A data-driven Koopman model predictive control framework for nonlinear flows

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.