用降阶估计器实现平板绕流不稳定稳态的反馈控制
Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators
摘要：
我们提出了一个基于估计的控制设计流程流量控制，使用降阶模型的控制方程，线性化的一个可能不稳定的稳态。降阶模型采用近似平衡截断方法得到，该方法保留了系统的最可控和最可观的模式。原来的方法只适用于稳定的线性系，我们提出了一个扩展到不稳定的线性系。不稳定子空间上的动力学表示投影到全球不稳定的本征模，假设是小的数量的原始方程。基于快照的算法开发，使用近似平衡截断，获得降阶模型的动力学稳定子空间。应用该算法研究了低雷诺数、大攻角二维平板绕流的反馈控制，其中自然流动是旋涡脱落，但也存在不稳定的定常状态。对于控制设计，我们推导出有效的降阶模型在这个不稳定的稳态附近。激励被建模为平板前缘附近的局部体积力，传感器是平板近尾流中的两个速度测量值。一个降阶卡尔曼滤波器的基础上开发这些模型，并示出准确地重建从传感器测量的流场，并由此产生的估计为基础的控制稳定不稳定的稳态。对于稳态的小扰动，该模型准确地预测了完整模拟的响应。此外，所得到的控制器甚至能够抑制稳定的周期性旋涡脱落，其中的非线性效应是强的，从而意味着一个大的吸引区域的稳定的稳态。
英文摘要：
We present an estimator-based control design procedure for flow control, using reduced-order models of the governing equations, linearized about a possibly unstable steady state. The reduced models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system. The original method is valid only for stable linear systems, and we present an extension to unstable linear systems. The dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes, assumed to be small in number. A snapshot-based algorithm is developed, using approximate balanced truncation, for obtaining a reduced-order model of the dynamics on the stable subspace. The proposed algorithm is used to study feedback control of 2-D flow over a flat plate at a low Reynolds number and at large angles of attack, where the natural flow is vortex shedding, though there also exists an unstable steady state. For control design, we derive reduced-order models valid in the neighborhood of this unstable steady state. The actuation is modeled as a localized body force near the leading edge of the flat plate, and the sensors are two velocity measurements in the near-wake of the plate. A reduced-order Kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements, and the resulting estimator-based control is shown to stabilize the unstable steady state. For small perturbations of the steady state, the model accurately predicts the response of the full simulation. Furthermore, the resulting controller is even able to suppress the stable periodic vortex shedding, where the nonlinear effects are strong, thus implying a large domain of attraction of the stabilized steady state.
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