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Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network

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Abstract

Differing from the traditional wavelet neural network, a special type of discrete-time multiscale wavelet neural network (MWNN) using mesh grid is presented and investigated to solve the problem of identification of the autonomous nonlinear dynamical system. Inspired by the multiscale perception of biological neurons and the concept of continuous wavelet theory, multiscale and mesh grid proposed in this paper can be regarded as scale transformation and time translation in the mechanism of MWNN. For the convenience of digital processor realization, discrete-time expressions of weights updating and errors iteration are inferred by the Taylor expansion. In order to ensure the convergence of performance of this discrete-time model, the relation between the constant C in the equation of error iteration and sampling interval has been discovered by applying Z transform theory. The tracking error of autonomous nonlinear dynamical system will converge to the neighborhood of zero, which has been testified by discrete-time Lyapunov stability theory. For comparative purposes, discrete-time MWNN, Raised-Cosine Radial Basis Function Neural Network (RCRBFNN) and Gaussian Radial Basis Function Neural Network (GRBFNN) are used for solving the problem of autonomous nonlinear dynamical system identification. The Lorenz system and clinical electrocardiogram (ECG) dynamical system are applied to test the efficacy and superiority of the proposed discrete-time MWNN, in comparison with GRBFNN and RCRBFNN.

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References

  1. Karimi-Ghartemani M, Iravani MR (2004) A method for synchronization of power electronic converters in polluted and variable-frequency environments. IEEE Trans Power Syst 19:1263–1270

    Article  Google Scholar 

  2. Liu X, Tao R, Tavakoli M (2014) Adaptive control of uncertain nonlinear teleoperation systems. Mechatronics 24:66–78

    Article  Google Scholar 

  3. Sayadi O, Shamsollahi MB (2008) Model-based fiducial points extraction for baseline wandered electrocardiograms. IEEE Trans Biomed Eng 55:347–351

    Article  Google Scholar 

  4. Noponen K, Kortelainen J, Seppänen T (2009) Invariant trajectory classification of dynamical systems with a case study on ECG. Pattern Recognit 42:1832–1844

    Article  Google Scholar 

  5. Al-Fahoum AS, Qasaimeh AM (2013) A practical reconstructed phase space approach for ECG arrhythmias classification. J Med Eng Technol 37:401–408

    Article  Google Scholar 

  6. Le W, Jiazhong Z, Wenfan Z (2015) Identify the rotating stall in centrifugal compressors by fractal dimension in reconstructed phase space. Entropy 17:7888–7899

    Article  Google Scholar 

  7. Zeng W, Ismail SA, Lim YP et al (2019) Classification of gait patterns using kinematic and kinetic features, gait dynamics and neural networks in patients with unilateral anterior cruciate ligament deficiency. Neural Process Lett 50:887–909

    Article  Google Scholar 

  8. Thammano A, Ruxpakawong P (2010) Nonlinear dynamic system identification using recurrent neural network with multi-segment piecewise-linear connection weight. Memet Comput 2:273–282

    Article  Google Scholar 

  9. Zhao H, Gao S, He Z et al (2014) Identification of nonlinear dynamic system using a novel recurrent wavelet neural network based on the pipelined architecture. IEEE Trans Ind Electron 61:4171–4182

    Article  Google Scholar 

  10. Zheng T, Wang C (2017) Relationship between persistent excitation levels and RBF network structures, with application to performance analysis of deterministic learning. IEEE Trans Cybern 47:3380–3392

    Article  Google Scholar 

  11. Ning H, Qing G, Jing X (2016) Identification of nonlinear spatiotemporal dynamical systems with nonuniform observations using reproducing-kernel-based integral least square regulation. IEEE Trans Neural Netw Learn Syst 27:2399–2412

    Article  MathSciNet  Google Scholar 

  12. Schilling RJ, Carroll JJ, Al-Ajlouni AF (2001) Approximation of nonlinear systems with radial basis function neural networks. IEEE Trans Neural Netw 12:1–15

    Article  Google Scholar 

  13. Lian J, Lee Y, Sudhoff SD et al (2008) Self-organizing radial basis function network for real-time approximation of continuous-time dynamical systems. IEEE Trans Neural Netw 19:460–74

    Article  Google Scholar 

  14. Lu Z, Sun J, Butts K (2017) Multiscale support vector learning with projection operator wavelet kernel for nonlinear dynamical system identification. IEEE Trans Neural Netw Learn Syst 28:231–243

    Article  Google Scholar 

  15. Cordova JJ, Yu W (2012) Two types of Haar wavelet neural networks for nonlinear system identification. Neural Process Lett 35:283–300

