Abstract
This chapter discusses theoretical foundations of modelling of nonlinear control systems with neural networks. Both feedforward and recurrent networks are described with emphasis on the practical implications of the mathematical results. The major approaches based on approximation and interpolation theories are presented: Stone-Weierstrass’ theorem, Kolmogorov’s theorem and multidimensional sampling. These are compared within a unified framework and the relevance for neural modelling of nonlinear control systems is stressed. Also, approximation of functionals with feedforward networks is briefly explained. Finally, approximation of dynamical systems with recurrent networks is described with emphasis on the concept of differential approximation.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V. I. Arnol’d. Some questions of approximation and representation of functions. In Proceedings of the International Congress of Mathematicians, pages 339–348, 1958. (English translation: American Mathematical Society Translations, Vol. 53).
J. L. Brown. Sampling reconstruction of n-dimensional bandlimited images after multilinear filtering. IEEE Trans. Circuits & Systems, CAS-36:1035–1038, 1989.
J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis. Cambridge University Press, Cambridge, England, 1970.
T. Chen and H. Chen. Approximation of continuous functionals by neural networks with application to dynamic systems. IEEE Transactions on Neural Networks, 4:910–918, 1993.
K. F. Cheung. A multidimensional extension of Papoulis’ Generalised Sampling Expansion with the application in minimum density sampling. In Robert J. Marks II, editor, Advanced Topics in Shannon Sampling and Interpolation Theory. Springer Verlag, New York, 1993.
G. Cybenko. Approximation by superposition of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2:303–314, 1989.
J. Dugundji. Topology. Allyn and Bacon, Boston, 1966.
A. Dzieliński. Optimal Filtering and Control of Two-dimensional Linear Discrete-Time Systems. Ph.D. Thesis, Faculty of Electrical Engineering, Warsaw University of Technology, Warsaw, Poland, February 1992.
H. G. Feichtinger and K. Gröchenig. Theory and practice of irregular sampling. In J. Benedetto and M. Frazier, editors, Wavelets: Mathematics and Applications. CRC Press, 1993.
R. Fletcher. Practical Methods of Optimization. Second Edition. Wiley, Chichester, 1987.
K. Funahashi. On the approximate realization of continuous mappings by neural networks. Neural Networks, 2:183–192, 1989.
K. Funahashi and Y. Nakamura. Approximation of dynamical systems by continuous time recurrent neural networks. Neural Networks, 6:801–806, 1993.
I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice-Hall, Englewood Cliffs, N.J., 1963.
F. Girosi and T. Poggio. Representation properties of networks: Kolmogorov’s theorem is irrelevant. Neural Computation, 1:465–469, 1989.
K. Gröchenig. Reconstruction algorithms in irregular sampling. Mathematics of Computations, 59:181–194, 1992.
R. Haber and H. Unbehauen. Structure identification of nonlinear dynamic systems—a survey on input/output approaches. Automatica, 26:651–677, 1990.
R. Hecht-Nielsen. Kolmogorov’s mapping neural network existence theorem. In Proceedings of the International Joint Conference on Neural Networks, volume 3, pages 11–14, New York, 1987. IEEE Press.
K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2:359–366, 1989.
K. J. Hunt, D. Sbarbaro, R. Żbikowski, and P. J. Gawthrop. Neural networks for control systems: A survey. Automatica, 28(6):1083–1112, November 1992.
A. Isidori. Nonlinear Control Systems: An Introduction. Springer-Verlag, New York, Second edition, 1989.
T. Kaczorek. Two-Dimensional Linear Systems. Springer-Verlag, Berlin, 1985.
A. N. Kolmogorov. On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables. Dokl. Akad. Nauk SSSR, 108:179–182, 1956. (in Russian).
A. N. Kolmogorov. On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR, 114:953–956, 1957. (English translation: American Mathematical Society Translations, Vol. 28).
J. S. Kowalik and M. R. Osborne. Methods for Unconstrained Optimization Problems. Elsevier, New York, 1968.
V. Kurková. Kolmogorov’s theorem is relevant. Neural Computation, 3:617–622, 1991.
V. Kurková. Kolmogorov’s theorem and multilayer neural networks. Neural Networks, 5:501–506, 1992.
J. N. Lin and R. Unbehauen. On the realization of Kolmogorov’s network. Neural Computation, 5:18–20, 1993.
S. Lipschutz. General Topology. McGraw-Hill, New York, 1965.
G. G. Lorentz. Approximation of Functions. Holt, Reinhart and Winston, New York, 1966.
F. Marvasti. Nonuniform sampling. In Robert J. Marks II, editor, Advanced Topics in Shannon Sampling and Interpolation Theory. Springer Verlag, New York, 1993.
D. P. Petersen and D. Middleton. Sampling and reconstruction of wave-number-limited functions in n-dimensional euclidean spaces. Information and Control, 5:279–323, 1962.
W. Rudin. Principles of Mathematical Analysis, Third Edition. McGraw-Hill, Auckland, 1976.
I. W. Sandberg. Approximation theorems for discrete-time systems. IEEE Transactions on Circuits and Systems, 38:564–566, 1991.
R. M. Sanner and J.-J. E. Slotine. Gaussian networks for direct adaptive control. IEEE Transactions on Neural Networks, 3:837–863, 1992.
D. Sbarbaro. Connectionist Feedforward Networks for Control of Nonlinear Systems. Ph.D. Thesis, Department of Mechanical Engineering, Glasgow University, Glasgow, Scotland, October 1992.
C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37:10–21, 1949.
I. N. Sneddon. Fourier Transforms. McGraw-Hill, New York, 1951.
E. Sontag. Neural nets as systems models and controllers. In Proc. Seventh Yale Workshop on Adaptive and Learning Systems, pages 73–79. Yale University, 1992.
D. A. Sprecher. On the structure of continuous functions of several variables. Transactions of the American Mathematical Society, 115:340–355, 1965.
M. H. Stone. The generalized Weierstrass approximation theorem. Mathematics Magazine, 21:167–184, 237–254, 1948.
E. Tzirkel-Hancock. Stable Control of Nonlinear Systems Using Neural Networks. Ph.D. thesis, Trinity College, Cambridge University, Cambridge, England, August 1992.
A. G. Vitushkin. On Hilbert’s thirteenth problem. Dokl. Akad. Nauk SSSR, 95:701–704, 1954. (in Russian).
R. Żbikowski. The problem of generic nonlinear control. In Proc. IEEE/SMC International Conference on Systems, Man and Cybernetics, Le Touquet, France, volume 4, pages 74–79, 1993.
R. Żbikowski. Recurrent Neural Networks: Some Control Problems. Ph.D. Thesis, Department of Mechanical Engineering, Glasgow University, Glasgow, Scotland, May 1994.
R. Żbikowski, K. J. Hunt, A. Dzieliński, R. Murray-Smith, and P. J. Gawthrop. A review of advances in neural adaptive control systems. Technical Report of the ESPRIT NACT Project TP-1, Glasgow University and Daimler-Benz Research, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag London Limited
About this chapter
Cite this chapter
Żbikowski, R., Dzieliński, A. (1995). Neural Approximation: A Control Perspective. In: Hunt, K.J., Irwin, G.R., Warwick, K. (eds) Neural Network Engineering in Dynamic Control Systems. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-3066-6_1
Download citation
DOI: https://doi.org/10.1007/978-1-4471-3066-6_1
Publisher Name: Springer, London
Print ISBN: 978-1-4471-3068-0
Online ISBN: 978-1-4471-3066-6
eBook Packages: Springer Book Archive