This work deals with the characterization of the closed-loop control performance aiming at the delay of transition. We focus on convective wavepackets, typical of the initial stages of transition to turbulence, starting with the linearized Kuramoto–Sivashinsky equation as a model problem representative of the transitional 2D boundary layer; its simplified structure and reduced order provide a manageable framework for the study of fundamental concepts involving the control of linear wavepackets. The characterization is then extended to the 2D Blasius boundary layer. The objective of this study is to explore how the sensor–actuator placement affects the optimal control problem, formulated using linear quadratic Gaussian (LQG) regulators. This is carried out by evaluating errors of the optimal estimator at positions where control gains are significant, through a proposed metric, labelled as $$\gamma $$ γ . Results show, in quantitative manner, why some choices of sensor–actuator placement are more effective than others for flow control: good (respectively, bad) closed-loop performance is obtained when estimation errors are low (respectively, high) in the regions with significant gains in the full-state-feedback problem. Unsatisfactory performance is further understood as dominant estimation error modes that overlap spatially with control gains, which shows directions for improvement of a given set-up by moving sensors or actuators. The proposed metric and analysis explain most trends in closed-loop performance as a function of sensor and actuator position, obtained for the model problem and for the 2D Blasius boundary layer. The spatial characterization of the $$\gamma $$ γ -metric provides thus a valuable and intuitive tool for the problem of sensor–actuator placement, targeting here transition delay but possibly extending to other amplifier-type flows.

中文翻译：

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用于对流不稳定性闭环控制的执行器和传感器放置

这项工作涉及针对转换延迟的闭环控制性能的表征。我们专注于对流波包，典型的过渡到湍流的初始阶段，从线性化的 Kuramoto-Sivashinsky 方程开始，作为代表过渡 2D 边界层的模型问题；它的简化结构和降阶为研究涉及线性波包控制的基本概念提供了一个可管理的框架。然后将表征扩展到 2D Blasius 边界层。本研究的目的是探索传感器 - 执行器放置如何影响使用线性二次高斯 (LQG) 调节器制定的最优控制问题。这是通过在控制增益显着的位置评估最优估计器的误差来实现的，通过建议的度量，标记为 $$\gamma $$ γ 。结果以定量的方式显示了为什么传感器 - 执行器放置的某些选择在流量控制方面比其他选择更有效：当估计误差低（分别高）时，在具有以下特征的区域中获得良好（分别，差）的闭环性能在全状态反馈问题上取得了重大进展。不满意的性能进一步被理解为与控制增益在空间上重叠的主要估计误差模式，这显示了通过移动传感器或执行器改进给定设置的方向。建议的度量和分析解释了作为传感器和执行器位置函数的闭环性能的大多数趋势，为模型问题和 2D Blasius 边界层获得。因此，$$\gamma $$ γ 度量的空间特征为传感器-执行器放置问题提供了一种有价值且直观的工具，针对此处的转换延迟，但可能扩展到其他放大器类型的流程。