# Reduced-order control using low-rank Dynamic Mode Decomposition

## TL;DR

This paper introduces low-rank Dynamic Mode Decomposition (lrDMD), a data-driven model reduction technique that improves control of fluid flows by providing lower reconstruction error and better stabilization with reduced model complexity.

## Contribution

The paper proposes lrDMD, a novel low-rank constrained DMD method that enhances model accuracy and control performance over standard DMD and OMD methods.

## Key findings

- lrDMD achieves lower data reconstruction error than DMD and OMD.
- lrDMD stabilizes complex fluid systems with lower-order controllers.
- lrDMD provides better estimates of the adjoint system at a given rank.

## Abstract

In this work we perform full-state LQR feedback control of fluid flows using non-intrusive data-driven reduced-order models. We propose a model reduction method called low-rank Dynamic Mode Decomposition (lrDMD) that solves for a rank-constrained linear representation of the dynamical system. lrDMD is shown to have lower data reconstruction error compared to standard Optimal Mode Decomposition (OMD) and Dynamic Mode Decomposition (DMD), but with an increased computational cost arising from solving a non-convex matrix optimization problem. We demonstrate model order reduction on the complex linearized Ginzburg-Landau equation in the globally unstable regime and on the unsteady flow over a flat plate at a high angle of attack. In both cases, low-dimensional full-state feedback controller is constructed using reduced-order models constructed using DMD, OMD and lrDMD. It is shown that lrDMD stabilizes the Ginzburg-Landau system with a lower order controller and is able to suppress vortex shedding from an inclined flat plate at a cost lower than either DMD or OMD. It is further shown that lrDMD yields an improved estimate of the adjoint system, for a given rank, relative to DMD and OMD.