Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

Frangos, Michalis; Jaimoukha, Imad M.

doi:10.3166/ejc.14.342-354

Cited by 23 publications

(18 citation statements)

References 32 publications

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“…Many existing model order reduction methods such as Padé approximation [11,28], balanced truncation [21], optimal Hankel norm [9,10] and Krylov subspace based methods In particular the Arnoldi algorithm [4,5,18,19] takes advantage of the sparsity of the large-scale model and has been extensively used for large problems; see [1,18,15]. When using block Krylov subspaces, one projects the system matrices of the original problem onto the subspace K m (A, B) = Range{B, AB, .…”

Section: (T) = a X(t) + B U(t) Y(t) = C X(t)mentioning

confidence: 99%

“…The use of rational Krylov spaces is recognized as a powerful tool within model order reduction techniques for linear dynamical systems, however its success has been hindered by the lack of a parameter-free procedure, which would effectively generate the sequence of shifts used to build the space. Major efforts have been devoted to this question in the recent years; see for example [6,7,8,18,20,22]. In the context of H 2 -optimality reduction, an interesting attempt to provide an automatic selection has been recently proposed in [13].…”

Section: (T) = a X(t) + B U(t) Y(t) = C X(t)mentioning

confidence: 99%

“…We recall that most model order reduction techniques, for example the moment-matching approaches, are based on the approximation of this transfer function; see [2,12,18]. If the number of state variables is very large, it would be very difficult to use the full system for simulation or run-on-time control.…”

Section: A New Adaptive Selection Of the Shiftsmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

Abidi¹,

Hached²,

Jbilou³

2016

ntmsci

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Abstract:In recent years, a great interest has been shown towards Krylov subspace techniques applied to model order reduction of large-scale dynamical systems. A special interest has been devoted to single-input single-output (SISO) systems by using moment matching techniques based on Arnoldi or Lanczos algorithms. In this paper, we consider multiple-input multiple-output (MIMO) dynamical systems and introduce the rational block Arnoldi process to design low order dynamical systems that are close in some sense to the original MIMO dynamical system. Rational Krylov subspace methods are based on the choice of suitable shifts that are selected a priori or adaptively. In this paper, we propose an adaptive selection of those shifts and show the efficiency of this approach in our numerical tests. We also give some new block Arnoldi-like relations that are used to propose an upper bound for the norm of the error on the transfer function.

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Section: (T) = a X(t) + B U(t) Y(t) = C X(t)mentioning

confidence: 99%

Section: (T) = a X(t) + B U(t) Y(t) = C X(t)mentioning

confidence: 99%

See 1 more Smart Citation

Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

Abidi¹,

Hached²,

Jbilou³

2016

ntmsci

View full text Add to dashboard Cite

show abstract

“…Some of them are based on Krylov subspace methods (moment matching) while others use balanced truncation; see [3,17,21] and the references therein. In particular, the Lanczos process has been used for the single-input and single-output (SISO) (the case p = 1) and MIMO dynamical systems; see [12,13,23,24] and the references therein. The standard version of the Lanczos algorithm builds reduced order models that poorly approximate some frequency dynamics and to overcome this problem, rational Krylov subspace methods have been developed these last years [11,14,15,19,20,29,31].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems

2015

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Mathematical modeling of complex electrical systems, has led us to linear mathematical models of higher order. Consequently, it is difficult to analyse and to design a control strategy of these systems. The order reduction is an important and effective tool to facilitate the handling and designing of a control strategy. In this paper we present, firstly, a reduction method which is based on the Krylov subspace and Lyapunov techniques, that we call Lyapunov-Global-Lanczos. This method minimizes the H∞ norm error, absolute error and preserves the stability of the reduced system. It also provides a better reduced system of order 1, with closer behaviour to the original system. This first order system is used to design PI (Proportional-Integral) controller. Secondly, we implement an adaptive digital PI controller in a microcontroller. It calculates the PI parameters in real time, referring to the error between the desired and measured outputs and the initial values of PI controller, that were determined from the first order system. Two simulation examples and a real time experimentation are presented to show the effectiveness of the proposed algorithms.

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“…Consider a class of descriptor linear dynamical electrical system in state space form given by [5], [6], [7], [8]:…”

Section: Introductionmentioning

confidence: 99%

Lyapunov-Global-Lanczos Algorithm for Model Order Reduction & Adaptive PI Controller of Large Scale Electrical Systems

2017

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Abstract-Mathematical modeling of complex electrical systems, has led us to linear mathematical models of higher order. Consequently, it is difficult to analyse and to design a control strategy of these systems. The order reduction is an important and effective tool to facilitate the handling and designing of a control strategy. In this paper we present, firstly, a reduction method which is based on the Krylov subspace and Lyapunov techniques, that we call Lyapunov-Global-Lanczos. This method minimizes the H∞ norm error, absolute error and preserves the stability of the reduced system. It also provides a better reduced system of order 1, with closer behaviour to the original system. This first order system is used to design PI (Proportional-Integral) controller. Secondly, we implement an adaptive digital PI controller in a microcontroller. It calculates the PI parameters in real time, referring to the error between the desired and measured outputs and the initial values of PI controller, that were determined from the first order system. Two simulation examples and a real time experimentation are presented to show the effectiveness of the proposed algorithms.

show abstract

Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

Cited by 23 publications

References 32 publications

Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems

Lyapunov-Global-Lanczos Algorithm for Model Order Reduction & Adaptive PI Controller of Large Scale Electrical Systems

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