Empirical least-squares fitting of parametrized dynamical systems

TL;DR

This paper introduces a two-phase empirical method for fitting low-order parametrized dynamical systems from response data, combining high-fidelity intermediate modeling with efficient model reduction.

Contribution

It extends non-parametric least-squares fitting to parametrized systems and integrates $ ext{ extbf{H}}_2$-optimal reduction for compact models, addressing practical constraints.

Findings

High-fidelity models achieved with low order

Effective reduction of model complexity

Numerical examples demonstrate accuracy and efficiency

Abstract

Given a set of response observations for a parametrized dynamical system, we seek a parametrized dynamical model that will yield uniformly small response error over a range of parameter values yet has low order. Frequently, access to internal system dynamics or equivalently, to realizations of the original system is either not possible or not practical; only response observations over a range of parameter settings might be known. Respecting these typical operational constraints, we propose a two phase approach that first encodes the response data into a high fidelity intermediate model of modest order, followed then by a compression stage that serves to eliminate redundancy in the intermediate model. For the first phase, we extend non-parametric least-squares fitting approaches so as to accommodate parameterized systems. This results in a (discrete) least-squares problem formulated with respect to both frequency and parameter that identifies "local" system response features. The second phase uses an $\mathbf{\mathcal{H}}_2$-optimal model reduction strategy accommodating the specialized parametric structure of the intermediate model obtained in the first phase. The final compressed model inherits the parametric dependence of the intermediate model and maintains the high fidelity of the intermediate model, while generally having dramatically smaller system order. We provide a variety of numerical examples demonstrating our approach.