Safe Navigation with Zonotopic Tubes: An Elastic Tube-based MPC Framework

Abstract

This paper presents an elastic tube-based model predictive control (MPC) framework for unknown discrete-time linear systems subject to disturbances. Unlike most existing elastic tube-based MPC methods, we do not assume perfect knowledge of the system model or disturbance realizations bounds. Instead, a conservative zonotopic disturbance set is initialized and iteratively refined using data and prior knowledge: data are used to identify matrix zonotope model sets for the system dynamics, while prior physical knowledge is employed to discard models and disturbances inconsistent with known constraints. This process yields constrained matrix zonotopes representing disturbance realizations and dynamics that enable a principled fusion of offline information with limited online data, improving MPC feasibility and performance. The proposed design leverages closed-loop system characterization to learn and refine control gains that maintain a small tube size. By separating open-loop model mismatch from closed-loop effects in the error dynamics, the method avoids dependence on the size of the state and input operating regions, thereby reducing conservatism. An adaptive co-design of the tube and ancillary feedback ensures $λ$-contractive zonotopic tubes, guaranteeing robust positive invariance, improved feasibility margins, and enhanced disturbance tolerance. We establish recursive feasibility conditions and introduce a polyhedral Lyapunov candidate for the error tube, proving exponential stability of the closed-loop error dynamics under the adaptive tube-gain updates. Simulations demonstrate improved robustness, enlarged feasibility regions, and safe closed-loop performance using only a small amount of online data.