Geodesic
Systems with associative dynamics
Kybernetika, Tome 38 (2002) no. 5, pp. 585-600 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 93C10, 93C55
Keywords: nonlinear discrete-time dynamic model; stability 
@article{KYB_2002_38_5_a6,
     author = {Pearson, Ronald Korin and Kotta, \"Ulle and N\={o}mm, Sven},
     title = {Systems with associative dynamics},
     journal = {Kybernetika},
     pages = {585--600},
     year = {2002},
     volume = {38},
     number = {5},
     mrnumber = {1966947},
     zbl = {1265.93178},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2002_38_5_a6/}
}
TY  - JOUR
AU  - Pearson, Ronald Korin
AU  - Kotta, Ülle
AU  - Nōmm, Sven
TI  - Systems with associative dynamics
JO  - Kybernetika
PY  - 2002
SP  - 585
EP  - 600
VL  - 38
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/KYB_2002_38_5_a6/
LA  - en
ID  - KYB_2002_38_5_a6
ER  - 
%0 Journal Article
%A Pearson, Ronald Korin
%A Kotta, Ülle
%A Nōmm, Sven
%T Systems with associative dynamics
%J Kybernetika
%D 2002
%P 585-600
%V 38
%N 5
%U http://geodesic.mathdoc.fr/item/KYB_2002_38_5_a6/
%G en
%F KYB_2002_38_5_a6
Pearson, Ronald Korin; Kotta, Ülle; Nōmm, Sven. Systems with associative dynamics. Kybernetika, Tome 38 (2002) no. 5, pp. 585-600. http://geodesic.mathdoc.fr/item/KYB_2002_38_5_a6/

[1] Aczel J.: A Short Course on Functional Equations. Reidel, Dordrecht 1987 | MR | Zbl

[2] Aczel J., Dhombres J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge 1989 | MR | Zbl

[3] Brockwell P. J., Davis R. A.: Time Series: Theory and Methods. Springer–Verlag, New York 1991 | MR | Zbl

[4] Cohen G., Gaubert, S., Quadrat J.-P.: Max-plus algebra and system theory: Where we are and where to go. Annual Reviews in Control 23 (1999), 207–219 | DOI

[5] Doyle F. J., Henson M. A.: Nonlinear systems theory. In: Nonlinear Process Controll (M. A. Hanson and D. E. Seborg, eds.), Prentice–Hall, Englewood Cliffs, N.J. 1997, pp. 111–147

[6] Elaydi S. N.: An Introduction to Difference Equations. Springer–Verlag, New York 1996 | MR | Zbl

[7] Farina L., Rinaldi S.: Positive Linear Systems: Theory and Applications. Wiley, New York 2000 | MR | Zbl

[8] 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 | Zbl

[9] 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

[10] Kwakernaak H., Sivan R.: Linear Optimal Control Systems. Wiley, New York 1972 | MR | Zbl

[11] Ljung L.: System Identification: Theory for the User. Prentice–Hall, Englewood Cliffs, N.J. 1999 | Zbl

[12] Oppenheim A. V., Schafer R. W.: Digital Signal Processing, Prentice-Hall, Englewood Cliffs, N. J. 1975

[13] Pearson R. K.: Discrete-Time Dynamic Models. Oxford Univ. Press, Oxford – New York 1999 | Zbl

[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.: Nonlinear mean filters in image processing. IEEE Trans. Acoustics Speach Signal Processing 34 (1986), 573–584 | DOI

[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

[18] Thomson D. J.: Spectrum estimation techniques for characterization and development of wt4 waveguide. Bell Syst. Tech. J. 56 (1977), 1769–1815 | DOI

[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 | Zbl

[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