Reduction Of Large-scale Systems Research Articles

Hyperbolic geometrical approach to model reduction

Model reduction of large scale systems is an actively researched area of modelling and control. The problem is more involved if uncertainties are also present and a computational tractable nominal model is needed for the design. Based on results of the Kolmogorov n-width theory the paper provides useful bounds for the worst case approximation error - both H2 and H∞ – in terms of the hyperbolic distance related to the sets of uncertain poles. A related model reduction strategy that uses only this a priori pole information is also proposed. The method is illustrated through numerical examples.

Reduction of large-scale dynamical systems by the Krylov subspaces method: Analysis of approaches

Approaches and numerical algorithms for reducing the order of mathematical models of multidimensional dynamical systems that are based on the Krylov's subspaces method are described. To calculate matrices representing the reduced models in the state space, Lanczos' and Arnoldi's methods are used. Practical examples of the reduction of large-scale systems are presented.

Stability Preservation in Projection-based Model Order Reduction of Large Scale Systems

In this paper, a new framework to address the stability preservation problem in projection-based model order reduction is presented. Sufficient conditions for obtaining stable reduced models are established and proven by using the notions of contractivity and matrix measure. Based on these results, we present two model reduction algorithms that preserve stability using any orthogonal projection. In addition, we show that for some system classes, stability preservation can be guaranteed just by choosing a suitable state space representation, which is applicable to the large class of models obtained by the Finite Element Method (FEM).

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

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 [A. Yousouff, R.E. Skelton, Covariance equivalent realizations with applications to model reduction of large-scale systems, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 22, Academic Press, 1985, pp. 273–348; A. Yousouff, D.A. Wagie, R.E. Skelton, Linear system approximation via covariance equivalent realizations, Journal of Math. Anal. and Appl. 196 (1985) 91–115] and the least-squares [S. Gugercin, A.C. Antoulas, Model reduction of large scale systems by least squares, Linear Algebra Appl. 415(2–3) (2006) 290–321] 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.

A Mixed Method for Large-Scale Systems Modelling using Eigen Spectrum Analysis and Cauer Second Form

The authors present a mixed method for the reduction of large-scale systems which combines the Eigen spectrum analysis with the Cauer second form. The denominator of the reduced order model/low order system is found by Eigen spectrum analysis while the numerator dynamics is determined by Cauer second form. This method guarantees stability of the reduced model if the original high order system is stable. The method is illustrated by numerical examples.

REDUCING SECOND ORDER SYSTEMS BY AN INTEGRATED STATE SPACE AND BACK CONVERSION PROCEDURE

The Arnoldi algorithm is modified to match the first Markov parameter and some of the first moments in order reduction of large scale systems while preserving the properties of the standard Arnoldi algorithm: an upper Hessenberg matrix as a coefficient of Xr, the identity matrix as a coefficient of xr and a multiple of the first unit vector as the input vector. These properties are then used for the reduction of large scale second order models by first reducing in state space and then converting into second order form by introducing a numerical algorithm.

Stopping Criterion in Order Reduction of Large Scale Systems Using Krylov Subspace Methods

AbstractWe show that order reduction by a two‐sided Krylov subspace method is equivalent to an approximate TBR in the sense of Galerkin type conditions. The Hankel Singular Values of the original system are thereby approximated by the Hankel singular values of the Krylov reduced model which are used as a stopping criterion to find a suitable order of the reduced model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Efficient model reduction of large scale systems using Krylov-subspace iterative methods

In this work, the use of Krylov-subspace iterative methods is examined to obtain low-rank approximate solutions of Lyapunov equations for use in contol-oriented model reduction of large space structures. In particular, we examine the application of these methods to the approximate component cost analysis (CCA) of large scale systems. It is shown that using the proposed methods approximate CCA reduced-order models can be obtained with a significant reduction in the computational effort and time. In addition, we derive closed-form expressions for approximate cost-equivalent, cost-decoupled and covariance-equivalent realizations using the proposed Krylov-subspace solutions.

A Simplified CFE Method for Large-Scale Systems Modelling about s = 0 and s = a

The authors present a new procedure for the reduction of large-scale systems by Continued-Fraction Expansion (CFE) about s = 0 and s = a. The method is simple and useful as it avoids necessity of application of reciprocal transformation and need to evaluate the time moments of the original system before hand. The method only needs the application of linear transformation and inverse linear transformation at a proposed points s = a. The method also overcomes the ambiguity in the selection of 'a' and always generates unique stable low order models for stable high order systems unlike other methods. The proposed method is computationally superior to many of the existing methods. Typical numerical examples illustrating the proposed method are provided.

A note on model reduction of large scale systems

A note on model reduction of large scale systems

On model reduction of large-scale systems

On model reduction of large-scale systems