Computation of Empirical Eigenfunctions and Order Reduction for Control of Time-Dependent Parabolic PDEs

Antonios Armaou, Stevan Dubljevic, Panagiotis D. Christofides

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

This article presents a methodology for the computation of empirical eigenfunctions and the construction of accurate low-dimensional approximations for control of nonlinear and time-dependent parabolic partial differential equations (PDE) systems. Initially, a representative (with respect to different initial conditions and input perturbations) ensemble of solutions of the time-varying PDE system is constructed by computing and solving a high-order discretization of the PDE. Then, the Karhunen-Loéve expansion is directly applied to the ensemble of solutions to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within a Galerkin's model reduction framework to derive low-order ordinary differential equation ODE systems that accurately describe the dominant dynamics of the PDE system. The method is applied to a diffusion-reaction process with nonlinearities, spatially-varying coefficients and time-dependent spatial domain, and is shown to lead to the construction of accurate low-order models and the synthesis of low-order controllers. The robustness of the predictions of the low-order models with respect to variations in the model parameters and different initial conditions, as well as the comparison of their performance with respect to low-order models which were constructed by using off-the-self basis function sets are successfully shown through computer simulations.

Original languageEnglish (US)
Pages (from-to)2089-2096
Number of pages8
JournalProceedings of the American Control Conference
Volume3
StatePublished - 2003

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Eigenvalues and eigenfunctions
Partial differential equations
Robustness (control systems)
Ordinary differential equations
Controllers
Computer simulation

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering

Cite this

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title = "Computation of Empirical Eigenfunctions and Order Reduction for Control of Time-Dependent Parabolic PDEs",
abstract = "This article presents a methodology for the computation of empirical eigenfunctions and the construction of accurate low-dimensional approximations for control of nonlinear and time-dependent parabolic partial differential equations (PDE) systems. Initially, a representative (with respect to different initial conditions and input perturbations) ensemble of solutions of the time-varying PDE system is constructed by computing and solving a high-order discretization of the PDE. Then, the Karhunen-Lo{\'e}ve expansion is directly applied to the ensemble of solutions to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within a Galerkin's model reduction framework to derive low-order ordinary differential equation ODE systems that accurately describe the dominant dynamics of the PDE system. The method is applied to a diffusion-reaction process with nonlinearities, spatially-varying coefficients and time-dependent spatial domain, and is shown to lead to the construction of accurate low-order models and the synthesis of low-order controllers. The robustness of the predictions of the low-order models with respect to variations in the model parameters and different initial conditions, as well as the comparison of their performance with respect to low-order models which were constructed by using off-the-self basis function sets are successfully shown through computer simulations.",
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Computation of Empirical Eigenfunctions and Order Reduction for Control of Time-Dependent Parabolic PDEs. / Armaou, Antonios; Dubljevic, Stevan; Christofides, Panagiotis D.

In: Proceedings of the American Control Conference, Vol. 3, 2003, p. 2089-2096.

Research output: Contribution to journalArticle

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