Reduced order models for control of fluids using the Eigensystem Realization Algorithm

Zhanhua Ma; Sunil Ahuja; Clarence W. Rowley

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Reduced order models for control of fluids using the Eigensystem Realization Algorithm

Zhanhua Ma, Sunil Ahuja, Clarence W. Rowley

arXiv:0907.1907v1Jul 10, 2009math.OC

TL;DR

High-dimensional flow-control models require reduction, but POD can perform poorly and balanced POD requires adjoint data unavailable in experiments. The paper shows that ERA theoretically reproduces balanced POD models from input-output data, with lower computational cost, while balanced POD retains useful modal advantages when adjoint information is available.

Problem
Flow-control models are often too high-dimensional for tractable linear control, while POD can perform poorly and balanced POD requires unavailable adjoint data for experiments.
Method
The paper compares ERA with snapshot-based balanced POD and analyzes their reduced-order models, experimental data requirements, computational costs, and modal properties.
Results
ERA and balanced POD theoretically produce exactly the same reduced-order models, while ERA avoids adjoint information and is cheaper to construct from simulation data.
Takeaways & Limitations
ERA can provide approximate balanced truncation from experimental input-output measurements and improve simulation-based model-reduction efficiency.
Takeaways & Limitations
Balanced POD provides bi-orthogonal modes useful for analysis and design, and it has been generalized to unstable systems, advantages ERA does not share.

Abstract

from arXiv · show

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.

1 Introduction

Reduced-order models are needed because flow-control systems can be prohibitively high-dimensional for linear control methods. Existing approaches trade experimental accessibility against model quality, motivating ERA as an adjoint-free alternative to balanced POD.

Flow-control models can reach dimensions of O(10^5∼9), making direct application of linear control techniques computationally infeasible.
Model-reduction methods obtain low-order approximations by projecting the dynamics onto selected modes, including eigenmodes, POD modes, and balanced-truncation modes.
Balanced POD approximates balanced truncation from primal and adjoint simulation snapshots, producing bi-orthogonal modes for projection.
Balanced truncation offers a priori error bounds and guaranteed reduced-model stability for stable full-order systems, but exact computation is intractable at very large state dimensions.
Balanced POD is unsuitable for experiments because adjoint impulse-response snapshots are unavailable, whereas ERA is proposed to eliminate that requirement.
The paper’s main result is that ERA theoretically produces the same reduced-order models as balanced POD without adjoint data, while reducing computational cost by an order of magnitude.

2 The eigensystem realization algorithm as snapshot-based approximate balanced truncation

The paper formulates ERA and balanced POD for stable discrete-time linear systems and shows that their reduced models are theoretically identical. ERA uses input-output impulse responses instead of adjoint snapshots, making it experimentally applicable and substantially cheaper to form, while balanced POD retains modal advantages when adjoint data exist.

Balanced truncation balances approximate controllability and observability Gramians before truncating the least controllable and observable states.
Both methods target a smaller state dimension while preserving the input-output relationship of a high-dimensional discrete-time linear system.
Balanced POD: Balanced POD collects primal and adjoint impulse-response snapshots, forms a generalized Hankel matrix, computes its SVD, constructs bi-orthogonal modes, and projects the dynamics.
ERA: ERA collects output responses from impulse simulations or experiments, arranges Markov parameters into a generalized Hankel matrix, computes its SVD, and defines the reduced matrices.
Theoretical equivalence: The reduced system matrices generated by balanced POD and ERA are theoretically identical, although numerical inaccuracies can produce slight practical differences.
Comparison: ERA requires no adjoint data and is about 1% as costly as balanced POD for Hankel construction when mc = mo = 200.
Comparison: Balanced POD additionally supplies bi-orthogonal primal and adjoint modes, which support system analysis, controller or observer design, nonlinear projection, and parameter retention.

3 A modified ERA method using pseudo-adjoint modes

The modified ERA procedure replaces true adjoint modes with pseudo-adjoint modes obtained from the primal modes, enabling projection without adjoint simulations. However, this transformation is generally not balancing, especially when controllable and observable directions differ.

