# A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem

Kybernetika (2009)

• Volume: 45, Issue: 3, page 427-444
• ISSN: 0023-5954

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## Abstract

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The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.

## How to cite

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Rehák, Branislav, et al. "A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem." Kybernetika 45.3 (2009): 427-444. <http://eudml.org/doc/37676>.

@article{Rehák2009,
abstract = {The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.},
author = {Rehák, Branislav, Čelikovský, Sergej, Ruiz, Javier, Orozco-Mora, Jorge},
journal = {Kybernetika},
keywords = {nonlinear output regulation; singularly perturbed equation; gyroscope; nonlinear output regulation; singularly perturbed equation; gyroscope},
language = {eng},
number = {3},
pages = {427-444},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem},
url = {http://eudml.org/doc/37676},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Rehák, Branislav
AU - Čelikovský, Sergej
AU - Ruiz, Javier
AU - Orozco-Mora, Jorge
TI - A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 3
SP - 427
EP - 444
AB - The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.
LA - eng
KW - nonlinear output regulation; singularly perturbed equation; gyroscope; nonlinear output regulation; singularly perturbed equation; gyroscope
UR - http://eudml.org/doc/37676
ER -

## References

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20. Asymptotic Expansions of Solutions of Singularly Perturbed Equations (in Russian), Nauka, Moscow 1973. MR0477344

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