  The Koopman operator theory is an increasingly popular formalism of dynamical
systems theory which enables analysis and prediction of the nonlinear dynamics
from measurement data. Building on the recent development of the Koopman model
predictive control framework (Korda and Mezic 2016), we propose a methodology
for closed-loop feedback control of nonlinear flows in a fully data-driven and
model-free manner. In the first step, we compute a Koopman-linear
representation of the control system using a variation of the extended dynamic
mode decomposition algorithm and then we apply model predictive control to the
constructed linear model. Our methodology handles both full-state and sparse
measurement; in the latter case, it incorporates the delay-embedding of the
available data into the identification and control processes. We illustrate the
application of this methodology on the periodic Burgers' equation and the
boundary control of a cavity flow governed by the two-dimensional
incompressible Navier-Stokes equations. In both examples the proposed
methodology is successful in accomplishing the control tasks with
sub-millisecond computation time required for evaluation of the control input
in closed-loop, thereby allowing for a real-time deployment.
