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Robust stability of recurrent neural networks with ISS learning algorithm

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Abstract

In this paper, an input-to-state stability (ISS) approach is used to derive a new robust weight learning algorithm for dynamic neural networks with external disturbance. Based on linear matrix inequality (LMI) formulation, the ISS learning algorithm is presented to not only guarantee exponential stability but also reduce the effect of an external disturbance. It is shown that the design of the ISS learning algorithm can be achieved by solving LMI, which can be easily facilitated by using some standard numerical packages. A numerical example is presented to demonstrate the validity of the proposed learning algorithm.

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References

  1. Gupta, M.M., Jin, L., Homma, N.: Static and Dynamic Neural Networks. Wiley-Interscience, New York (2003)

    Book  Google Scholar 

  2. Kelly, D.G.: Stability in contractive nonlinear neural networks. IEEE Trans. Biomed. Eng. 3, 241–242 (1990)

    Google Scholar 

  3. Matsouka, K.: Stability conditions for nonlinear continuous neural networks with asymmetric connections weights. Neural Netw. 5, 495–500 (1992)

    Article  Google Scholar 

  4. Liang, X.B., Wu, L.D.: A simple proof of a necessary and sufficient condition for absolute stability of symmetric neural networks. IEEE Trans. Circuits Syst. I 45, 1010–1011 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sanchez, E.N., Perez, J.P.: Input-to-state stability (ISS) analysis for dynamic neural networks. IEEE Trans. Circuits Syst. I 46, 1395–1398 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chu, T., Zhang, C., Zhang, Z.: Necessary and sufficient condition for absolute stability of normal neural networks. Neural Netw. 16, 1223–1227 (2003)

    Article  Google Scholar 

  7. Chu, T., Zhang, C.: New necessary and sufficient conditions for absolute stability of neural networks. Neural Netw. 20, 94–101 (2007)

    Article  MATH  Google Scholar 

  8. Rovithakis, G.A., Christodoulou, M.A.: Adaptive control of unknown plants using dynamical neural networks. IEEE Trans. Syst. Man Cybern. 24, 400–412 (1994)

    Article  MathSciNet  Google Scholar 

  9. Jagannathan, S., Lewis, F.L.: Identification of nonlinear dynamical systems using multilayered neural networks. Automatica 32, 1707–1712 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Suykens, J.A.K., Vandewalle, J., De Moor, B.: Lur’e systems with multilayer perceptron and recurrent neural networks; absolute stability and dissipativity. IEEE Trans. Autom. Control 44, 770–774 (1999)

    Article  MATH  Google Scholar 

  11. Yu, W., Li, X.: Some stability properties of dynamic neural networks. IEEE Trans. Circuits Syst. I 48, 256–259 (2001)

    Article  MATH  Google Scholar 

  12. Chairez, I., Poznyak, A., Poznyak, T.: New sliding-mode learning law for dynamic neural network observer. IEEE Trans. Circuits Syst. II 53, 1338–1342 (2006)

    Article  Google Scholar 

  13. Yu, W., Li, X.: Passivity analysis of dynamic neural networks with different time-scales. Neural Process. Lett. 25, 143–155 (2007)

    Article  MathSciNet  Google Scholar 

  14. Rubio, J.J., Yu, W.: Nonlinear system identification with recurrent neural networks and dead-zone Kalman filter algorithm. Neurocomputing 70, 2460–2466 (2007)

    Article  Google Scholar 

  15. Suykens, J.A.K., Vandewalle, J., De Moor, B.: NL q theory: checking and imposing stability of recurrent neural networks for nonlinear modelling. IEEE Trans. Signal Process. 45, 2682–2691 (1997)

    Article  Google Scholar 

  16. Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34, 435–443 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sontag, E.D.: Further facts about input to state stabilization. IEEE Trans. Autom. Control 35, 473–476 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, Z.P., Teel, A., Praly, L.: Small-gain theorem for ISS systems and applications. Math. Control Signals Syst. 7, 95–120 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sontag, E.D., Wang, Y.: On characterizations of the input-to-state stability property. Syst. Control Lett. 24, 351–359 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Christofides, P.D., Teel, A.R.: Singular perturbations and input-to-state stability. IEEE Trans. Autom. Control 41, 1645–1650 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sontag, E.D.: Comments on integral variants of ISS. Syst. Control Lett. 34, 93–100 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Teel, A.R.: Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans. Autom. Control 43, 960–964 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xie, W., Wen, C., Li, Z.: Input-to-state stabilization of switched nonlinear systems. IEEE Trans. Autom. Control 46, 1111–1116 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Angeli, D., Nesic, D.: Power characterizations of input-to-state stability and integral input-to-state stability. IEEE Trans. Autom. Control 46, 1298–1303 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishinan, V.: Linear Matrix Inequalities in Systems and Control Theory. SIAM, Philadelphia (1994)

    Google Scholar 

  26. Gahinet, P., Nemirovski, A., Laub, A.J., Chilali, M.: LMI Control Toolbox. The Mathworks Inc., New York, (1995)

    Google Scholar 

  27. Hopfield, J.J.: Neurons with grade response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)

    Article  Google Scholar 

  28. Albert, A.E.: Regression and the Moore-Penrose Pseudoinverse. Academic Press, New York (1972)

    MATH  Google Scholar 

  29. Strang, G.: Introduction to Applied Mathematics. Wellesley Cambridge Press, Cambridge (1986)

    MATH  Google Scholar 

  30. Hunt, K.J., Sbarbaro, D., Zbikowski, R., Gawthrop, P.J.: Neural networks for control systems—a survey. Automatica 28, 1083–1112 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Automotive Engineering, Seoul National University of Science & Technology, Seoul, South Korea

    Choon Ki Ahn

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  1. Choon Ki Ahn
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Correspondence to Choon Ki Ahn.

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Ahn, C.K. Robust stability of recurrent neural networks with ISS learning algorithm. Nonlinear Dyn 65, 413–419 (2011). https://doi.org/10.1007/s11071-010-9901-5

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