  In feedback flow control, 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)
\citep{JuPa-85}, 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.
