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Krylov Projection Methods For Model Reduction
, 1997
"... This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov p ..."
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This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczosbased methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete modelreduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multipleinput multipleoutput systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.
Padé Approximation Of LargeScale Dynamic Systems With Lanczos Methods
, 1994
"... The utility of Lanczos methods for the approximation of largescale dynamical systems is considered. In particular, it is shown that the Lanczos method is a technique for yielding Pad'e approximants which has several advantages over more traditional explicit moment matching approaches. An exten ..."
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Cited by 33 (1 self)
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The utility of Lanczos methods for the approximation of largescale dynamical systems is considered. In particular, it is shown that the Lanczos method is a technique for yielding Pad'e approximants which has several advantages over more traditional explicit moment matching approaches. An extension of the Lanczos algorithm is developed for computing multipoint Pad'e approximations of descriptor systems. Keywords: Dynamic system, Pad'e approximation, Lanczos algorithm, model reduction. 1. Introduction This paper explores the use of Lanczos techniques for the reducedorder modeling and simulation of largescale, SISO dynamical systems. One can define such a system through the set of state space equations ae E x(t) = Ax(t) + bu(t) y(t) = cx(t) + du(t): (1) The scalar functions u(t) and y(t) are the system's input and output while x(t) is the state vector of dimension n. For simplicity, the directcoupling term, d, will be assumed to be zero. The system matrix, A 2 R n\Thetan ...
The Lanczos algorithm and Padé approximations
 Short Course, Benelux Meeting on Systems and Control
, 1995
"... Introduction In these two lectures we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. These notes are based on material in the papers [10, 17, 11, 12] for which a lot of credit ought to be ..."
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Cited by 10 (0 self)
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Introduction In these two lectures we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. These notes are based on material in the papers [10, 17, 11, 12] for which a lot of credit ought to be given to the respective coauthors. For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case. Let a nth order dynamical system be described by x = Ax + bu (1) y = cx + du (2) where A is a square, b is a column vector, c is a row vector, and d is a scalar. It is wellknown that the transfer function of this system : h(s) = c(sI \Gamma A) \Gamma1 b +<F29
On some modifications of the Lanczos algorithm and the relation with Padé approximations
, 1995
"... In this paper we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. 1 1 Introduction For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case ..."
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Cited by 5 (0 self)
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In this paper we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. 1 1 Introduction For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case. Let a nth order dynamical system be described by x = Ax + bu (1.1) y = cx + du (1.2) where A is a square, b is a column vector, c is a row vector, and d is a scalar. It is wellknown that the transfer function of this system : h(s) = c(sI \Gamma A) \Gamma1 b + d has a Taylor expansion around s = 1 that looks like : h(s) = d + cbs \Gamma1 + cAbs \Gamma2 + cA 2 bs \Gamma3 + cA 3 bs \Gamma4 + : : : The coefficients m \Gammai of the powers of s \Gammai satisfy thus m 0 = d ; m \Gammai = cA i\Gamma1 b ; i 1: For i 1 these are also called moments or Markov parameters of the system fA; b; cg. It follows already from the work of Hankel that the first 2n moments 1 To appea...
Rational approximations of prefiltered transfer functions via the Lanczos algorithm
, 1999
"... this paper trivially extends to the complex case. As described in the introduction, we are interested in approximating this transfer function by another one of lower degree and such that both match in their first 2k terms (called moments) of an expansion around s = #. Since prefiltering amounts to p ..."
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this paper trivially extends to the complex case. As described in the introduction, we are interested in approximating this transfer function by another one of lower degree and such that both match in their first 2k terms (called moments) of an expansion around s = #. Since prefiltering amounts to premultiplying a transfer function, the following lemma will be useful. Lemma 1 (see [11] for a proof) Let {A i , B i , C i , D i } for i = 1, 2 be realizations of transfer functions R i (s) then a realization for the product R 1 (s)R 2 (s) is specified by {A, B, C, D} where R 1 (s)R 2 (s) # # A B C D # = # # # # A 1 B 1 C 2 B 1 D 2 0 A 2 B 2 C 1 D 1 C 2 D 1 D 2 # # # # = # # # # A 1 B 1 I n 2 C 1 D 1 # # # # # # # # I n 1 A 2 B 2 C 2 D 2 # # # # . (12) The realization quadruple can thus be obtained by a mere product of "expanded " realizations and we can expect sparsity to be preserved to a large extent, especially in the case that B i and C i are vectors and D i scalars. As an example 4 K. Gallivan and P. Van Dooren / Prefiltering rational approximations let us take the singleinput singleoutput (SISO) case and let us choose # = 1, i.e. f(s) = (s  z)/(s  p). A realization for f(s) is easily seen to be f(s) # # A 1 b 1 c 1 d 1 # = # p 1 p  z 1 # . (13) Assume g(s) also scalar and g(#) = 0, then g(s) # # A 2 b 2 c 2 0 # (14) and the following realization for f(s)g(s) is obtained from Lemma 1 : f(s)g(s) # # A b c 0 # = # # # # p c 2 0 0 A 2 b 2 p  z c 2 0 # # # # , (15) which clearly preserves sparsity. We show later on that this simple case of a first degree filter f(s) is not restrictive since we can repeat the same argument to add on additional first order prefilters and build recursively one of arbitrary degree #. How can this ...