Data-Driven Feedback Control of an Oscillating Flow Using Low-Order Linear Time-Invariant Models

Core Concepts

This paper presents a data-driven approach to stabilize an oscillating flow, such as the flow past a cylinder, from its natural limit cycle to its equilibrium state using linear time-invariant (LTI) controllers. The method iteratively identifies an LTI model of the oscillating flow from input-output data, designs an LQG controller, and stacks the controllers to drive the flow to equilibrium.

Abstract

The paper presents a data-driven approach to stabilize an oscillating flow, such as the flow past a cylinder at Reynolds number 100, from its natural limit cycle to its equilibrium state using linear time-invariant (LTI) controllers. The key steps are:

Identification of an LTI mean transfer function model G(s) of the oscillating flow using input-output data and the mean resolvent framework. Multisine excitations are used to extract the frequency response H0(jω), which is then fitted to a low-order state-space model G(s).

Design of an LQG controller K+(s) for the identified model G(s) to reduce the oscillations. The LQG controller is chosen for its ease of automation and predictable controller structure.

Iterative stabilization of the flow by stacking the controllers K̃i(s) = BT(K̃i-1(s) + K+i(s)), where BT denotes balanced truncation to manage the increasing order of the compound controller. The flow reaches a new dynamical equilibrium with lower perturbation kinetic energy at each iteration.

The procedure is repeated until the flow is fully stabilized at the base flow equilibrium. Care is taken to ensure the method can be fully automated and applied in experiments, unlike previous approaches.

The method is demonstrated on the classic 2D flow past a cylinder at Re=100, showing the ability to efficiently drive the flow from its natural limit cycle to the stabilized base flow equilibrium using only a single sensor and actuator.

Stats

The flow past a cylinder at Re=100 has an unstable base flow and a periodic attractor (the Von Kármán vortex street).
The sensor measures the cross-stream velocity v2 at (x1=3, x2=0).
The actuator provides vertical velocity injection/suction at the upper and lower poles of the cylinder.

Quotes

"The proposed approach directly builds upon Leclercq et al. (2019) and provides several improvements for an efficient online implementation, aimed at being applicable in experiments."
"Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment."

Key Insights Distilled From

Data-driven stabilization of an oscillating flow with LTI controllers

by W. J... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08487.pdf

Deeper Inquiries

How could the method be extended to handle more complex flow configurations, such as 3D flows or flows with multiple attractors

To extend the method to handle more complex flow configurations like 3D flows or flows with multiple attractors, several adjustments and enhancements can be made:

Higher-dimensional State-Space Models: Incorporating higher-dimensional state-space models to capture the additional complexities of 3D flows. This would involve expanding the state vector to include variables that describe the flow in the additional dimensions.

Multiple Model Approach: Implementing a multiple model approach where different LTI models are used to represent different attractors or flow regimes. By switching between these models based on the current state of the flow, the controller can adapt to the presence of multiple attractors.

Nonlinear Extensions: Introducing nonlinear elements into the model to better capture the behavior of complex flows. This could involve using nonlinear state-space models or incorporating nonlinear terms into the LTI models.

Adaptive Control Strategies: Implementing adaptive control strategies that can adjust the controller parameters in real-time based on the evolving dynamics of the flow. This adaptability is crucial for handling the uncertainties and complexities of 3D flows or flows with multiple attractors.

What are the limitations of the LTI modeling approach, and how could more advanced data-driven modeling techniques be incorporated to improve the controller performance

The limitations of the LTI modeling approach include:

Limited Representation: LTI models are inherently limited in their ability to capture the full complexity of nonlinear and time-varying systems. This can lead to inaccuracies in the model representation, especially for highly dynamic or turbulent flows.

Model Mismatch: LTI models may not fully capture the intricacies of the flow dynamics, leading to model mismatch and suboptimal controller performance.

To improve controller performance, more advanced data-driven modeling techniques can be incorporated:

Nonlinear System Identification: Utilizing nonlinear system identification techniques to capture the nonlinear behavior of the flow more accurately. This could involve using neural networks or Gaussian processes to model the complex relationships in the flow dynamics.

Data-Driven Reinforcement Learning: Implementing data-driven reinforcement learning algorithms to learn optimal control policies directly from data. This approach can adapt to the nonlinearities and uncertainties present in the flow dynamics, leading to improved controller performance.

Hybrid Models: Combining LTI models with more advanced data-driven models to create hybrid models that capture both the linear and nonlinear aspects of the flow dynamics. This hybrid approach can provide a more comprehensive representation of the system and enhance controller performance.

What other control objectives beyond stabilization, such as drag reduction or mixing enhancement, could be targeted using this iterative data-driven control framework

Beyond stabilization, the iterative data-driven control framework can be applied to achieve various control objectives in fluid flows, such as:

Drag Reduction: By optimizing the control inputs to minimize drag forces on objects in the flow, the framework can be used to reduce drag and improve the overall efficiency of the system.

Vortex Shedding Suppression: Targeting the suppression of vortex shedding in flows around objects to reduce vibrations and noise, enhancing the stability and performance of the system.

Flow Mixing Enhancement: Implementing control strategies to enhance mixing in fluid flows, which can be beneficial in chemical processes, combustion systems, and other applications where efficient mixing is crucial for performance.

By adapting the controller design and optimization objectives, the iterative data-driven control framework can be tailored to achieve a wide range of control objectives beyond stabilization in fluid flows.