    Article  Google Scholar 

  16. Zhang Y, Mu B, Zheng H (2013) Link between and comparison and combination of Zhang neural network and quasi-Newton BFGS method for time-varying quadratic minimization. IEEE Trans Cybern 43:490–503

    Article  Google Scholar 

  17. Zhang Y, Li Z, Guo D et al (2013) Discrete-time ZD, GD and NI for solving nonlinear time-varying equations. Numer Algorithms 64:721–740

    Article  MathSciNet  Google Scholar 

  18. Jin L, Zhang Y (2017) Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulator motion generation. IEEE Trans Neural Netw Learn Syst 26:1525–1531

    MathSciNet  Google Scholar 

  19. Übeyli ED (2009) Detecting variabilities of ECG signals by Lyapunov exponents. Neural Comput Appl 18:653–662

    Article  Google Scholar 

  20. Uebeyli ED (2010) Recurrent neural networks employing Lyapunov exponents for analysis of ECG signals. Expert Syst Appl 37:1192–1199

    Article  Google Scholar 

  21. Salisbury JI, Sun Y (2004) Assessment of chaotic parameters in nonstationary electrocardiograms by use of empirical mode decomposition. Ann Biomed Eng 32:1348–1354

    Article  Google Scholar 

  22. Cubero RJ, Marsili M, Roudi Y et al (2020) Multiscale relevance and informative encoding in neuronal spike trains. J Comput Neurosci 48:85–102

    Article  Google Scholar 

  23. Stein RB, Gossen ER, Jones KE (2005) Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6:389–397

    Article  Google Scholar 

  24. Zhang Q (1997) Using wavelet network in nonparametric estimation. IEEE Trans Neural Netw 8:227–36

    Article  Google Scholar 

  25. Pindoriya NM, Singh SN, Singh SK (2008) An adaptive wavelet neural network-based energy price forecasting in electricity markets. IEEE Trans Power Syst 23:1423–1432

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Liu YJ, Tong S (2014) Adaptive fuzzy control for a class of nonlinear discrete-time systems with backlash. IEEE Trans Fuzzy Syst 22:1359–1365

    Article  Google Scholar 

  28. Kreiseler D, Bousseliot R (1995) Automatisierte EKG-Auswertung mit Hilfe der EKG-Signaldatenbank CARDIODAT der PTB. Biomedizinische Technik/Biomed Eng 40:319–320

    Google Scholar 

  29. Goldberger AL, Amaral LA, Glass L, Hausdorff JM, Ivanov PC, Mark RG, Mietus JE, Moody GB, Peng CK, Stanley HE (2020) Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation 101:215–220

    Google Scholar 

  30. Wang Z, Ning X, Zhang Y et al (2000) Nonlinear dynamic characteristics analysis of synchronous 12-lead ECG signals. IEEE Eng Med Biol Mag Qu Mag Eng Med Biol Soc 19:110–115

    Article  Google Scholar 

  31. Deng M, Tang M, Wang C et al (2017) Cardiodynamicsgram as a new diagnostic tool in coronary artery disease patients with nondiagnostic electrocardiograms. Am J Cardiol 119:698–704

    Article  Google Scholar 

  32. Cuomo KM, Oppenheim AV, Strogatz SH (2002) Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circuits Syst II Analog Digital Signal Process 40:626–633

    Article  Google Scholar 

  33. Al-Ajlouni A, Schilling R, Harris S (2004) Synchronization of Lorenz-based chaotic circuits with applications to communications. Int J Syst Sci 35:211–221

    Article  Google Scholar 

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Acknowledgements

This work was supported by Science and Technology Program of Guangzhou, China (201904010224)

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Authors and Affiliations

  1. School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, China

    Guo Luo

  2. School of Intelligent Systems Engineering, Sun Yat-Sen University, Shenzhen, 518107, China

    Zhi Yang

  3. School of Computing, South China Normal University, Guangzhou, China

    Qizhi Zhang

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  1. Guo Luo
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  2. Zhi Yang
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  3. Qizhi Zhang
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Correspondence to Zhi Yang.

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The authors (Guo Luo, Zhi Yang and Qizhi Zhang) declare that they have no conflict of interests in relation to the work in this article.

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Luo, G., Yang, Z. & Zhang, Q. Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network. Neural Comput & Applic 33, 15191–15203 (2021). https://doi.org/10.1007/s00521-021-06142-z

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  • DOI: https://doi.org/10.1007/s00521-021-06142-z

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