Dominant Subspaces of High-Fidelity Nonlinear Structured Parametric Dynamical Systems and Model Reduction

TL;DR

This paper introduces a model reduction method for high-fidelity nonlinear structured parametric dynamical systems using Volterra series, enabling efficient reduced models that preserve system structure and interpolate transfer functions at selected frequencies.

Contribution

It presents a novel approach combining Volterra series and dominant subspace extraction for structured nonlinear parametric systems, including delay systems, with efficient algorithms and numerical validation.

Findings

Reduced models accurately interpolate original transfer functions.

Method preserves system structure in reduced models.

Numerical benchmarks demonstrate efficiency and accuracy.

Abstract

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.