A probabilistic approach to modeling and controlling fluid flows

We extend cluster-based reduced-order modeling (CROM) (Kaiser et al, 2014) to include control inputs in order to determine optimal control laws with respect to a cost function for unsteady flows. The proposed methodology frames high-dimensional, nonlinear dynamics into low- dimensional, probabilistic, linear dynamics which considerably simplifies the optimal control problem while preserving nonlinear actuation mechanisms. The data-driven approach builds upon the unsupervised partitioning of the data into few kinematically similar flow states using a clustering algorithm. The coarse-grained dynamics are then described by a Markov model which is closely related to the approximation of Perron-Frobenius operators. The Markov model can be used as predictor for the ergodic probability distribution for a particular control law approximating the long-term behavior of the system on which basis the optimal control law is determined. Moreover, we combine CROM with a recently developed approach for optimal sparse sensor placement for classification (Brunton et al., 2013) as a critical enabler for in-time control and for the systematic identification of dynamical regimes from few measurements. The approach is applied to a separating flow and a mixing layer exhibiting vortex pairing.