  Koopman operators are of infinite dimension and capture the characteristics
of nonlinear dynamics in a lifted global linear manner. The finite data-driven
approximation of Koopman operators results in a class of linear predictors,
useful for formulating linear model predictive control (MPC) of nonlinear
dynamical systems with reduced computational complexity. However, the
robustness of the closed-loop Koopman MPC under modeling approximation errors
and possible exogenous disturbances is still a crucial issue to be resolved.
Aiming at the above problem, this paper presents a robust tube-based MPC
solution with Koopman operators, i.e., r-KMPC, for nonlinear discrete-time
dynamical systems with additive disturbances. The proposed controller is
composed of a nominal MPC using a lifted Koopman model and an off-line
nonlinear feedback policy. The proposed approach does not assume the
convergence of the approximated Koopman operator, which allows using a Koopman
model with a limited order for controller design. Fundamental properties, e.g.,
stabilizability, observability, of the Koopman model are derived under standard
assumptions with which, the closed-loop robustness and nominal point-wise
convergence are proven. Simulated examples are illustrated to verify the
effectiveness of the proposed approach.
