[Paper Review] Reduced order models for control of fluids using the Eigensystem Realization Algorithm

Zhanhua Ma, Sunil Ahuja+1 more|Jul 10, 2009

Model Reduction and Neural Networks3 citations

TL;DR

This paper demonstrates that the Eigensystem Realization Algorithm (ERA) produces reduced-order models for fluid control that are theoretically equivalent to those from balanced proper orthogonal decomposition (balanced POD), enabling accurate, computationally efficient models without requiring adjoint simulations. Unlike standard POD, ERA achieves balanced model accuracy using only experimental or simulation data, offering a practical alternative for real-world flow control applications with minimal computational cost and no need for adjoint information.

ABSTRACT

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.

Motivation & Objective

• To develop a method for constructing accurate, balanced reduced-order models of fluid flows without requiring adjoint simulation data.
• To address the limitation of balanced POD, which relies on adjoint information unavailable in experimental settings.
• To improve computational efficiency in generating reduced-order models from high-dimensional fluid simulations.
• To compare ERA with balanced POD and standard POD in terms of accuracy, computational cost, and applicability to experimental data.
• To explore the feasibility of using ERA for unstable or nonlinear systems, extending its utility beyond current methods.

Proposed method

• Apply the Eigensystem Realization Algorithm (ERA) to snapshot data from fluid simulations or experiments to construct a reduced-order model.
• Construct a generalized Hankel matrix from impulse response snapshots to extract system modes and dynamics.
• Use singular value decomposition (SVD) on the Hankel matrix to extract balanced modes, ensuring balanced Gramians and stability.
• Compare ERA results directly with balanced POD by computing identical reduced-order models using the same data and projection basis.
• Implement a modified ERA variant using pseudo-inverse to generate bi-orthogonal modes without adjoint data, though without guaranteed balancing.
• Project full-system outputs onto leading POD modes to compare frequency responses and validate model accuracy.

Experimental results

Research questions

• RQ1 Can ERA produce reduced-order models that are theoretically equivalent to those from balanced POD for linear fluid systems?
• RQ2 Does ERA achieve the same accuracy as balanced POD while eliminating the need for adjoint simulation data?
• RQ3 How does ERA’s computational cost compare to balanced POD when forming reduced-order models from simulation data?
• RQ4 Can ERA be effectively applied to experimental data where adjoint information is unavailable?
• RQ5 What are the limitations of a modified ERA approach that uses pseudo-inverse to generate bi-orthogonal modes without adjoint data?

Key findings

• ERA produces reduced-order models that are theoretically identical to those from balanced POD, confirming that ERA performs approximate balanced truncation without requiring adjoint systems.
• ERA achieves a 10-fold reduction in computational cost compared to balanced POD when forming reduced-order models from simulation data.
• ERA models match the full system’s frequency response nearly perfectly, outperforming both standard POD and modified ERA with pseudo-adjoint modes.
• The 30-mode ERA model captures transient growth in a channel flow nearly as accurately as the 30-mode balanced POD model, while a 17-mode POD model fails to capture it.
• ERA with pseudo-adjoint modes generates spurious peaks in the frequency response, indicating poor performance despite producing bi-orthogonal modes.
• The small differences between ERA and balanced POD models arise from numerical inaccuracies in the discrete adjoint operator used in balanced POD, not from the methods themselves.