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On Model Order Reduction of Perturbed Nonlinear Neural Networks with Feedback

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Scientific Computing in Electrical Engineering SCEE 2008

Part of the book series: Mathematics in Industry ((TECMI,volume 14))

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Abstract

The paper addresses the dynamical properties of large-scale perturbed nonlinear systems of the Hopfield type with feedback. In particular, it focuses on the hyperstability of the equilibria of the system. It proceeds to examine the effect of the empirical balanced truncation model reduction technique on the hyperstability properties. Finally, estimates of the additional conditions for preserving hyperstability when perturbations are present are derived.

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References

  1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. (Advances in Design and Control Series), SIAM, Philadelphia (2003)

    Google Scholar 

  2. Borisyuk, A., Friedman, A., Ermentrout, B., Terman, D.: Tutorials in Mathematical Biosciences I: Mathematical Neuroscience. Lecture Notes in Mathematics 1860, Springer-Verlag, Berlin (2005)

    Google Scholar 

  3. Condon, M., Ivanov, R.: Nonlinear systems-algebraic gramians and model reduction. J. Nonl. Sci. 14, 405–414 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hopfield, J.J.: Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proc. Nat. Acad. Sci. USA 79, 2554–2558 (1982);Neurons with graded response have collective computationa properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA 81, 3088–3092 (1984)

    Google Scholar 

  5. Hopfield, J.J., Tank, D.W.: Computing with neural circuits: A model. Science 233, 625–633 (1986)

    Article  Google Scholar 

  6. Edelstein-Keshet, L.: Mathematical Models in Biology. Classics in Applied Mathematics 46, SIAM, Philadelphia (2005)

    Google Scholar 

  7. Michel, A.N., Liu, D.: Qualitative Analysis and Synthesis of Recurrent Neural Networks. Marcel Dekker Inc., New York (2004)

    Google Scholar 

  8. Popov, V.M.: Hyperstability of Control Systems. Die Grundlehren der Mathematischen Wissenschaften 245, Springer-Verlag, Berlin (1973)

    Google Scholar 

  9. Moore, B.: Principal component analysis in linear systems: Controllability, Observability and model reduction. IEEE Trans. on Automatic Control AC-26(1) (1981)

    Google Scholar 

  10. Scherpen, J.M.A.: Balancing of nonlinear systems. Systems and Control Letters 21, 143–153 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gray, W.S., Scherpen, J.M.A.: Hankel Operators and Gramians for Nonlinear Systems. Proceedings of the 37th IEEE Conference on Decision and Control (CDC’98), pp. 1416–1421, Tampa, Fl, USA (1998)

    Google Scholar 

  12. Lall, S., Marsden, J.E., Glavaski S.: A subspace approach to balanced truncation for model reduction of nonlinear control systems. International Journal of Robust and Nonlinear Control 12, 519–535 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hahn, J., Edgar, T.F.: An Improved Method for Nonlinear Model Reduction Using Balancing of Empirical Gramians. Computers and Chemical Engineering 16, 1379–1397 (2002)

    Article  Google Scholar 

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Correspondence to Marissa Condon .

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Condon, M., Grahovski, G.G. (2010). On Model Order Reduction of Perturbed Nonlinear Neural Networks with Feedback. In: Roos, J., Costa, L. (eds) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry(), vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12294-1_71

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