Empirical state-space representations for Theodorsen's lift model

S. L. Brunton and C. W. Rowley

Journal of Fluids and Structures (submitted), April 2012.

Abstract:

In this work, we cast Theodorsen's unsteady aerodynamic model into a general form that allows for the introduction of empirically determined quasi-steady and added-mass coefficients as well as an empirical Theodorsen function. An empirically determined Theodorsen model is constructed using data from direct numerical simulations of a flat plate airfoil pitching at low Reynolds number, Re=100. Next, we develop low-dimensional, state-space realizations that are useful for either the classical Theodorsen lift model or the empirical model. The resulting model is parameterized by pitch-axis location and has physically meaningful states that isolate the effect of added-mass and quasi-steady forces, as well as the effect of the wake. A low-order approximation of Theodorsen's function is developed based on balanced truncation of a model fit to the analytical frequency response, and it is shown that this approximation outperforms other models from the literature. We demonstrate the utility of these state-space lift models by constructing a robust controller that tracks a reference lift coefficient by varying pitch angle while rejecting gust disturbances.

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