Balanced realization and model reduction for unstable systems

Zhou, Kemin; Salomon, Gregory; Wu, Eva

doi:10.1002/(sici)1099-1239(199903)9:3<183::aid-rnc399>3.0.co;2-e

Cited by 143 publications

(68 citation statements)

References 8 publications

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“…Even the optimal rational interpolation-based model reduction methods have been extended to unstable systems; see, e.g., [143,165]. Balanced truncation has been also generalized to reducing unstable systems, by either appropriately redefining the system Gramians [25,232,233] or by using different balancing techniques, such as LQG balancing [185]. For the POD, the frequency domain formulation will be the appropriate choice for unstable systems since the time-domain snapshots will grow exponentially in this case.…”

Section: 7mentioning

confidence: 99%

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: 7mentioning

confidence: 99%

“…For the POD, the frequency domain formulation will be the appropriate choice for unstable systems since the time-domain snapshots will grow exponentially in this case. This would be a similar approach to the balanced truncation method for unstable systems presented in [232].…”

Section: 7mentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…(g) Model reduction of unstable or descriptor systems can be carried out by various means, e.g., LQG balancing, frequency domain grammians [83], or normalized coprime factorizations [78].…”

Section: Remark 31 (A)mentioning

confidence: 99%

Approximation of Large-Scale Dynamical Systems: An Overview

Antoulas

2004

IFAC Proceedings Volumes

144

179

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In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.

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“…A difficulty related to balanced truncation is that the linearized system must be stable. An extension to unstable linear systems was proposed in [132] by decoupling the dynamics on the stable and unstable subspaces, and then truncating the relatively uncontrollable and unobservable modal representations on each subspace (see e.g. [132] for further details).…”

Section: Model Reduction Of Linearized Time-invariant Systemsmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

148

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Balanced realization and model reduction for unstable systems

Cited by 143 publications

References 8 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Approximation of Large-Scale Dynamical Systems: An Overview

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

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