# Reduced-order control using low-rank dynamic mode decomposition

## 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 in 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.

## Keywords

Flow control Dynamic mode decomposition Model reduction## Notes

### Acknowledgements

The authors would like to thank Mr. Daniel Floryan and Dr. Clarence Rowley for their help with the code used to simulate flow past an inclined flat plate. This work was sponsored, in part, by the Office of Naval Research (ONR) as part of the Multidisciplinary University Research Initiatives (MURI) Program, under Grant Number N00014-16-1-2617.

## Supplementary material

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