An investigation of optimal feedback controllers' performance and robustness is carried out for vortex shedding behind a 2D cylinder at low Reynolds numbers. To facilitate controller design, we present an efficient modelling approach in which we utilise the resolvent operator to recast the linearised Navier-Stokes equations into an input-output form from which frequency responses can be computed. The difficulty of applying modern control design techniques to complex, high-dimensional flow systems is thus overcome by using low-order models identified from these frequency responses. The low-order models are used to design optimal control laws using $\mathcal{H}_{\infty}$ loop shaping. Two distinct control arrangements are considered, both of which employ a single-input and a single-output. In the first control arrangement, a velocity sensor located in the wake drives a pair of body forces near the cylinder. Complete suppression of shedding is observed up to a Reynolds number of $Re=110$. Due to the convective nature of vortex shedding and the corresponding time delays, we observe a fundamental trade-off: the sensor should be close enough to the cylinder to avoid any excessive time lag, but it should be kept sufficiently far from the cylinder to measure any unstable modes developing downstream. It is found that these two conflicting requirements become more difficult to satisfy for larger Reynolds numbers. In the second control arrangement, we consider a practical setup with a body-mounted force sensor and an actuator that oscillates the cylinder according to the lift measurement. It is shown that the system is stabilised only up to $Re=100$, and we demonstrate why the performance of the resulting feedback controllers deteriorates much more rapidly with increasing Reynolds number. The challenges of designing robust controllers for each control setup are also analysed and discussed.

中文翻译：

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使用基于溶剂的建模方法的涡旋脱落反馈控制

对低雷诺数下 2D 圆柱体后面的涡旋脱落进行了最佳反馈控制器的性能和鲁棒性研究。为了便于控制器设计，我们提出了一种有效的建模方法，在该方法中，我们利用求解算子将线性化的 Navier-Stokes 方程重铸为可以计算频率响应的输入-输出形式。因此，通过使用从这些频率响应中识别的低阶模型，克服了将现代控制设计技术应用于复杂的高维流动系统的困难。低阶模型用于使用 $\mathcal{H}_{\infty}$ 循环整形设计最优控制律。考虑了两种不同的控制安排，它们都采用单输入和单输出。在第一个控制安排中，位于尾流中的速度传感器在圆柱体附近驱动一对体力。观察到完全抑制脱落直至雷诺数 $Re=110$。由于涡旋脱落的对流性质和相应的时间延迟，我们观察到一个基本的权衡：传感器应该足够靠近圆柱体以避免任何过度的时间滞后，但它应该离圆柱体足够远以进行测量任何在下游发展的不稳定模式。发现对于较大的雷诺数，这两个相互矛盾的要求变得更加难以满足。在第二种控制方案中，我们考虑了一个实际设置，它带有一个安装在车身上的力传感器和一个根据升力测量值来摆动气缸的执行器。结果表明，系统仅在 $Re=100$ 时才稳定，我们展示了为什么随着雷诺数的增加，所产生的反馈控制器的性能恶化得更快。还分析和讨论了为每个控制设置设计鲁棒控制器的挑战。