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.

中文翻译：

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使用低秩动态模式分解的降阶控制

在这项工作中，我们使用非侵入式数据驱动的降阶模型对流体流动进行全状态 LQR 反馈控制。我们提出了一种称为低秩动态模式分解 (lrDMD) 的模型缩减方法，该方法求解动态系统的秩约束线性表示。与标准最优模式分解 (OMD) 和动态模式分解 (DMD) 相比，lrDMD 被证明具有更低的数据重构误差，但由于解决非凸矩阵优化问题而增加了计算成本。我们证明了在全局不稳定状态下复杂线性化的 Ginzburg-Landau 方程的模型阶数减少，以及在大迎角下平板上的不稳定流动。在这两种情况下，使用使用 DMD、OMD 和 lrDMD 构建的降阶模型构建低维全状态反馈控制器。结果表明，lrDMD 使用低阶控制器稳定了 Ginzburg-Landau 系统，并且能够以低于 DMD 或 OMD 的成本抑制来自倾斜平板的涡旋脱落。进一步表明，对于给定的秩，相对于 DMD 和 OMD，lrDMD 产生了对伴随系统的改进估计。