Low-dimensional linearized models for systems with periodic orbits, with application to the Ginzburg-Landau equation

Zhanhua Ma, Clarence W. Rowle

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

We introduce a general procedure for obtaining a low-dimensional linear time-periodic model from a very high-dimensional nonlinear system that has an asymptotically stable periodic orbit. Our goal is to develop models that are suitable for designing feedback con-trollers for fluids systems with periodic orbits, such as periodically shedding wakes, or flow control problems where periodic actuation is introduced. In our method, we first linearize the nonlinear system about its asymptotically stable periodic orbit. We then compute a projection to project out the one-dimensional neutrally stable eigenspace appearing in the linear model corresponding to perturbations along the direction of the periodic orbit. Finally, we apply the method of snapshot-based balanced truncation for the high-dimensional linear periodic system to obtain a reduced-order model. We illustrate the method by developing reduced-order models for the complex Ginzburg-Landau equation.

Original languageEnglish (US)
Title of host publication4th AIAA Flow Control Conference
PublisherAmerican Institute of Aeronautics and Astronautics Inc.
ISBN (Print)9781563479427
DOIs
StatePublished - 2008

Publication series

Name4th AIAA Flow Control Conference

All Science Journal Classification (ASJC) codes

  • Fluid Flow and Transfer Processes
  • Aerospace Engineering
  • Control and Systems Engineering

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    Ma, Z., & Rowle, C. W. (2008). Low-dimensional linearized models for systems with periodic orbits, with application to the Ginzburg-Landau equation. In 4th AIAA Flow Control Conference [2008-4196] (4th AIAA Flow Control Conference). American Institute of Aeronautics and Astronautics Inc.. https://doi.org/10.2514/6.2008-4196