Approximation of linear constant systems

Meier, L.; Luenberger, David G.

doi:10.1109/tac.1967.1098680

Cited by 173 publications

(115 citation statements)

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“…Meier and Luenberger in [173] proved these conditions for SISO systems originally; generalizations to MIMO systems were given in [64,116,214]. The optimality conditions in (3.9) mean that an H 2 -optimal reduced model H r (s) is a bitangential Hermite interpolant to H(s).…”

Section: Optimal-h 2 Tangential Interpolation For Nonparametric Systemsmentioning

confidence: 99%

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: Optimal-h 2 Tangential Interpolation For Nonparametric Systemsmentioning

confidence: 99%

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

2015

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“…the problem in which also the interpolation points are parameters of the minimization, are known since the '60s and are given in [25]. In [17] the IRKA algorithm is proposed: the output of the algorithm converges to an optimal (not necessarily stable) model.…”

Section: B Problem Formulationmentioning

confidence: 99%

“…which yields (25). Equation (24) follows by comparison with equation (9) written for the reduced-order model.…”

Section: A the Input Of The System Is U = Lωmentioning

confidence: 99%

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Constrained optimal reduced-order models from input/output data

Scarciotti

Jiang

Astolfi

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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Abstract-Model reduction by moment matching does not preserve, in a systematic way, the transient response of the system to be reduced, thus limiting the use of this model reduction technique in control problems. With the final goal of designing reduced-order models which can effectively be used (not just for analysis but also) for control purposes, we determine, using a data-driven approach, an estimate of the moments and of the transient response of an unknown system. We compute the unique, up to a change of coordinates, reducedorder model which possesses the estimated transient and, simultaneously, achieves moment matching at the prescribed interpolation points. The error between the output of the system and the output of the reduced-order model is minimized and we show that the resulting system is a constrained optimal (in a sense to be specified) reduced-order model. The results of the paper are illustrated by means of a simple numerical example.

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“…However, the method in [35] does not use the observability gramian Q unlike the proposed method; both Z and V are rational Krylov subspaces. The main difference is that while the starting point for ISRK is the projection structure in (3.1) and (3.2), [35] uses the interpolation based first-order necessary conditions for the optimal H 2 model reduction problem [45] as a starting point and generates a reduced model satisfying these conditions. Moreover, even though an unstable reduced model has been observed extremely rarely, [35] does not guarantee stability since Q is not used in the reduction process.…”

Section: Remark 31mentioning

confidence: 99%

An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems

Gugercin

Proceedings of the 44th IEEE Conference on Decision and Control

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In this paper, we propose a model reduction algorithm for approximation of large-scale linear time-invariant dynamical systems. The method is a two-sided projection combining features of the singular value decomposition (SVD)-based and the Krylov-based model reduction techniques. While the SVD-side of the projection depends on the observability gramian, the Krylov-side is obtained via iterative rational Krylov steps. The reduced model is asymptotically stable, matches certain moments and solves a restricted H2 minimization problem. We present modifications to the proposed approach for employing low-rank gramians in the reduction step and also for reducing discrete-time systems. Several numerical examples from various disciplines verify the effectiveness of the proposed approach. It performs significantly better than the q-cover [60,59] and the least-squares [33] methods that have a similar projection structure to the proposed method. Also, in terms of both the H2 and H∞ error measures, its performance is comparable to or sometimes better than that of balanced truncation. Moreover, the method proves to be robust with respect to the perturbations due to usage of approximate gramians.

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Approximation of linear constant systems

Cited by 173 publications

References 2 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

Constrained optimal reduced-order models from input/output data

An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems

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