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Solving LargeScale Control Problems
, 2004
"... Sparsity and parallel algorithms: two approaches to beat the curse of dimensionality. By Peter Benner I n this article we discuss sparse matrix algorithms and parallel algorithms, as well as their application to largescale systems. For illustration, we solve the linearquadratic regulator (LQR) pro ..."
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Cited by 54 (26 self)
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Sparsity and parallel algorithms: two approaches to beat the curse of dimensionality. By Peter Benner I n this article we discuss sparse matrix algorithms and parallel algorithms, as well as their application to largescale systems. For illustration, we solve the linearquadratic regulator (LQR) problem and apply balanced truncation model reduction using either parallel computing or sparse matrix algorithms. We conclude that modern tools from numerical linear algebra, along with careful investigation and exploitation of the problem structure, can be used to derive algorithms capable of solving large control problems. Since these approaches are implemented in productionquality software, control engineers can employ complex models and use computational tools to analyze and design feedback control laws. Background
Perturbation analysis for the eigenvalue problem of a formal product of matrices
 BIT
"... We study the perturbation theory for the eigenvalue problem of a formal matrix product A s1 1 · · · Asp p, where all Ak are square and sk ∈ {−1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eige ..."
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Cited by 21 (7 self)
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We study the perturbation theory for the eigenvalue problem of a formal matrix product A s1 1 · · · Asp p, where all Ak are square and sk ∈ {−1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skewHamiltonian pencils. AMS subject classification: 65F15, 93B40, 93B60, 65H17.
Model reduction based on spectral projection methods
 Dimension Reduction of LargeScale Systems
, 2005
"... We discuss the efficient implementation of model reduction methods such as modal truncation, balanced truncation, and other balancingrelated truncation techniques, employing the idea of spectral projection. Mostly, we will be concerned with the sign function method which serves as the major computa ..."
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Cited by 13 (6 self)
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We discuss the efficient implementation of model reduction methods such as modal truncation, balanced truncation, and other balancingrelated truncation techniques, employing the idea of spectral projection. Mostly, we will be concerned with the sign function method which serves as the major computational tool of most of the discussed algorithms for computing reducedorder models. Implementations for largescale problems based on parallelization or formatted arithmetic will also be discussed. This chapter can also serve as a tutorial on Gramianbased model reduction using spectral projection methods. 1
SkewHamiltonian and Hamiltonian eigenvalue problems: Theory, algorithms and applications
 Proceedings of ApplMath03, Brijuni (Croatia
"... SkewHamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computation ..."
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Cited by 12 (4 self)
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SkewHamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skewHamiltonian eigenproblems are briefly described.
Structured condition numbers for invariant subspaces
 SIAM J. Matrix Anal. Appl
, 2005
"... Abstract. Invariant subspaces of structured matrices are sometimes better conditioned with respect to structured perturbations than with respect to general perturbations. Sometimes they are not. This paper proposes an appropriate condition number cS, for invariant subspaces subject to structured per ..."
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Cited by 11 (2 self)
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Abstract. Invariant subspaces of structured matrices are sometimes better conditioned with respect to structured perturbations than with respect to general perturbations. Sometimes they are not. This paper proposes an appropriate condition number cS, for invariant subspaces subject to structured perturbations. Several examples compare cS with the unstructured condition number. The examples include block cyclic, Hamiltonian, and orthogonal matrices. This approach extends naturally to structured generalized eigenvalue problems such as palindromic matrix pencils. Key words. Structured eigenvalue problem, invariant subspace, perturbation theory, condition
Perturbation bounds for isotropic invariant subspaces of skewHamiltonian matrices
 SIAM J. Matrix Anal. Appl
"... Abstract. We investigate the behavior of isotropic invariant subspaces of skewHamiltonian matrices under structured perturbations. It is shown that finding a nearby subspace is equivalent to solving a certain quadratic matrix equation. This connection is used to derive meaningful error bounds and c ..."
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Cited by 6 (4 self)
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Abstract. We investigate the behavior of isotropic invariant subspaces of skewHamiltonian matrices under structured perturbations. It is shown that finding a nearby subspace is equivalent to solving a certain quadratic matrix equation. This connection is used to derive meaningful error bounds and condition numbers that can be used to judge the quality of invariant subspaces computed
Structured Eigenvalue Problems
, 2005
"... Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetrie ..."
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Cited by 3 (1 self)
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Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skewsymmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices.
The sensitivity of computational control problems  What factors contribute to the accurate and efficient numerical solution of problems in control systems analysis and design?
 IEEE CONTROL SYST. MAG
, 2004
"... Although numerical methods have been used for many centuries to solve problems in science and engineering, the importance of computation grew tremendously with the advent of digital computers. It became immediately clear that many of the classical analytical and numerical methods and algorithms cou ..."
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Although numerical methods have been used for many centuries to solve problems in science and engineering, the importance of computation grew tremendously with the advent of digital computers. It became immediately clear that many of the classical analytical and numerical methods and algorithms could not be implemented directly as computer codes, although they were well suited for hand computations. What was the reason? When doing computations by hand a person can choose the accuracy of each elementary calculation and then estimate, based on intuition and experience, its influence on the final result. In contrast, when computations are done automatically, intuitive error control is usually not possible and the effect of errors on the intermediate calculations must be estimated in a more systematic way. Due to this observation, starting
Random Aspects of Beam Physics and LaserPlasma Interactions
, 2007
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let In denote the n × n identity matrix
"... Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or cl ..."
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Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle. We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability as well as the index of stability for these subspaces.