Stabilization of linear time‐varying reduced‐order models: A feedback controller approach

Mojgani, Rambod; Balajewicz, Maciej

doi:10.1002/nme.6489

Cited by 9 publications

(3 citation statements)

References 61 publications

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“…Conversely, the library can be expanded to systematically "explore the computational universe", e.g., using gene expression programming [62]. Even further, additional constrained, for example on stability of the corrected model, can be imposed [63]. Effective strategies for the selection of an adequate and concise library should be investigated in future work using more complex test cases.…”

Section: B Noisy Observationsmentioning

confidence: 99%

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

Mojgani,

Chattopadhyay,

Hassanzadeh

2021

Preprint

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Models used for many important engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial difference between the numerical solutions of the model and the observations of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is substantial interest in reducing model errors, particularly through understanding their physics and sources and leveraging the rapid growth of observational data. Here we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation (DA) technique such as ensemble Kalman filter (EnKF) is first used to provide a noise-free analysis state of the system, which is then used in estimating the model error. Finally, an equation-discovery technique, such as the relevance vector machine (RVM), a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, closedform representation of the model error. Using the chaotic Kuramoto-Sivashinsky (KS) system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.

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Section: B Noisy Observationsmentioning

confidence: 99%

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

Mojgani,

Chattopadhyay,

Hassanzadeh

2021

Preprint

Self Cite

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“…Stabilization methods and closure terms have improved the performance of POD-Galerkin ROMs in various applications [5,[18][19][20]. Eigenvalue reassignment methods [17,[21][22][23], subspace rotation [22,24,25], linear quadratic regulation and numerical dissipation [26] recover ROMs by post-processing non-intrusive stabilization. Leveraging energy stability in linear POD-Galerkin ROMs through symmetrization by an energy-based inner product [22,27], and transformation of nonlinear POD-Galerkin ROMs using entropy-conservative variables [28,29] on the other hand, stabilize ROMs through an intrusive framework that modifies and reformulates ROM equations.…”

Section: Introductionmentioning

confidence: 99%

Non-intrusive balancing transformation of highly stiff systems with lightly damped impulse response

Rezaian

Huang

Duraisamy

2022

Phil. Trans. R. Soc. A.

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Balanced truncation (BT) is a model reduction method that uses a coordinate transformation to retain eigen-directions that are highly observable and reachable. To address realizability and scalability of BT applied to highly stiff and lightly damped systems, a non-intrusive data-driven method is developed for balancing discrete-time linear systems via the eigensystem realization algorithm (ERA). The advantage of ERA for balancing transformation makes full-state outputs tractable. Further, ERA enables balancing despite stiffness, by eliminating computation of balancing modes and adjoint simulations. As a demonstrative example, we create balanced reduced-order models (ROMs) for a one-dimensional reactive flow with pressure forcing, where the stiffness introduced by the chemical source term is extreme (condition number 10 13 ), preventing analytical implementation of BT. We investigate the performance of ROMs in prediction of dynamics with unseen forcing inputs and demonstrate stability and accuracy of balanced ROMs in truly predictive scenarios, whereas without ERA, proper orthogonal decomposition-Galerkin and least-squares Petrov–Galerkin projections fail to represent the true dynamics. We show that after the initial transients under unit impulse forcing, the system undergoes lightly damped oscillations, which magnifies the influence of sampling properties on predictive performance of the balanced ROMs. We propose an output domain decomposition approach and couple it with tangential interpolation to resolve sharp gradients at reduced computational costs. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

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“…Following the success of eigenvalue assignment 66 in the stabilization of linear time‐invariant (LTI) systems, Mojgani and Balajewicz 67 extended this approach to the stabilization of linear ROMs with time‐varying operators. To further extend the idea to a general framework including more complicated dynamical systems such as a fully nonlinear flow equation, our current study proposes an eigenvalue reassignment method for the stabilization of nonlinear ROMs (ERN).…”

Section: Introductionmentioning

confidence: 99%

A global eigenvalue reassignment method for the stabilization of nonlinear reduced‐order models

Rezaian

Wei

2021

Numerical Meth Engineering

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The systematic and physics‐infused construction of a projection‐based reduced‐order model (ROM) shows the capability to replicate the original high‐dimensional system's dynamical evolution but with a fractional computational cost. However, certain nonlinear features and high‐frequency contributions may be lost throughout the aggressive order reduction. Thus, ROMs in a broad category of fluid dynamics applications require stabilization and closure methods to compensate for the key contributions missed through the model order reduction. A new stabilization method for nonlinear ROMs is proposed in this study that learns a linear control law to drive the nonlinear ROM toward maximum agreement with the full‐order model, where a total power constraint guarantees the stability of the nonlinear ROM. This new method achieves both stability and accuracy for nonlinear proper orthogonal decomposition‐Galerkin ROMs in two chosen applications with strong shock‐wake interactions and unsteady oscillations, which trigger strong instabilities in the original ROMs before stabilization. A multistage layout is designed to further enhance the proposed stabilization method for more efficient and robust stabilization of nonlinear ROMs with a large number of unstable modes.

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Stabilization of linear time‐varying reduced‐order models: A feedback controller approach

Cited by 9 publications

References 61 publications

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

Non-intrusive balancing transformation of highly stiff systems with lightly damped impulse response

A global eigenvalue reassignment method for the stabilization of nonlinear reduced‐order models

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