• Corpus ID: 17778728

A Rational Krylov Iteration for Optimal H 2 Model Reduction

  title={A Rational Krylov Iteration for Optimal H 2 Model Reduction},
  author={Serkan Gugercin and Athanasios C. Antoulas and Christopher A. Beattie},
In the sequel, we will construct the reduced order models Gr(s) through Krylov projection methods. Toward this end, we construct matrices V ∈ R and Z ∈ R that span certain Krylov subspaces with the property that Z V = Ir. The reduced order model Gr(s) will then be obtained as Ar = Z T AV, Br = Z T B, and Cr = CV. (2) The corresponding oblique projection is given by V Z . Many researchers have worked on the problem (1); see [18], [16], [7], [5], [3], [17], [4] and references therein. Since… 

Figures from this paper

Realization-independent H 2-approximation
Iterative Rational Krylov Algorithm (IRKA) of [11] is an effective tool for tackling the H2-optimal model reduction problem. However, so far it has relied on a first-order state-space realization of
Realization-independent ℌ2-approximation
  • C. Beattie, S. Gugercin
  • Mathematics, Computer Science
    2012 IEEE 51st IEEE Conference on Decision and Control (CDC)
  • 2012
A Loewner-matrix approach for interpolation is employed, and a new formulation of IRKA is developed that only uses transfer function evaluations, without requiring any particular realization, to be extended to H2 approximation of irrational, infinite-dimensional dynamical systems.
On the ADI method for the Sylvester Equation and the optimal-$\mathcal{H}_2$ points
This paper shows that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods, and calls these shifts pseudo H"2-optimal shifts, which are optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to therational Krylov projection subspace.
On the ADI method for the Sylvester Equation and the optimal-H 2 points
The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equation. In this paper we show that the ADI and
Rational Krylov decompositions : theory and applications
Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. In this thesis we study rational Krylov spaces by considering rational Krylov
An algorithm for model reduction of large-scale systems via equality constrained least squares
  • Yu'e An, Chuanqing Gu
  • Mathematics, Computer Science
    2010 Sixth International Conference on Natural Computation
  • 2010
A new SVD-Krylov based method is proposed, which is equivalent to compute an equality constrained least-squares problem and is numerically effective and suited for large-scale problem, which can be verified in the numerical examples.
Control and Cybernetics Iterative-interpolation Algorithms for L 2 Model Reduction *
This paper is concerned with the construction of reduced–order models for high–order linear systems in such a way that the L2 norm of the impulse–response error is minimized. Two convergent
Méthodes de type Lanczos rationnel pour la réduction de modèles
Numerical solution of dynamical systems have been a successful means for studying complex physical phenomena. However, in large-scale setting, the system dimension makes the computations infeasible
Comparison of Model Reduction Methods with Applications to Circuit Simulation
Summary. We compare different model reduction methods applied to the dynamical system of a coupled transmission line: balanced truncation (BT), truncation by balancing one gramian (or PMTBR - poor
The RKFIT Algorithm for Nonlinear Rational Approximation
This paper derives a strategy for the degree reduction of the approximants, as well as methods for their conversion to partial fraction form, for the efficient evaluation, and root-finding, and puts RKFIT into a general framework.


An approximate approach to H2 optimal model reduction
The problem is formulated as that of minimizing the H/sub 2/ model-reduction cost over the Stiefel manifold so that the stability constraint on reduced-order models is automatically satisfied and thus totally avoided in the new problem formulation.
Krylov Projection Methods for Model Reduction
The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation, based on which three algorithms for model reduction are proposed, which are suited for parallel or approximate computations.
The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore
First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an
Second-order algorithm for optimal model order reduction
A second-order algorithm is given for optimal model order reduction of continuous single-input/single-output systems. Given a transfer function of order n, it finds the transfer function of a model
A new algorithm for L2 optimal model reduction
It is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and, on this basis, a two-step iterative algorithm is developed combining a least squares solver with a gradient minimizer.
Lectures on complex approximation
I: Approximation by Series Expansions and by Interpolation.- I. Representation of complex functions by orthogonal series and Faber series.- 1. The Hilbert space L2(G).- A. Definition of L2(G).- B.
Rational L/sub 2/ approximation: a non-gradient algorithm
The problem of determining the best rational approximant of a given rational transfer function of higher order according to the L/sub 2/-norm criterion is considered. An efficient algorithm is
Solving large-scale control problems
  • P. Benner
  • Computer Science
    IEEE Control Systems
  • 2004
It is concluded 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.
Efficient numerical solution of the LQR‐problem for the heat equation
We discuss how the theory developed by Banks and Kunisch can be applied to a modified version of a controlled heat transfer model introduced by Troltzsch and Unger. In the numerical implementation we
A comparative study of 7 algorithms for model reduction
  • S. Gugercin, A. Antoulas
  • Mathematics
    Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
  • 2000
Compares seven model reduction algorithms by applying them to four different dynamical systems. There are four singular value decomposition (SVD) based methods, and three moment matching based