A model-order reduction method based on Krylov subspaces for mimo bilinear dynamical systems

Lin, Yiqin; Li, Bao; Wei, Yimin

doi:10.1007/bf02832354

Cited by 19 publications

(27 citation statements)

References 10 publications

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“…Note that only SISO systems are considered in [16,159]. For MIMO systems, the expression of the transfer function will be different, see [84,137]. In [137], a method similar to [159] was extended to MIMO systems.…”

Section: Multimoment-matching Methodsmentioning

confidence: 99%

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

Baur

Benner

Feng

2014

Arch Computat Methods Eng

294

270

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In the past decades, Model Order Reduction (MOR) has demonstrated its robustness and wide applicability for simulating large-scale mathematical models in engineering and the sciences. Recently, MOR has been intensively further developed for increasingly complex dynamical systems. Wide applications of MOR have been found not only in simulation, but also in optimization and control. In this survey paper, we review some popular MOR methods for linear and nonlinear large-scale dynamical systems, mainly used in electrical and control engineering, in computational electromagnetics, as well as in micro-and nanoelectro-mechanical systems design. This complements recent surveys on generating reduced-order models for parameterdependent problems (Benner et al. in 2013; Boyaval et al. in Arch Comput Methods Eng 17(4):435-454, 2010; Rozza et al. Arch Comput Methods Eng 15(3):229-275, 2008) which we do not consider here. Besides reviewing existing methods and the computational techniques needed to implement them, open issues are discussed, and some new results are proposed.

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Section: Multimoment-matching Methodsmentioning

confidence: 99%

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

Baur

Benner

Feng

2014

Arch Computat Methods Eng

294

270

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“…In the following we consider several medium-sized numerical examples and compare the approximation property of balanced truncation with that of a Krylov subspace projection method, developed in [4], which we briefly summarize below. (Other Krylov subspace methods for bilinear systems can be found, e.g., in [47,38,21,12,8].) Since it is not obvious to define a transfer norm for nonlinear systems, we compare just the outputs of the original system and the reduced systems for a given input function.…”

Section: Covariance Approximationmentioning

confidence: 99%

Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems

Benner¹,

Damm²

2011

SIAM J. Control Optim.

214

222

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Abstract. We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems.

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“…We do so by means of a multiscale analysis of the balanced equations of motion which are shown to collapse to a dimension reduced system when some of the Hankel singular values go to zero; see, e.g., [35,36] for a related approach or [22,47,59,29] in which low-rank perturbative approximations of transfer functions of linear systems are sought. To the best of our knowledge our approach is new, and, although it is based on computing an appropriate balanced form of the system equations which certainly becomes infeasible if the system is extremely high-dimensional (n ∼ 10 6 or larger), we see it not merely as an alternative, but rather as an extension to existing projectionbased methods such as Krylov subspace [54,4,15,45,46,12,10] or interpolation (moment-matching) methods [9,23,67], and empirical POD [37,44,13,58]; see also [7,60]. A common feature of these methods is that they identify a subspace which contains the "essential" part of the dynamics, and we suggest to combine the numerically cheap identification of Krylov or POD subspaces with singular perturbation methods that have certain advantages in terms of structure preservation.…”

Section: Introductionmentioning

confidence: 99%

Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

Hartmann¹,

Schäfer-Bung²,

Thöns‐Zueva³

2013

SIAM J. Control Optim.

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Abstract. We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics can be split into a fast and a slow subspace and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.

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A model-order reduction method based on Krylov subspaces for mimo bilinear dynamical systems

Cited by 19 publications

References 10 publications

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems

Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

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