Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment

Kalashnikova, Irina; Waanders, Bart van Bloemen; Arunajatesan, Srinivasan; Barone, Matthew Franklin

doi:10.1016/j.cma.2014.01.011

Cited by 81 publications

(79 citation statements)

References 27 publications

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“…However, it has been observed, e.g., [2][3][4][5][6] , that POD-ROMs can lose stability. This loss of stability is due to the truncation of higherorder modes, which are responsible for the dissipation of energy and thus have a stabilizing effect.…”

Section: Introductionmentioning

confidence: 99%

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Benosman¹,

Borggaard²,

San³

et al. 2017

Applied Mathematical Modelling

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a b s t r a c tWe present some results on the stabilization of reduced-order models (ROMs) for thermal fluids. The stabilization is achieved using robust Lyapunov control theory to design a new closure model that is robust to parametric uncertainties. Furthermore, the free parameters in the proposed ROM stabilization method are optimized using a data-driven multiparametric extremum seeking (MES) algorithm. The 2D and 3D Boussinesq equations provide challenging numerical test cases that are used to demonstrate the advantages of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Benosman¹,

Borggaard²,

San³

et al. 2017

Applied Mathematical Modelling

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“…Kalashnikova et al have shown that for the LTI system in Equation , ER is equivalent to full‐state feedback control with control input u M =− Kx M and the specific choice of B and K matrices:

B = [\begin{matrix} {bold-italicV}_{1} & {bold-italicV}_{2} & \dots & {bold-italicV}_{M} \end{matrix}],

K = [\begin{matrix} λ_{1}^{u} - {\overset{λ}{true}}_{1} & 0 & 0 & \dots & 0 \\ 0 & λ_{2}^{u} - {\overset{λ}{true}}_{2} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & λ_{M}^{u} - {\overset{λ}{true}}_{M} & 0 \end{matrix}] V^{- 1},

where λ u s are the unstable eigenvalues and λ∼s are the new eigenvalues assigned by ER. This definition, that is not immediately available for the ISC and therefore Hybrid methods, has made it possible to compute the controllability matrix and gramian for the L2 and symmetry ROMs that are stabilized by ER in this study.…”

Section: Controllability Analysismentioning

confidence: 99%

“…The new Hybrid method developed here aims to take the strengths of both the ISC method and the ER method . Since the ISC method, though being efficient and robust, tends to excessively alter low‐frequency (originally more unstable) modes and reduce the overall accuracy, the Hybrid approach takes two steps: first, uses the ISC method to stabilize all modes in an efficient manner; then, applies the ER method to fine tune the eigenvalues of a small group of low‐frequency modes with a cost function defined to directly control the accuracy of the overall system outputs.…”

Section: Introductionmentioning

confidence: 99%

A hybrid stabilization approach for reduced‐order models of compressible flows with shock‐vortex interaction

Rezaian

Wei

2019

Numerical Meth Engineering

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SUMMARY Model order reduction approaches, such as proper orthogonal decomposition (POD)‐Galerkin projection, provide a systematic manner to construct Reduced‐Order Models (ROM) from pregenerated high‐fidelity datasets. The current study focuses on the stabilization of ROMs built from high‐fidelity simulation data of a supersonic flow passing a circular cylinder, which features strong interactions between shockwaves and vortices. As shown in previous literatures and the current study, an implicit subspace correction (ISC) method is efficient in the stabilization of similar problems, but its accuracy is not consistent when applied on different ROMs; on the other hand, an eigenvalue reassignment (ER) method delivers superb accuracy when the mode number is small, but becomes too expensive and less robust as the number increases. A Hybrid method is proposed here to balance the computational cost while improving the overall robustness/accuracy in ROM stabilization. The Hybrid method first handles the majority of the modes using the ISC method and then applies the ER method to fine tune a smaller number of modes under a constraint for accuracy. Furthermore, when the typical L2 inner product is changed to a symmetry inner product in both POD computation and Galerkin projection, the performance of the stabilized ROMs is substantially improved for all methods.

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“…In [37], a set of linear constraints for the left-projection matrix, given the right-projection matrix, are derived by Bond et al to yield a projection framework that is guaranteed to generate a stable ROM. An approach for stabilizing unstable ROMs for LTI systems, termed ROM stabilization via optimization-based eigenvalue reassignment, was proposed by Kalashnikova et al in the recent work [53]. In this approach, the unstable eigenvalues of an unstable ROM are modified through the numerical solution of a constrained nonlinear leastsquares optimization problem formulated such that the error in the stabilized ROM output is minimal.…”

Section: Introductionmentioning

confidence: 99%

Construction of energy-stable projection-based reduced order models

Kalashnikova

Barone

Arunajatesan

et al. 2014

Applied Mathematics and Computation

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a r t i c l e i n f o Keywords: Reduced order model (ROM) Proper orthogonal decomposition (POD)/ Galerkin projection Linear hyperbolic/incompletely parabolic systems Linear time-invariant (LTI) systems Numerical stability Lyapunov equation a b s t r a c tAn approach for building energy-stable Galerkin reduced order models (ROMs) for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. This method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow. The key idea is to apply to the PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. For linear problems, the desired transformation is induced by a special inner product, termed the ''symmetry inner product'', which is derived herein for several systems of physical interest. Connections are established between the proposed approach and other stability-preserving model reduction methods, giving the paper a review flavor. More specifically, it is shown that a discrete counterpart of this inner product is a weighted L 2 inner product obtained by solving a Lyapunov equation, first proposed by Rowley et al. and termed herein the ''Lyapunov inner product''. Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases.Published by Elsevier Inc.

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Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment

Cited by 81 publications

References 27 publications

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

A hybrid stabilization approach for reduced‐order models of compressible flows with shock‐vortex interaction

Construction of energy-stable projection-based reduced order models

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