Systems with associative dynamics

Ronald Korin Pearson; Ülle Kotta; Sven Nōmm

Kybernetika (2002)

  • Volume: 38, Issue: 5, page [585]-600
  • ISSN: 0023-5954

Abstract

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This paper introduces a class of nonlinear discrete-time dynamic models that generalize familiar linear model structures; our motivation is to explore the extent to which known results for the linear case do or do not extend to this nonlinear class. The results presented here are based on a complete characterization of the solution of the associative functional equation F [ F ( x , y ) , z ] = F [ x , F ( y , z ) ] due to J. Aczel, leading to a class of invertible binary operators that includes addition, multiplication, and infinitely many others. We present some illustrative examples of these dynamic models, give a simple explicit representation for their inverses, and present sufficient conditions for bounded-input, bounded-output stability. Finally, we propose a generalization of this model class and we demonstrate that these models have classical state-space realizations, unlike arbitrarily structured NARMA models.

How to cite

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Pearson, Ronald Korin, Kotta, Ülle, and Nōmm, Sven. "Systems with associative dynamics." Kybernetika 38.5 (2002): [585]-600. <http://eudml.org/doc/33605>.

@article{Pearson2002,
abstract = {This paper introduces a class of nonlinear discrete-time dynamic models that generalize familiar linear model structures; our motivation is to explore the extent to which known results for the linear case do or do not extend to this nonlinear class. The results presented here are based on a complete characterization of the solution of the associative functional equation $F[F(x,y),z] = F[x,F(y,z)]$ due to J. Aczel, leading to a class of invertible binary operators that includes addition, multiplication, and infinitely many others. We present some illustrative examples of these dynamic models, give a simple explicit representation for their inverses, and present sufficient conditions for bounded-input, bounded-output stability. Finally, we propose a generalization of this model class and we demonstrate that these models have classical state-space realizations, unlike arbitrarily structured NARMA models.},
author = {Pearson, Ronald Korin, Kotta, Ülle, Nōmm, Sven},
journal = {Kybernetika},
keywords = {nonlinear discrete-time dynamic model; stability; nonlinear discrete-time dynamic model; stability},
language = {eng},
number = {5},
pages = {[585]-600},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Systems with associative dynamics},
url = {http://eudml.org/doc/33605},
volume = {38},
year = {2002},
}

TY - JOUR
AU - Pearson, Ronald Korin
AU - Kotta, Ülle
AU - Nōmm, Sven
TI - Systems with associative dynamics
JO - Kybernetika
PY - 2002
PB - Institute of Information Theory and Automation AS CR
VL - 38
IS - 5
SP - [585]
EP - 600
AB - This paper introduces a class of nonlinear discrete-time dynamic models that generalize familiar linear model structures; our motivation is to explore the extent to which known results for the linear case do or do not extend to this nonlinear class. The results presented here are based on a complete characterization of the solution of the associative functional equation $F[F(x,y),z] = F[x,F(y,z)]$ due to J. Aczel, leading to a class of invertible binary operators that includes addition, multiplication, and infinitely many others. We present some illustrative examples of these dynamic models, give a simple explicit representation for their inverses, and present sufficient conditions for bounded-input, bounded-output stability. Finally, we propose a generalization of this model class and we demonstrate that these models have classical state-space realizations, unlike arbitrarily structured NARMA models.
LA - eng
KW - nonlinear discrete-time dynamic model; stability; nonlinear discrete-time dynamic model; stability
UR - http://eudml.org/doc/33605
ER -

References

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  14. Pearson R. K., Pottmann M., Combining linear dynamics and static nonlinearities, In: Proc. ADCHEM 2000, Pisa 2000, pp. 485–490 
  15. Pearson R. K., Kotta Ü., Associative dynamic models, In: Proc. 1st IFAC Symposium on System Structure and Control, Prague 2001 
  16. Pitas I., Venetsanopoulos A. N., 10.1109/TASSP.1986.1164857, IEEE Trans. Acoustics Speach Signal Processing 34 (1986), 573–584 (1986) DOI10.1109/TASSP.1986.1164857
  17. Qin S. J., Badqwell T. A., An overview of nonlinear model predictive control applications, In: International Symposium on Nonlinear Model Predictive Control: Assessment and Future Directions (F. Allgöwer and A. Zheng, eds.), Ascona 1989, pp. 128–145 (1989) 
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  19. Kotta Ü., Sadegh N., Two approaches for state space realization of NARMA models: bridging the gap, In: Proc. 3rd IMACS Conference “Mathmod” (I. Troch and F. Breitenecker, eds.), Vienna 2000, pp. 415–419 Zbl1004.93008
  20. Verriest E. I., Linear systems over projective fields, In: Proc. 5th IFAC Symposium on Nonlinear Control Systems “NOLCOS’01”, Saint-Petersburg 2001, pp. 207–212 
  21. Pearson R. K., Kotta Ü., Associative dynamic models, In: Proc. 1st IFAC Symposium on System Structure and Control, Prague 2001 
  22. Kotta Ü., Zinober A. S. I., Nõmm S., On realizability of bilinear input-output models, In: Proc. 3rd International Conference on Control Theory and Applications “ICCTA’01”, Pretoria 2001 

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