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

Branislav Rehák; Sergej Čelikovský; Javier Ruiz; Jorge Orozco-Mora

Kybernetika (2009)

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

Abstract

top
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

top

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

top
  1. Structurally stable output regulation of nonlinear systems, Automatica 33 (1997), 369–285. MR1442555
  2. Output regulation problem with nonhyperbolic zero dynamics: a FEMLAB-based approach, In: Proc. 2nd IFAC Symposium on System, Structure and Control 2004, Oaxaca 2004, pp. 700–705. 
  3. FEMLAB-based error feedback design for the output regulation problem, In: Proc. Asian Control Conference 2006. Bandung 2006. 
  4. Nonlinear inversion-based output tracking, IEEE Trans. Automat. Control 41 (1996), 930–942. Zbl0859.93006MR1398777
  5. The internal model principle of control theory, Automatica 12 (1976), 457–465. MR0429257
  6. The linear multivariable regulator problem, SIAM J. Control Optim. 15 (1977), 486–505. MR0446631
  7. Error feedback and internal models on differentiable manifolds, IEEE Trans. Automat. Control 29 (1981), 397–403. MR0748204
  8. Nonlinear Output Regulation: Theory and Applications, SIAM, New York 2004. Zbl1087.93003MR2308004
  9. Output regulation of nonlinear systems with nonhyperbolic zero dynamics, IEEE Trans. Automat. Control 40 (1995), 1497–1500. Zbl0841.93022MR1343825
  10. On the solvability of the regulator equations for a class of nonlinear systems, IEEE Trans. Automat. Control 48 (2003), 880–885. MR1980601
  11. On a nonlinear multivariable servomechanism problem, Automatica 26 (1990), 963–972. MR1080983
  12. Output regulation of nonlinear systems, IEEE Trans. Automat. Control 35 (1990), 131–140. MR1038409
  13. Nonlinear Control Systems, Third edition. Springer, Berlin – Heidelberg – New York 1995. Zbl0931.93005
  14. Nonlinear Systems, Pearson Education Inc., Upper Saddle River 2000. Zbl1140.93456
  15. FEMLAB-based output regulation of nonhyperbolically nonminimum phase system and its real-time implementation, In: Proc. 16th IFAC World Congress, Prague, IFAC 2005. 
  16. Real-time error-feedback output regulation of nonhyperbolically nonminimum phase system, In: Proc. American Control Conference 2007, New York 2007. 
  17. Numerical method for the solution of the regulator equation with application to nonlinear tracking, Automatica 44 (2008), 1358–1365. MR2531803
  18. Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin 1996. MR1477665
  19. [unknown], J. Ruiz-León, J. L. Orozco-Mora, and D. Henrion: Real-time and control of a gyroscope using Polynomial Toolbox 2.5. In: Latin-American Conference on Automatic Control CLCA 2002, Guadalajara 2002. 
  20. Asymptotic Expansions of Solutions of Singularly Perturbed Equations (in Russian), Nauka, Moscow 1973. MR0477344

Citations in EuDML Documents

top
  1. Shutang Liu, Yuan Jiang, Ping Liu, Rejection of nonharmonic disturbances in nonlinear systems
  2. Jianglin Lan, Weijie Sun, Yunjian Peng, Constrained robust adaptive stabilization for a class of lower triangular systems with unknown control direction
  3. Yuan Jiang, Ke Lu, Jiyang Dai, Global robust output regulation of a class of nonlinear systems with nonlinear exosystems
  4. Yuan Jiang, Jiyang Dai, Robust control of chaos in modified FitzHugh-Nagumo neuron model under external electrical stimulation based on internal model principle
  5. Volodymyr Lynnyk, Štěpán Papáček, Branislav Rehák, Biochemical network of drug-induced enzyme production: Parameter estimation based on the periodic dosing response measurement
  6. Branislav Rehák, Finite element-based observer design for nonlinear systems with delayed measurements
  7. Ctirad Matonoha, Štěpán Papáček, Volodymyr Lynnyk, On an optimal setting of constant delays for the D-QSSA model reduction method applied to a class of chemical reaction networks

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.