### 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 language | English (US) |
---|---|

Pages (from-to) | 2089-2096 |

Number of pages | 8 |

Journal | Proceedings of the American Control Conference |

Volume | 3 |

State | Published - 2003 |

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### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering

### Cite this

*Proceedings of the American Control Conference*,

*3*, 2089-2096.

}

*Proceedings of the American Control Conference*, vol. 3, pp. 2089-2096.

**Computation of Empirical Eigenfunctions and Order Reduction for Control of Time-Dependent Parabolic PDEs.** / Armaou, Antonios; Dubljevic, Stevan; Christofides, Panagiotis D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Computation of Empirical Eigenfunctions and Order Reduction for Control of Time-Dependent Parabolic PDEs

AU - Armaou, Antonios

AU - Dubljevic, Stevan

AU - Christofides, Panagiotis D.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

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M3 - Article

VL - 3

SP - 2089

EP - 2096

JO - Proceedings of the American Control Conference

JF - Proceedings of the American Control Conference

SN - 0743-1619

ER -