Empirical Reduced‐Order Modeling for  Boundary Feedback Flow Control

Djouadi, Seddik M.; Camphouse, Russell Chris; Myatt, James

doi:10.1155/2008/154956

Cited by 14 publications

(8 citation statements)

References 20 publications

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“…This result implies that one can realize approximate balanced truncation even in experiments, and can also improve computational efficiency in simulations. We note that ERA and snapshot-based approximate balanced truncation have been applied together in a model reduction procedure in [10]. However, the theoretical equivalence between these two algorithms was not explored in that work.…”

Section: Introductionmentioning

confidence: 99%

“…Modes are illustrated using contour plots of the vorticity field. Comparison of Gramians computed using (a) ERA, (b) balanced POD, and (c) ERA with pseudo-adjoint modes: The empirical Hankel singular values ( ) and the diagonal elements of the controllability ( , •) and observability ( , ×) Gramians with different order of modes (e.g.,4,10,20) in output projection.…”

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confidence: 99%

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Reduced-order models for control of fluids using the eigensystem realization algorithm

Ahuja

Rowley

2010

Theor. Comput. Fluid Dyn.

205

151

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As sensors and flow control actuators become smaller, cheaper, and more pervasive, the use of feedback control to manipulate the details of fluid flows becomes increasingly attractive. One of the challenges is to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglecting irrelevant details of the flow in order to remain computationally tractable. A number of techniques are presently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), and approximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengths and weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can be computed directly from experimental data; approximate balanced truncation often produces vastly superior models to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data.In this paper, we show that using the Eigensystem Realization Algorithm (ERA) [15], one can theoretically obtain exactly the same reduced order models as by balanced POD. Moreover, the models can be obtained directly from experimental data, without the use of adjoint information. The algorithm can also substantially improve computational efficiency when forming reduced-order models from simulation data. If adjoint information is available, then balanced POD has some advantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the method has been generalized to unstable systems. We also present a modified ERA procedure that produces modes without adjoint information, but for this procedure, the resulting models are not balanced, and do not perform as well in examples. We present a detailed comparison of the methods, and illustrate them on an example of the flow past an inclined flat plate at a low Reynolds number.

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

Reduced-order models for control of fluids using the eigensystem realization algorithm

Ahuja

Rowley

2010

Theor. Comput. Fluid Dyn.

205

151

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“…The computed Markov parameters that do not agree with the other data can effectively be filtered out of the frequency response function. The Hankel matrix containing the Markov parameters is of the following form [67] 1 2…”

Section: Eigensystem Realization Algorithm (Era)mentioning

confidence: 99%

Optical Model Reduction and Robust Feedback Control for Aerodynamics

Djouadi¹

2010

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In this project, methods of model reduction that integrate feedback active flow control with applications to nonlinear convection and turbulent flows governed by Navier-Stokes equations are developed. A new methodology which extracts boundary conditions in reduced order proper orthogonal decomposition (POD) and finite difference models is developed. A new model reduction method based on empirical data and balanced truncation was developed and applied to nonlinear Galerkin models. Based on this method a new empirical Hankel norm model reduction algorithm is proposed. These methods are applied to a prototype nonlinear convective problem governed by the two-dimensional (2D) Burgers' equation. The reduced models are used in the design of robust boundary controllers that achieve tracking, and implemented on the full order Computational Fluid Dynamics (CFD) models. POD and balanced truncation are shown to be optimal in the sense of distance minimizations in spaces of Hilbert-Schmidt (HS) operators. POD has been shown to be optimal in a broader sense than is reported in the literature. The optimality is in the sense of distance minimization in a space of integral operators under the Hilbert-Schmidt norm. A connection with balanced truncation is found. In particular, balanced truncation is shown to be optimal in the sense of distance minimization albeit in a different space of integral operators under the Hilbert-Schmidt norm when the so-called 'Curtain-Glover' balanced realization is used. This is a novel discovery as balanced truncation is usually known to be not optimal in any sense.An algorithm that combines the extended Kalman filter and expectation maximization to estimate the model coefficients from particle image velocimetry (PIV) measurements for turbulent 2D flows is developed. The algorithm is recursive and convenient for on-line implementation. The method is applied to a 2D flow control problem over the NACA 4412 airfoil using experimental data obtained from Prof. Glauser's flow control group at the University of Syracuse. The motivation for this problem is minimization of aero-optic distortion in a bluff-body flow.POD and balanced truncation are shown to minimize different n-widths of partial differential equation solutions including the Kolmogorov, Gelfand, linear and Bernstein n-widths. The nwidths are notions from metric complexity theory. They quantify inherent and representation errors due to lack of data and loss of information. Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis ...

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“…This argument is problematic in the feedback control setting. Energy of POD modes generated from snapshots incorporating open-loop actuation might not correlate to the energy of the system under feedback control [2], [9]. POD fails to capture the nonlinear degrees of freedom in nonlinear systems, since it assumes that data belongs to a linear space and therefore relies on the Euclidean distance as the metric to minimize [7].…”

Section: Introductionmentioning

confidence: 99%