Abstract
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.
Highlights
Model order reduction (MOR) is used to approximate large-scale dynamical systems with smaller ones that ideally have similar response characteristics to the original
In our formulation, data correspond to frequency domain samples of the input/output mapping of the underlying linear systems with quadratic output (LQO) system, in the form of samples of its two transfer functions: the first transfer function being a single-variable one and the second a bivariate one
For the LQO system (2.1), the nonlinearity is present in the state-to-output equation only and one can write the input-output mapping of the system in the frequency domain using two transfer functions: (i) one corresponding to the linear part of the output, i.e., y1(t) = cT x(t) and (ii) one corresponding to the quadratic part of the output, i.e., y2(t) = K(x(t) ⊗ x(t))
Summary
Model order reduction (MOR) is used to approximate large-scale dynamical systems with smaller ones that ideally have similar response characteristics to the original. This has been an active research area and many approaches to MOR have been proposed. We refer the reader to [1, 3, 6, 9, 37, 39] and the references therein for an overview of MOR methods for both linear and nonlinear dynamical systems. MOR, as the name implies, assumes access to a full order model to be reduced; in most cases, in the form of a state-space formulation obtained via, e.g., a spatial discretization of the underlying partial differential equations.
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