Estimation and Control of Fluid Flows Using Sparsity-Promoting Dynamic Mode Decomposition

Abstract

Control and estimation of fluid systems is a challenging problem that requires approximating high-dimensional, nonlinear dynamics with computationally tractable models. A number of techniques, such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have been developed to derive such reduced-order models. In this letter, the problem of selecting the dynamically important modes of dynamic mode decomposition with control (DMDc) is addressed. Similar to sparsity-promoting DMD, the method described in this letter solves a convex optimization problem in order to determine the most important modes. The proposed algorithm produces sparse dynamical models for systems with inputs by solving a regularized least-squares problem that minimizes the reweighted $\\mathcal {L}_{1}$ norm of the relative mode weights and can work even with snapshot data that are not sequential. In addition, the process of estimating the modeling errors and designing a Kalman filter for flow estimation from limited measurements is presented. The method is demonstrated in the feedback control and estimation of the unsteady wake past an inclined flat plate in a high-fidelity direct numerical simulation.