用于降低结构化输入-输出稳定性分析中人工能量产生的改进不确定性表示
Improved uncertainty representation for reducing artificial energy production in structured input-output stability analysis
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- The Stephen B. Klein Faculty of Aerospace Engineering, Technion – Israel Institute of Technology（以色列理工学院航空航天工程学院斯蒂芬·B·克莱因学院）
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中文总结 AI 辅助
本研究将新型结构化不确定性应用于结构化小增益定理，推导得重复对角结构不确定性表示，用于Couette流和平面泊肃叶流的稳定性分析，所得阈值更准确且人工能量产生项最小。
中文摘要 AI 辅助
本研究采用一种新型的固定结构化不确定性，应用于Frank-Shapir与Gluzman（《流体力学杂志》，第1030卷，2026年，第A8页）提出的结构化小增益定理方法，对受有限幅值扰动的不可压缩剪切流进行稳定性分析。在该框架内，纳维-斯托克斯方程中的非线性平流项被替换为与线性化动力学相连的结构化反馈不确定性，以体现非线性反馈的影响。本研究通过对输入和输出通道进行线性变换推导得到一种新的不确定性表示，该变换将反馈回路转化为具有重复对角结构的结构化不确定性。此结构旨在保留非线性平流项的分量路径，同时保持结构化奇异值计算的可处理性。我们将该方法应用于Couette流和平面泊肃叶流这两种典型基流，得到的保持稳定性的扰动幅值阈值更保守程度更低、精度更高。我们将本研究提出的新方法与先前提出的不确定性结构的重复块和非重复块近似方法进行比较，结果显示本研究的稳定性阈值与先前的数值和实验研究结果最为吻合。研究表明，过去研究中提出的重复块和非重复块结构会因在结构化输入-输出公式中使用恒定结构化不确定性而产生人工能量产生项，违反无散度假设；而采用本研究提出的方法时，该能量产生项最小，能更忠实地体现非线性反馈与纳维-斯托克斯系统线性化动力学互联的影响。
英文摘要
This work employs a new form of fixed structured uncertainty within the structured small-gain theorem approach proposed by Frank-Shapir & Gluzman (J. Fluid Mech., vol. 1030, 2026, pp A8) for the stability analysis of incompressible shear flows subject to finite-magnitude disturbances. Within this framework, the nonlinear advection term in the Navier-Stokes equations is replaced by a structured feedback uncertainty interconnection with the linearized dynamics to account for the impact of nonlinear feedback. Herein, a new uncertainty representation is derived via linear transformations of the input and output channels, transforming the feedback loop such that the resulting structured uncertainty has a repeated-diagonal structure. This structure aims to preserve the component-wise pathways of the nonlinear advection term while keeping the structured singular value computation tractable. We apply the method to two canonical base flows: Couette and plane Poiseuille flows. The resulting thresholds on disturbance magnitude to preserve stability are less conservative and more accurate. We compare the novel methodology presented here with previously proposed repeated and non-repeated block approximations of the uncertainty structure, where our stability threshold provided the closest agreement with previous numerical and experimental studies. We show that repeated and non-repeated block structures that were proposed in past studies result in an artificial energy-production term arising from using constant structured uncertainty in the structured input-output formulation, violating the divergencefree assumption. This energy-production term is smallest when using the methodology presented in this work, providing a more faithful representation of the impact of nonlinear feedback interconnection with the linearized dynamics of the Navier-Stokes system.
