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

By Sunil Ahuja, Clarence W. Rowley. Journal of Fluid Mechanics, 2009

155 citations recorded by the source corpus.

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-order 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 in this paper, 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 two-dimensional 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 neighbourhood of this unstable steady state. The actuation is modelled as a localized body force near the trailing 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.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b79826ca36466d5ecc75f3bfe5e4201b960a10ae

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Citations in the corpus

Modal Analysis of Fluid Flows: An Overview (2017, methodology): “VI) were given by Rowley et al. [105], which explained the meaning of DMD for a nonlinear system (see also the review articles by Mezić [106] and Tu et al. [107]).”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, background): “to propose reduced-order controllers around linearly unstable steady states for flow around a cylinder in [122] and for flow past a flat plate in [1].”
Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison (2011, background): “…et al. 1999; Cohen et al. 2003; Bergmann et al. 2005; Lehmann et al. 2005; Hoepffner et al. 2006; Bagheri et al. 2009; Barbagallo et al. 2009; Ahuja & Rowley 2010; Akhtar & Nayfeh 2010) and data assimilation of atmospheric and oceanic flows (Luo et al. 2007; Daescu & Navon 2008; Fang et al.…”
Model Reduction for Flow Analysis and Control (2017, methodology): “Variants of these approaches have been used in examples by Barbagallo et al. (2009) and Ahuja & Rowley (2010).”
On the Unsteady Behavior of the Flow around NACA 0012 Airfoil with Steady External Conditions at Re=1000 (2015, methodology): “Ahuja and Rowley [18] have performed a study on a flat plate at Re=100 for angles of attack ranging from 0° to 90° and proposed an estimator-based control design procedure using reduced-order models for flow control of unstable linear systems.”