Data-driven stabilization of an oscillating flow with LTI controllers

TL;DR

This work presents a fully data-driven strategy to stabilize oscillatory flows by modeling the dynamics around the limit cycle with a mean resolvent-based LTI transfer function $G(s)$. An LQG controller is synthesized for this ROM and, due to growth in controller order across iterations, online balanced truncation plus a two-step state initialization/manage switching are employed to keep the controller compact and automatable. The method is demonstrated on two-dimensional cylinder wake flow at $Re=100$, achieving iterative convergence from the natural limit cycle to the base flow while maintaining low control effort and moderate transient in the actuation. Key insights include the interpretability of the implicit closed-loop model $G^I$ versus the mean-flow model, the role of pole dynamics in nonlinear relaxation, and the practical considerations for experimental deployment such as sensor/actuator noise, saturations, and three-dimensional effects. Overall, the approach offers an accessible, data-driven pathway to stabilizing oscillator flows in experiments with minimal prior modeling.

Abstract

This paper presents advances towards the data-based control of periodic oscillator flows, from their fully-developed regime to their equilibrium stabilized in closed-loop, with linear time-invariant (LTI) controllers. 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. First, we use input-output data to construct an LTI mean transfer functions of the flow. The model is subsequently used for the design of an LTI controller with Linear Quadratic Gaussian (LQG) synthesis, that is practical to automate online. Then, using the controller in a feedback loop, the flow shifts in phase space and oscillations are damped. The procedure is repeated until equilibrium is reached, by stacking controllers and performing balanced truncation to deal with the increasing order of the compound controller. In this article, we illustrate the method on the classic flow past a cylinder at Reynolds number Re=100. Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment.

Figures (24)

• Figure 1: Streamlines of a snapshot of the incompressible flow past a two-dimensional cylinder at Reynolds number $\Rey=100$. Colored by velocity magnitude.
• Figure 2: Graphical summary of the method : data-based stabilization of an oscillating flow with LTI controllers, using the mean resolvent framework leclercq2023.
• Figure 3: At each iteration $i$, a time simulation is performed in closed-loop with the controller $\tilde{K}_i(s)$; then, an exogenous signal $u_{\mathbf{\boldsymbol{\Phi}}}(t)$ is injected for the identification of an LTI model $G_i(s)$, for which an LTI controller $K_i^+(s)$ is synthesized. This corresponds to the start of iteration $i+1$, where the controller in the loop is $\tilde{K}_{i+1} = \mathcal{B}_T(\tilde{K}_i + K_i^+)$ that should drive the flow to a new dynamical equilibrium with lower perturbation kinetic energy. The process is then repeated.
• Figure 4: Domain geometry for the flow past a cylinder. Dimensions are in black, while boundary conditions are in light gray. Drawing is not to scale.
• Figure 5: Cylinder flow regimes (velocity magnitude) - Unstable base flow (\ref{['fig_baseflow']}) and snapshot of the attractor (\ref{['fig_lco']}). Domain is cut for clarity.