Reviews: Deep Dynamical Modeling and Control of Unsteady Fluid Flows

The dynamics describing such flows are nonlinear, making modeling and control challenging. The approach adopted in this paper is based on spectral analysis of the Koopman operator (a popular approach that has received considerable attention in the literature), which (roughly speaking) models the finite-dimensional nonlinear dynamics with infinite-dimensional linear dynamics. For tractable analysis it is desirable to find a finite-dimensional approximation of these infinite-dimensional dynamics (i.e., a Koopman invariant subspace). The strategy presented in this paper is dynamic mode decomposition (DMD), another popular approach that has received much attention in the literature. The key challenge in DMD is coming up with a basis for the Koopman invariant subspace (i.e., a collection of nonlinear functions referred to as'observables'). Here lies the main contribution of the paper: a data-driven way to learn appropriate observables, where each observable function is modeled by a deep neural network (encoder). The resulting finite-dimensional linear model is then used for control design (standard MPC for linear systems). It is observed that the MPC strategy coincides with proportional feedback (i.e.