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Exploiting Chaos in Learning System Identification for Nonlinear State Space Models

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The paper presents two learning methods for nonlinear system identification. Both methods employ neural network models for representing state and output functions. The first method of learning nonlinear state space is based on using chaotic or noise signals in the training of state neural network so that the state neural network is designed to produce a sequence in a recursive way under the excitement of the system input. The second method of learning nonlinear state space has an observer neural network devoted to estimate the states as a function of the system inputs and the outputs of the output neural network. This observer neural network is trained to produce a state sequence when the output neural network is forced by the same sequence and then the state neural network is trained to produce the estimated states in a recursive way under the excitement of the system input. The developed identification methods are tested on a set of benchmark plants including a non-autonomous chaotic system, i.e. Duffing oscillator. Both proposed methods are observed much superior than well-known identification methods including nonlinear ARX, nonlinear ARMAX, Hammerstein, Wiener, Hammerstein–Wiener, Elman network, state space models with subspace and prediction error methods.

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  1. Ljung L (1987) System identification: theory for the user. PTR Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  2. Narendra KS, Parthasarathy K (1990) Identification and control of dynamical systems using neural networks. IEEE Trans Neural Netw 1(1):4–27

    Article  Google Scholar 

  3. Chen S, Billings SA (1992) Neural networks for nonlinear dynamic system modelling and identification. Int J Control 56:319–346

    Google Scholar 

  4. Ljung L, Sjöberg J (1992) A system identification perspective on neural nets. IEEE Neural Netw Signal Process. 423–435

  5. Sjöberg J, Hjalmarsson H, Ljung L (1994) Neural networks in system identification. In: Proceedings of the 10th IFAC symposium on system identification, vol. 2:49–71

  6. Goethals I, Pelckmans K, Suykens JAK, De Moor B (2005a) Identification of MIMO Hammerstein models using least squares support vector machines. Automatica 41:1263–1272

    Article  MATH  Google Scholar 

  7. Goethals I, Pelckmans K, Suykens JAK, De Moor B (2005b) Subspace identification of Hammerstein systems using least squares support vector machines. IEEE Trans Autom Control 50(10):1509–1519

    Article  Google Scholar 

  8. Hong X, Chen S (2012) The system identification and control of Hammerstein system using non-uniform rational B-spline neural network and particle swarm optimization. Neurocomputing 82:216–223

    Article  Google Scholar 

  9. Ljung L (2010) Perspectives on system identification. Annu Rev Control 34(1):1–12

    Article  MathSciNet  Google Scholar 

  10. Billings SA, Wei HL (2005) A new class of wavelet networks for nonlinear system identification. IEEE Trans Neural Netw 16(4):862–874

    Article  Google Scholar 

  11. Martínez-Ramón M, Rojo-Álvarez JL, Camps-Valls G, Muñoz-Marí J, Navia-Vázquez A, Soria-Olivas E, Figueiras-Vidal AR (2006) Support vector machines for nonlinear kernel ARMA system identification. IEEE Trans Neural Netw 17(6):1617–1622

    Article  Google Scholar 

  12. Hong X, Mitchell RJ (2007) A Hammerstein model identification algorithm using Bezier–Bernstein approximation. IET Proc Control Theory Appl 1(4):1149–1159

    Article  MathSciNet  Google Scholar 

  13. Elman JL (1990) Finding structure in time. Cogn Sci 14:179–211

    Article  Google Scholar 

  14. Suykens JAK, De Moor B, Vandewalle J (1995) Nonlinear system identification using neural state space models, applicable to robust control design. Int J Control 62(1):129–152

    Article  MATH  Google Scholar 

  15. Suykens JAK, Vandewalle J (1995) Learning a simple recurrent neural state space model to behave like Chua’s double scroll. IEEE Trans Circuits Syst Part I 42(8):499–502

    Article  Google Scholar 

  16. Gao XZ, Gao XM, Ovaska SJ (1996) A modified Elman neural network model with application to dynamical systems identification. Proc. IEEE Syst Man Cybern Int Conf 2:1376–1381

    Google Scholar 

  17. Yu W, Poznyak AS, Li X (2001) Multilayer dynamic neural networks for nonlinear system on-line identification. Int J Control 74(18):1858–1864

    Article  MATH  MathSciNet  Google Scholar 

  18. Yu W (2005) State-space recurrent fuzzy neural networks for nonlinear system identification. Neural Process Lett 22:391–404

    Article  Google Scholar 

  19. Ölmez M (2013) Exploiting chaos in system identification and control. PhD Thesis, Graduate School of Natural and Applied Sciences, Dokuz Eylül University

  20. Cuomo KM, Oppenheim AV (1993) Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett 71(1):65–68

    Article  Google Scholar 

  21. Chen G, Chen Y, Öğmen H (1997) Identifying chaotic systems via a Wiener-type cascade model. IEEE Control Syst Mag 8:29–36

    Article  Google Scholar 

  22. Narendra KS (1996) Neural networks for control theory and practice. Proc IEEE 84(10):1385–1406

    Article  Google Scholar 

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Correspondence to Mehmet Ölmez.

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Ölmez, M., Güzeliş, C. Exploiting Chaos in Learning System Identification for Nonlinear State Space Models. Neural Process Lett 41, 29–41 (2015).

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