R-3 Repository :: Browsing by Author "Antoulas, A.C."

Browsing by Author "Antoulas, A.C."

Now showing 1 - 10 of 10

A Modified Low-Rank Smith Method for Large-Scale Lyapunov Equations
(2001-05) Antoulas, A.C.; Sorensen, D.C.; Gugercin, S.
In this note we present a modified cyclic low-rank Smith method to compute low-rank approximations to solutions of Lyapunov equations arising from large-scale dynamical systems. Unlike the original cyclic low-rank Smith method introduced by Penzl in [18], the number of the columns in the approximate solutions does not necessarily increase at each step. The number of columns required by the modified method is usually much lower than the original cyclic low-rank Smith method. The modified method never requires more columns than the original. Upper bounds are established for the errors in the low-rank approximate solutions and also for the errors in the resulting approximate Hankel singular values. Numerical results are given to verify the efficiency and accuracy of the new algorithm.
A Survey of Model Reduction Methods for Large-Scale Systems
(2000-12) Antoulas, A.C.; Sorensen, D.C.; Gugercin, S.
An overview of model reduction methods and a comparison of the resulting algorithms are presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better.
Approximation of large-scale dynamical systems: An Overview
(2001-02) Antoulas, A.C.; Sorensen, D.C.
In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.
Data-Driven Model Order Reduction of Linear Switched Systems in the Loewner Framework
(Society for Industrial and Applied Mathematics, 2018) Gosea, I.V.; Petreczky, M.; Antoulas, A.C.
The Loewner framework for model reduction is extended to the class of linear switched systems. One advantage of this framework is that it introduces a trade-off between accuracy and complexity. Moreover, through this procedure, one can derive state-space models directly from data which is related to the input-output behavior of the original system. Hence, another advantage of the framework is that it does not require the initial system matrices. More exactly, the data used in this framework consists in frequency domain samples of input-output mappings of the original system. The definition of generalized transfer functions for linear switched systems resembles the one for bilinear systems. A key role is played by the coupling matrices, which ensure the transition from one active mode to another.
Gramians of Structured Systems and an Error Bound for Structure-Preserving Model Reduction
(2004-09) Sorensen, D.C.; Antoulas, A.C.
In this paper a general framework is posed for defining the reachability and controllability gramians of structured linear dynamical systems. The novelty is that a formula for the gramian is given in the frequency domain. This formulation is surprisingly versatile and may be applied in a variety of structured problems. Moreover, this formulation enables a rather straightforward development of apriori error bounds for model reduction in the H2 norm. The bound applies to a reduced model derived from projection onto the dominant eigenspace of the appropriate gramian. The reduced models are structure preserving because they arise as direct reduction of the original system in the reduced basis. A derivation of the bound is presented and verified computationally on a second order system arising from structural analysis.
Lyapunov, Lanczos, and Inertia
(2000-05) Antoulas, A.C.; Sorensen, D.C.
We present a new proof of the inertia result associated with Lyapunov equations. Furthermore we present a connection between the Lyapunov equation and the Lanczos process which is closely related to the Schwarz form of a matrix. We provide a method for reducing a general matrix to Schwarz form in a finite number of steps (O(n3)). Hence, we provide a finite method for computing inertia without computing eigenvalues. This scheme is unstable numerically and hence is primarily of theoretical interest.
Model reduction by iterative error system approximation
(Taylor & Francis, 2018) Antoulas, A.C.; Benner, Peter; Feng, Lihong
The analysis of a posteriori error estimates used in reduced basis methods leads to a model reduction scheme for linear time-invariant systems involving the iterative approximation of the associated error systems. The scheme can be used to improve reduced-order models (ROMs) with initial poor approximation quality at a computational cost proportional to that for computing the original ROM. We also show that the iterative approximation scheme is applicable to parametric systems and demonstrate its performance using illustrative examples.
On the Decay Rate of Hankel Singular Values and Related Issues
(2001-05) Antoulas, A.C.; Sorensen, D.C.; Zhou, Y.
This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues like the Hankel singular values, are decaying rapidly. This fact has motivated the development of several algorithms for computing low rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low rank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions.
Projection Methods for Balanced Model Reduction
(2001-03) Sorensen, D.C.; Antoulas, A.C.
The purpose of this paper is to investigate projection methods for the iterative computation of partially balanced reduced order systems. This approach is both completely automatic once an error tolerance is specified and yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Padé via Lanczos) and rational Krylov, which do not satisfy these properties. Our approach is based on the computation and appoximation of certain system grammians and in particular on the cross grammian.
State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories
(SIAM, 2016) Rapisarda, P.; Antoulas, A.C.
We solve two problems in modeling polynomial vector-exponential trajectories dependent on two independent variables. In the first one we assume that the data-generating system has no inputs, and we compute a state representation of the most powerful unfalsified model for this data. In the second instance we assume that the data-generating system is controllable and quarter-plane causal, and we compute a Roesser input-state-output model. We provide procedures for solving these identification problems, both based on the factorization of constant matrices directly constructed from the data, from which state trajectories can be computed.

Browsing by Author "Antoulas, A.C."

Results Per Page

Sort Options