Closed-Loop Control of Fluid Flow: a Review of Linear Approaches and Tools for the Stabilization of Transitional Flows

International audienceFlow control is concerned with the targeted manipulation of intrinsic flow behavior to optimally satisfy prescribed objectives. This article will give an overview of the most common tools for the design of control strategies. We focus on linear control that is aimed at stabilizing fixed points of the Navier-Stokes equations, such as those existing in the case of transitional flows. Key steps to build a Galerkin-based and a data-based model will be presented and illustrated on two generic flow configurations: flow over an open cavity and over a backward-facing step. In the former case, a feedback configuration will result and particular attention will be paid to performance measures and robustness analyses. For the latter case, a feedforward setup has been chosen and a system-identification method will be employed to extract the fluid behavior from data sequences, via statistical learning techniques

Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators

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.Comment: 36 pages, 17 figure

Performance of a linear robust control strategy on a nonlinear model of spatially developing flows

International audienceThis paper investigates the control of self-excited oscillations in spatially developing flow systems such as jets and wakes using H 8 control theory on a complex Ginzburg-Landau (CGL) model. The coefficients used in this one-dimensional equation, which serves as a simple model of the evolution of hydrodynamic instability waves, are those selected by Roussopoulos & Monkewitz (Physica D 1996, vol. 97, p. 264) to model the behaviour of the near-wake of a circular cylinder. Based on noisy measurements at a point sensor typically located inside the cylinder wake, the compensator uses a linear H 8 filter based on the CGL model to construct a state estimate. This estimate is then used to compute linear H 8 control feedback at a point actuator location, which is typically located upstream of the sensor. The goal of the control scheme is to stabilize the system by minimizing a weighted average of the 'system response' and the 'control effort' while rigorously bounding the response of the controlled linear system to external disturbances. The application of such modern control and estimation rules stabilizes the linear CGL system at Reynolds numbers far above the critical Reynolds number Rec ˜ 47 at which linear global instability appears in the uncontrolled system. In so doing, many unstable modes of the uncontrolled CGL system are linearly stabilized by the single actuator/sensor pair and the model-based feedback control strategy. Further, the linear performance of the closed-loop system, in terms of the relevant transfer function norms quantifying the linear response of the controlled system to external disturbances, is substantially improved beyond that possible with the simple proportional measurement feedback proposed in previous studies. Above Re ˜ 84, the control designs significantly outperform the corresponding control designs in terms of their ability to stabilize the CGL system in the presence of worst-case disturbances. The extension of these control and estimation rules to the nonlinear CGL system on its attractor (a simple limit cycle) stabilizes the full nonlinear system back to the stationary state at Reynolds numbers up to Re ˜ 97 using a single actuator/sensor pair, fixed-gain linear feedback and an extended Kalman filter incorporating the system nolinearity. © 2004 Cambridge University Press

Input-output measures for model reduction and closed-loop control: Application to global modes

International audienceFeedback control applications for flows with a large number of degrees of freedom require the reduction of the full flow model to a system with significantly fewer degrees of freedom. This model-reduction process is accomplished by Galerkin projections using a reduction basis composed of modal structures that ideally preserve the input-output behaviour between actuators and sensors and ultimately result in a stabilized compensated system. In this study, global modes are critically assessed as to their suitability as a reduction basis, and the globally unstable, two-dimensional flow over an open cavity is used as a test case. Four criteria are introduced to select from the global spectrum the modes that are included in the reduction basis. Based on these criteria, four reduced-order models are tested by computing open-loop (transfer function) and closed-loop (stability) characteristics. Even though weak global instabilities can be suppressed, the concept of reduced-order compensators based on global modes does not demonstrate sufficient robustness to be recommended as a suitable choice for model reduction in feedback control applications. The investigation also reveals a compelling link between frequency-restricted input-output measures of open-loop behaviour and closed-loop performance, which suggests the departure from mathematically motivated H-measures for model reduction toward more physically based norms; a particular frequency-restricted input-output measure is proposed in this study which more accurately predicts the closed-loop behaviour of the reduced-order model and yields a stable compensated system with a markedly reduced number of degrees of freedom. © 2011 Cambridge University Press

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

Robust feedback control of Rayleigh-Bénard convection

We investigate the application of linear-quadratic-Gaussian (LQG) feedback control, or, in modern terms, H2 control, to the stabilization of the no-motion state against the onset of Rayleigh-Bénard convection in an infinite layer of Boussinesq fluid. We use two sensing and actuating methods: The planar sensor model (Tang & Bau 1993, 1994), and the shadowgraph model (Howle 1997a). By extending the planar sensor model to the multi-sensor case, it is shown that a LQG controller is capable of stabilizing the no-motion state up to 14.5 times the critical Rayleigh number. We characterize the robustness of the controller with respect to parameter uncertainties, unmodelled dynamics. Results indicate that the LQG controller provides robust performances even at high Rayleigh numbers

Closed-Loop Control of Leading Edge Vorticity on a 3D Wing: Simulations and Low-Dimensional Models

We study model-based feedback control of the low-Reynolds-number flow over a flat plate at large angles of attack, in both two and three dimensions. Our long-term goal is to be able to manipulate the leading-edge vortices that form on low-aspect-ratio wings at high angles of attack, and that often contribute to exceptionally large lift coefficients. Intwo-dimensional simulations, we present a model-based feedback controller that uses an observer to reconstruct the entire flow field from velocity measurements at three locations, and stabilizes the flow at an angle of attack for which the natural flow state is periodic shedding. In three-dimensional simulations, we use open-loop forcing to study actuator placement, and conclude that trailing-edge actuation is more effective than leading-edge actuation in influencing the forces on the plate, as well as the wake structures. Finally, we present initial results towards extending our model-based control design to the 3D setting, and apply a selective frequency damping method to find unstable equilibrium flow fields in 3D simulations