples illustrate the effectiveness of the proposed approach.