通过Wiener-Hopf形式主义进行最优估计和控制的基于分解的工具
Resolvent-based tools for optimal estimation and control via the Wiener-Hopf formalism
摘要：
控制工具在复杂流中的应用通常需要近似，例如降阶模型和/或简化的强迫假设，这些假设可能被认为是低秩的或根据简化的统计量（例如白噪声）定义的。在这项工作中，我们提出了一种基于解析的控制方法，通过维纳-霍普夫形式主义施加因果关系。线性最优因果估计和控制律直接从满秩、全局稳定的系统中得到，避免了其他方法的许多缺点。我们使用有效的，矩阵的方法来构建矩阵Wiener-Hopf问题，我们实现了一个定制的方法来解决这个问题的数值。该方法自然地处理强迫项与时空颜色;它允许廉价的参数调查的传感器/执行器的位置在干扰/目标的情况下是低秩的;它直接适用于复杂的流扰动高阶强迫;它具有较低的成本相比，标准方法;它可以用于伴随求解器不可用的情况下;或者它可以完全基于实验数据。该方法特别适用于放大器流量的控制，对于放大器流量，最优控制方法通常是鲁棒的。验证的方法是使用线性化的Ginzburg-Landau方程。然后考虑高阶强迫扰动下后向台阶上的流动。传感器和致动器的位置进行了研究，这种情况下，我们表明，虽然下游的步骤的流量响应主要是由开尔文-亥姆霍兹机制，它有一个复杂的，高秩的接收传入上游扰动，需要多个传感器进行控制。
英文摘要：
The application of control tools to complex flows frequently requires approximations, such as reduced-order models and/or simplified forcing assumptions, where these may be considered low-rank or defined in terms of simplified statistics (e.g. white noise). In this work, we propose a resolvent-based control methodology with causality imposed via a Wiener-Hopf formalism. Linear optimal causal estimation and control laws are obtained directly from full-rank, globally stable systems with arbitrary disturbance statistics, circumventing many drawbacks of alternative methods. We use efficient, matrix-free methods to construct the matrix Wiener-Hopf problem, and we implement a tailored method to solve the problem numerically. The approach naturally handles forcing terms with space-time colour; it allows inexpensive parametric investigation of sensor/actuator placement in scenarios where disturbances/targets are low rank; it is directly applicable to complex flows disturbed by high-rank forcing; it has lower cost in comparison to standard methods; it can be used in scenarios where an adjoint solver is not available; or it can be based exclusively on experimental data. The method is particularly well-suited for the control of amplifier flows, for which optimal control approaches are typically robust. Validation of the approach is performed using the linearized Ginzburg-Landau equation. Flow over a backward-facing step perturbed by high-rank forcing is then considered. Sensor and actuator placement are investigated for this case, and we show that while the flow response downstream of the step is dominated by the Kelvin-Helmholtz mechanism, it has a complex, high-rank receptivity to incoming upstream perturbations, requiring multiple sensors for control.
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