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Reduction methods for feedback stabilization of fluid flows.

机译：减少流体流量反馈稳定性的方法。

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摘要

This dissertation deals with reduction of two classes of dynamical systems in order to make them suitable for control design using linear systems theory. The first class consists of the large-dimensional systems governing fluid flows, for which we employ dimension reduction techniques for stabilization of unstable steady states. The second class consists of systems with a continuous symmetry, for which we use a symmetry reduction method for stabilization of unstable relative equilibria.;Numerical discretization of fluid flows results in a large system of equations of O(105--8), while linear systems tools are limited to dimensions of O(102--4). Model reduction has played an important role in making these tools available for flow control. An attractive method is the approximate balanced truncation, in which the governing equations (linearized about a steady state) are projected onto a small number (≤100) of dynamically important modes, and the resulting models accurately capture the input-output (actuation to sensing) behavior. A limitation is that this method is restricted to linearizations about stable steady states. In this work, we extend its applicability to unstable steady states, assuming a small number of unstable modes. The unstable dynamics is treated exactly while reduced models are obtained of the (large dimensional) stable dynamics. We show a theoretical equivalence between approximate balanced truncation and a system identification technique called eigensystem realization algorithm (ERA). We extend ERA to simulations and obtain an order-of-magnitude cost reduction over balanced truncation.;The reduction techniques are applied to a model problem of the two-dimensional flow past a at plate at a low Reynolds number and a large angle of attack. The natural (uncontrolled) flow is periodic vortex shedding, although there also exists an unstable steady state that we seek to stabilize. The control actuation is modeled using a localized body-force actuator close to the leading or trailing edge and velocities are measured at two near-wake sensor locations. We obtain reduced models of the input-output dynamics linearized about the unstable steady state and show that 20-30 order models accurately capture the full system dynamics. We use the models to develop sensor-based feedback controllers and include them in the full nonlinear simulations. Even though the models are valid in a local neighborhood of the steady state, we show that they are capable of suppressing the periodic vortex shedding, which is a nonlinear phenomenon.;We also consider systems with a continuous symmetry and use a template-based approach to reduce the equations to a frame in which the symmetry is factored out. Relative equilibria are steady states in the symmetry-reduced frame; an example is traveling waves in systems with translational symmetry. The control goal is to stabilize unstable relative equilibria and the control design is based on linearization of the reduced equations about these steady states. A systematic reconstruction procedure is provided to obtain the form of the controller in the original coordinates. The control design relies on standard linear systems tools and the controlled system retains the symmetry of the original system. The control is demonstrated using various examples, including stabilization of unstable traveling waves in the 1D Kuramoto-Sivashinsky equation.
机译：本文针对两类动力学系统的简化，使其适用于线性系统理论的控制设计。第一类由控制流体流动的大尺寸系统组成，为此，我们采用降维技术来稳定不稳定的稳态。第二类由具有连续对称性的系统组成，为此我们使用对称性简化方法来稳定不稳定的相对平衡。流体流动的数值离散化导致了O（105--8）方程组的增大，而线性方程组系统工具限于O（102--4）的尺寸。模型简化在使这些工具可用于流量控制中发挥了重要作用。一种有吸引力的方法是近似平衡截断，其中将控制方程式（围绕稳态线性化）投影到少量（≤100）动态重要的模式上，然后得到的模型准确地捕获输入-输出（致动感应） ）的行为。局限性在于该方法仅限于关于稳定稳态的线性化。在这项工作中，假设少量不稳定模式，我们将其适用性扩展到不稳定稳态。精确地处理了不稳定的动力学，同时获得了（大尺寸）稳定动力学的简化模型。我们展示了近似平衡截断与称为特征系统实现算法（ERA）的系统识别技术之间的理论等效性。我们将ERA扩展到模拟中，并获得平衡截断的数量级成本降低。;该降低技术应用于二维流过低Reynolds数和大攻角的at平板的模型问题。自然（不受控制的）流动是周期性的涡旋脱落，尽管也存在我们寻求稳定的不稳定稳态。使用靠近前缘或后缘的局部身体力致动器对控制致动进行建模，并在两个接近尾迹的传感器位置测量速度。我们获得了关于不稳定稳态线性化的输入-输出动力学的简化模型，并显示了20-30阶模型可以准确地捕获整个系统动力学。我们使用这些模型来开发基于传感器的反馈控制器，并将其包括在完整的非线性仿真中。即使这些模型在稳态的局部邻域中都是有效的，我们也表明它们能够抑制周期性涡旋脱落，这是一种非线性现象。；我们还考虑了具有连续对称性的系统，并使用基于模板的方法将等式简化为一个考虑了对称性的框架。相对平衡是对称减少框架中的稳态；一个例子是具有平移对称性的系统中的行波。控制目标是稳定不稳定的相对平衡，控制设计基于关于这些稳态的简化方程的线性化。提供了系统的重建过程，以获取原始坐标中控制器的形式。控制设计依赖于标准线性系统工具，而受控系统则保留了原始系统的对称性。使用各种示例对控制进行了演示，包括稳定一维Kuramoto-Sivashinsky方程中的不稳定行波。

著录项

  • 作者

    Ahuja, Sunil.;

  • 作者单位

    Princeton University.;

  • 授予单位 Princeton University.;
  • 学科 Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 178 p.
  • 总页数 178
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 机械、仪表工业;
  • 关键词

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