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Data-driven selection of actuators for optimal control of airfoil separation

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Abstract

We present a systematic approach for determining the optimal actuator location for separation control from input–output response data, gathered from numerical simulations or physical experiments. The Eigensystem realization algorithm is used to extract state-space descriptions from the response data associated with a candidate set of actuator locations. These system realizations are then used to determine the actuator location among the set that can drive the system output to an arbitrary value with minimal control effort. The solution of the corresponding minimum energy optimal control problem is evaluated by computing the generalized output controllability Gramian. We use the method to analyze high-fidelity numerical simulation data of the lift and separation angle responses to a pulse of localized body-force actuation from six distinct locations on the upper surface of a NACA 65(1)-412 airfoil. We find that the optimal location for controlling lift is different from the optimal location for controlling separation angle. In order to explain the physical mechanisms underlying these differences, we conduct controllability analyses of the flowfield by leveraging the dynamic mode decomposition with control algorithm. These modal analyses of flowfield response data reveal that excitation of coherent structures in the wake benefits lift control, whereas excitation of coherent structures in the shear layer benefits separation angle control.

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Notes

  1. Although finite time-horizons can be considered, we choose to focus on the infinite time-horizon case in order to maintain objectivity in the optimality measure; the solution to the finite time-horizon problem is dependent on the final time, which can be undesirable because the final time can always be chosen to influence the outcome of the optimality measure.

  2. Tustin’s approximation allows for conversion between continuous-time and discrete-time representations of a dynamic system. Given the sampling time \(T_s\), the approximation leverages a bilinear Tustin transformation to map between all points in the s-plane and the z-plane:

    $$\begin{aligned} z = e^{sT_s} \approx \frac{1+sT_s/2}{1-sT_s/2}. \end{aligned}$$

    The transformation is sometimes used to relate optimal control problems formulated in continuous-time to counterparts in discrete-time and vice versa.

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Acknowledgements

This material is based upon work supported by the Air Force Office of Scientific Research under awards FA9550-16-1-0392, FA9550-17-1-0252, and FA9550-19-1-0034 monitored by Drs. Douglas R. Smith and Gregg Abate. The authors thank Dr. Kevin K. Chen for initial discussions related to optimal actuator selection.

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Authors and Affiliations

  1. Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, 55455, USA

    Debraj Bhattacharjee

  2. Aerospace Engineering, San Diego State University, San Diego, CA, 92182, USA

    Bjoern Klose & Gustaaf B. Jacobs

  3. Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    Maziar S. Hemati

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  1. Debraj Bhattacharjee
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  2. Bjoern Klose
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  3. Gustaaf B. Jacobs
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  4. Maziar S. Hemati
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Correspondence to Maziar S. Hemati.

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Bhattacharjee, D., Klose, B., Jacobs, G.B. et al. Data-driven selection of actuators for optimal control of airfoil separation. Theor. Comput. Fluid Dyn. 34, 557–575 (2020). https://doi.org/10.1007/s00162-020-00526-y

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