3 A modified ERA method using pseudo-adjoint modes: Pseudo-adjoint modes are constructed from the Moore–Penrose pseudo-inverse of the primal modes and used to form reduced-order system matrices.The resulting matrices use the first r primal and pseudo-adjoint modes as projection bases.
3 A modified ERA method using pseudo-adjoint modes: The pseudo-adjoint procedure performs poorly when controllable and observable directions differ, whereas balanced POD and ERA provide their greatest improvement over standard POD/Galerkin in such systems.Its performance is better when the most controllable and observable directions coincide.
3 A modified ERA method using pseudo-adjoint modes: True balanced POD uses primal and adjoint snapshots to construct a transformation whose approximate Gramians are block diagonal with equal, diagonal leading blocks.The remaining lower-right states are either unobservable or uncontrollable in the approximate Gramian product.
3.1 Transformed approximate Gramians: When pseudo-adjoint modes are used, M3 can be large and the transformed observability Gramian and Gramian product show significant off-diagonal structure.By contrast, true adjoint modes yield equal, diagonal transformed Gramians in the illustrated random-matrix example.
3.1 Transformed approximate Gramians: With pseudo-adjoint modes, the transformed observability Gramian is generally not block diagonal, so its eigenvalues and eigenvectors do not match those of the transformed controllability Gramian.The pseudo-adjoint transformation therefore does not represent balancing in the sense required by balanced truncation.
3.1 Transformed approximate Gramians: The matrix M3 measures failure to balance the approximate Gramians and becomes largest when dominant adjoint and primal modes are nearly orthogonal.Such misalignment commonly occurs in nonnormal systems, where primal and adjoint directions do not coincide.

4 Example: flow past an inclined flat plate

The flat-plate example compares ERA, balanced POD, pseudo-adjoint ERA, and standard POD for reduced modeling of a stable low-Reynolds-number flow. ERA substantially reduces construction cost while producing models that closely match balanced POD and outperform the alternatives in several performance measures.

4.1 Model problem and parameters: The example models two-dimensional flow over a flat plate inclined at α = 25° and Re = 100, which asymptotically reaches a stable steady state.The numerical method solves for the vorticity field using an immersed boundary formulation with nested grids.
4.2 Input and output: The linearized model uses discrete vorticity as its state, a localized leading-edge body force as input, and the full velocity field as output.ERA and balanced POD project the high-dimensional velocity output onto leading POD modes from impulse-response snapshots.
4.3 Reduced-order models: ERA constructs the reduced model from 400 Markov parameters, whereas balanced POD uses primal and adjoint snapshots to assemble its generalized Hankel matrix.For a 10-mode output-projected system, balanced POD requires 4m × 10^4 inner products for H, compared with 4m × 10^2 for each ERA H and H′ construction.
4.3 Reduced-order models: ERA models produce nearly indistinguishable Gramian and empirical Hankel-singular-value curves from balanced POD across output-projection orders.Balanced POD observability Gramians show inaccuracies for some leading modes because of an approximation in the adjoint formulation.
4.3 Reduced-order models: ERA with pseudo-adjoint modes produces poorly balanced Gramians because primal modes poorly approximate the spatially different true adjoint modes.The pseudo-adjoint modes more closely resemble leading primal modes than the true adjoint modes.
4.4 Model performance: ERA and balanced POD error norms generally converge to the output-projected lower bound, while pseudo-adjoint ERA and POD converge more slowly or to a larger bound.A 16-mode ERA model accurately predicts the impulse response, whereas 30-mode POD and pseudo-adjoint ERA models over-predict it, especially after t ≈ 80; their frequency responses also show spurious peaks in [0.1, 2].

5 Discussion

ERA and balanced POD theoretically produce identical reduced-order models, while ERA avoids adjoint data and is computationally cheaper. In the flat-plate example, both outperform standard POD and pseudo-adjoint ERA, but balanced POD retains advantages for bi-orthogonal modes and unstable-system generalization.

ERA and balanced POD theoretically produce exactly the same reduced-order models.
ERA balances approximate Gramians, avoids adjoint-system data, and constructs the generalized Hankel matrix at an order-of-magnitude lower computational cost.
Balanced POD provides bi-orthogonal primal and adjoint modes useful for nonlinear projection and retaining parameters such as Reynolds number.
Balanced POD has been generalized to unstable systems, whereas the analysis here considers only stable, linear models.
In the inclined-flat-plate example, balanced POD and ERA perform nearly identically and both significantly outperform standard POD and pseudo-adjoint ERA.