Balanced Truncation Model Reduction of a Nonlinear Cable-Mass PDE System with Interior Damping

TL;DR

This paper develops a balanced truncation method for reducing a nonlinear cable-mass PDE system with interior damping, demonstrating its effectiveness through detailed numerical experiments for various parameters.

Contribution

It introduces a balanced truncation approach for nonlinear PDE systems with interior damping and provides numerical validation of its accuracy.

Findings

ROM accurately approximates system displacement and velocity

Method performs well across different parameter sets

Provides insights into nonlinear model reduction stability

Abstract

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.