# Linear time-periodic dynamical systems: An H2 analysis and a model reduction framework

## TL;DR

This paper introduces a new model reduction framework for linear time-periodic systems using an extended H2 analysis, enabling efficient simulation of complex systems in fluid dynamics, electronics, and mechanics.

## Contribution

It generalizes H2 optimal model reduction methods from LTI to LTP systems, providing an algorithm that preserves periodic structure and includes an a posteriori error bound.

## Key findings

- Successfully applied to numerical examples demonstrating effectiveness.
- Retains periodic structure in reduced models.
- Provides an a posteriori error bound for model accuracy.

## Abstract

Linear time-periodic (LTP) dynamical systems frequently appear in the modeling of phenomena related to fluid dynamics, electronic circuits, and structural mechanics via linearization centered around known periodic orbits of nonlinear models. Such LTP systems can reach orders that make repeated simulation or other necessary analysis prohibitive, motivating the need for model reduction. We develop here an algorithmic framework for constructing reduced models that retains the linear time-periodic structure of the original LTP system. Our approach generalizes optimal approaches that have been established previously for linear time-invariant (LTI) model reduction problems. We employ an extension of the usual H2 Hardy space defined for the LTI setting to time-periodic systems and within this broader framework develop an a posteriori error bound expressible in terms of related LTI systems. Optimization of this bound motivates our algorithm. We illustrate the success of our method on two numerical examples.