83 research outputs found
Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework
We extend the Adaptive Antoulas-Anderson (AAA) algorithm to develop a data-driven modeling framework for linear systems with quadratic output (LQO). Such systems are characterized by two transfer functions: one corresponding to the linear part of the output and another one to the quadratic part. We first establish the joint barycentric representations and the interpolation theory for the two transfer functions of LQO systems. This analysis leads to the proposed AAA-LQO algorithm. We show that by interpolating the transfer function values on a subset of samples together with imposing a least-squares minimization on the rest, we construct reliable data-driven LQO models. Two numerical test cases illustrate the efficiency of the proposed method
Interpolatory Weighted-H2 Model Reduction
This paper introduces an interpolation framework for the weighted-H2 model
reduction problem. We obtain a new representation of the weighted-H2 norm of
SISO systems that provides new interpolatory first order necessary conditions
for an optimal reduced-order model. The H2 norm representation also provides an
error expression that motivates a new weighted-H2 model reduction algorithm.
Several numerical examples illustrate the effectiveness of the proposed
approach
Interpolatory methods for model reduction of multi-input/multi-output systems
We develop here a computationally effective approach for producing
high-quality -approximations to large scale linear
dynamical systems having multiple inputs and multiple outputs (MIMO). We extend
an approach for model reduction introduced by Flagg,
Beattie, and Gugercin for the single-input/single-output (SISO) setting, which
combined ideas originating in interpolatory -optimal model
reduction with complex Chebyshev approximation. Retaining this framework, our
approach to the MIMO problem has its principal computational cost dominated by
(sparse) linear solves, and so it can remain an effective strategy in many
large-scale settings. We are able to avoid computationally demanding
norm calculations that are normally required to monitor
progress within each optimization cycle through the use of "data-driven"
rational approximations that are built upon previously computed function
samples. Numerical examples are included that illustrate our approach. We
produce high fidelity reduced models having consistently better
performance than models produced via balanced truncation;
these models often are as good as (and occasionally better than) models
produced using optimal Hankel norm approximation as well. In all cases
considered, the method described here produces reduced models at far lower cost
than is possible with either balanced truncation or optimal Hankel norm
approximation
An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations
We discuss model reduction for a particular class of quadratic-bilinear (QB)
descriptor systems. The main goal of this article is to extend the recently
studied interpolation-based optimal model reduction framework for QBODEs
[Benner et al. '16] to a class of descriptor systems in an efficient and
reliable way. Recently, it has been shown in the case of linear or bilinear
systems that a direct extension of interpolation-based model reduction
techniques to descriptor systems, without any modifications, may lead to poor
reduced-order systems. Therefore, for the analysis, we aim at transforming the
considered QB descriptor system into an equivalent QBODE system by means of
projectors for which standard model reduction techniques for QBODEs can be
employed, including aforementioned interpolation scheme. Subsequently, we
discuss related computational issues, thus resulting in a modified algorithm
that allows us to construct \emph{near}--optimal reduced-order systems without
explicitly computing the projectors used in the analysis. The efficiency of the
proposed algorithm is illustrated by means of a numerical example, obtained via
semi-discretization of the Navier-Stokes equations
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