Skip to main content
SpringerLink
Log in

Modeling of nonlinear system based on deep learning framework

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A novel modeling based on deep learning framework which can exactly manifest the characteristics of nonlinear system is proposed in this paper. Specifically, a Deep Reconstruction Model (DRM) is defined integrating with the advantages of the deep learning and Elman neural network (ENN). The parameters of the model are initialized by performing unsupervised pre-training in a layer-wise fashion using restricted Boltzmann machines (RBMs) to provide a faster convergence rate for modeling. ENN can be used to manifest the memory effect of system. To validate the proposed approach, two different nonlinear systems are used for experiments. The first one corresponds to the class-D power amplifier which operates in the ohmic and cutoff regions. According to error of time domain and spectrum, back propagation neural network model improved by RBMs (BP-RBMs) and ENN are compared with different input signals which are the simulated two-tone signal and actual square wave signal. The second system is a permanent magnet synchronous motor servo control system based on fuzzy PID control strategy. In terms of simulated and actual speed curves, BP-RBMs, DRM and ENN models are adopted on comparison, respectively. It is shown by experimental results that the proposed model with fewer parameters and iteration number can reconstruct the nonlinear system accurately and depict the memory effect, the nonlinear distortion and the dynamic performance of system precisely.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Zhu, Q., Wang, Y., Zhao, D.: Review of rational (total) nonlinear dynamic system modelling, identification, and control. Int. J. Syst. Sci. (2013)

  2. Liu, T., Ye, Y., Zeng, X., Ghannouchi, F.M.: Memory effect modeling of wideband wireless transmitters using neural networks. In: IEEE International Conference on Circuits and Systems for Communications, pp. 703–707. IEEE (2008)

  3. Amin, S., Landin, P.N., Handel, P., Ronnow, D.: Behavioral modeling and linearization of crosstalk and memory effects in RF MIMO transmitters. IEEE Trans. Microw. Theory Tech. 62(4), 810–823 (2014)

    Article  Google Scholar 

  4. Jie, S., Peng, T., Yuan, X.: An improved synchronous control strategy based on fuzzy controller for PMSM. Elektronika Ir Elektrotechnika. 20(6), 17–23 (2014)

    Article  Google Scholar 

  5. Xia, Z., Qiao, G., Zheng, T., Zhang, W.: Nonlinear modeling and dynamic analysis of the rotor-bearing system. Nonlinear Dyn. 57(4), 559–577 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo, Y., Guo, L.Z., Billings, S.A., Coca, D., Lang, Z.Q.: Volterra series approximation of a class of nonlinear dynamical systems using the Adomian decomposition method. Nonlinear Dyn. 74(1–2), 359–371 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fehri, B., Boumaiza, S.: Baseband equivalent Volterra series for digital predistortion of dual-band power amplifiers. IEEE Trans. Microw. Theory Tech. 62(3), 700–714 (2014)

    Article  Google Scholar 

  8. Bengio, Y., Courville, A., Vincent, P.: Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1798–1828 (2013)

    Article  Google Scholar 

  9. Langkvist, M., Karlsson, L., Loutfi, A.: A review of unsupervised feature learning and deep learning for time-series modeling. Pattern Recogn. Lett. 42(6), 11–24 (2014)

    Article  Google Scholar 

  10. Xu, J., Li, H., Zhou, S.: Improving mixing rate with tempered transition for learning restricted Boltzmann machines. Neurocomputing 139, 328–335 (2014)

    Article  Google Scholar 

  11. Huang, W., Song, G., Hong, H., Xie, K.: Deep architecture for traffic flow prediction: deep belief networks with multitask learning. IEEE Trans. Intell. Transp. Syst. 15(5), 2191–2201 (2014)

    Article  Google Scholar 

  12. Bengio, Y.: Learning deep architectures for AI. Found. Trends Mach. Learn. (2009)

  13. Erhan, D., Courville, A., Bengio, Y., Vincent, P.: Why does unsupervised pre-training help deep learning. J. Mach. Learn. Res. 11(3), 625–660 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Salakhutdinov, R., Hinton, G.: An efficient learning procedure for deep Boltzmann machines. Neural Comput. 24(8), 1967–2006 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Song, Qing: On the weight convergence of Elman networks. IEEE Trans. Neural Netw. 21(3), 463–480 (2010)

    Article  Google Scholar 

  16. Mkadem, F., Boumaiza, S.: Physically inspired neural network model for RF power amplifier behavioral modeling and digital predistortion. IEEE Trans. Microw. Theory Tech. 59(4), 913–923 (2011)

    Article  Google Scholar 

  17. Wisell, D.H., Rudlund, B., Ronnow, D.: Characterization of memory effects in power amplifiers using digital two-tone measurements. IEEE Trans. Instrum. Meas. 56(6), 2757–2766 (2007)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the Foundation of Key Laboratory of China’s Education Ministry and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Author information

Authors and Affiliations

  1. Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing Univ. Aeronaut. Astronaut.), Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xiangjun Jin, Jie Shao, Xin Zhang & Wenwei An

  2. Key Laboratory of Underwater Acoustic Signal Processing, Ministry of Education, Southeast University, Nanjing, 210096, China

    Xiangjun Jin, Jie Shao, Xin Zhang & Wenwei An

  3. Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, 0002, South Africa

    Reza Malekian

Authors
  1. Xiangjun Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wenwei An
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Reza Malekian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reza Malekian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jin, X., Shao, J., Zhang, X. et al. Modeling of nonlinear system based on deep learning framework. Nonlinear Dyn 84, 1327–1340 (2016). https://doi.org/10.1007/s11071-015-2571-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2571-6

Keywords

Search

Navigation