Lyapunov stability-Dynamic Back Propagation-based comparative study of different types of functional link neural networks for the identification of nonlinear systems

  • Rajesh KumarEmail author
  • Smriti Srivastava
  • Amit Mohindru
Methodologies and Application


In this paper, the performance comparison of various types of functional link neural networks (FLNNs) has been done for the nonlinear system identification. The FLNNs being compared in the present study are: trigonometry FLNN, Legendre FLNN (LeFLNN), Chebyshev FLNN, power series FLNN (PSFLNN) and Hermite FLNN. The recursive weights adjustment equations are derived using the combination of Lyapunov stability criterion and dynamic back propagation algorithm. In the simulation study, a total of three nonlinear systems (both static and dynamic systems) are considered for testing and comparing the approximation ability and computational complexity of the above-mentioned FLNNs. From the simulation results, it is observed that the LeFLNN has given better approximation accuracy and PSFLNN offered least computational load as compared to the rest models.


Functional link neural network Nonlinear systems Dynamic back propagation algorithm Identification Lyapunov stability analysis Adaptive learning rate 



This study is not funded by any agency.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Anders U, Korn O (1999) Model selection in neural networks. Neural Netw 12(2):309–323CrossRefGoogle Scholar
  2. Belli M, Conti M, Crippa P, Turchetti C (1999) Artificial neural networks as approximators of stochastic processes. Neural Netw 12(4–5):647–658CrossRefGoogle Scholar
  3. Cass R, Radl B (1996) Adaptive process optimization using functional-link networks and evolutionary optimization. IFAC Proc Vol 29(7):253–258CrossRefGoogle Scholar
  4. Castro JL, Mantas CJ, Benıtez J (2000) Neural networks with a continuous squashing function in the output are universal approximators. Neural Netw 13(6):561–563CrossRefGoogle Scholar
  5. Chen CP, LeClair SR, Pao Y-H (1998) An incremental adaptive implementation of functional-link processing for function approximation, time-series prediction, and system identification. Neurocomputing 18(1–3):11–31CrossRefGoogle Scholar
  6. Cui M, Liu H, Li Z, Tang Y, Guan X (2014) Identification of hammerstein model using functional link artificial neural network. Neurocomputing 142:419–428CrossRefGoogle Scholar
  7. Dash PK, Liew A, Satpathy HP (1999) A functional-link-neural network for short-term electric load forecasting. J Intell Fuzzy Syst 7(3):209–221Google Scholar
  8. Emrani S, Salehizadeh SA, Dirafzoon A, Menhaj M (2010) Individual particle optimized functional link neural network for real time identification of nonlinear dynamic systems. In: 5th IEEE conference on industrial electronics and applications. IEEE, pp 35–40Google Scholar
  9. Giles CL, Maxwell T (1987) Learning, invariance, and generalization in high-order neural networks. Appl Opt 26(23):4972–4978CrossRefGoogle Scholar
  10. Haring S, Kok JN et al (1995) Finding functional links for neural networks by evolutionary computation. In: BENELEARN1995, proceedings of the fifth Belgian–Dutch conference on machine learning, Brussels, Belgium, pp 71–78Google Scholar
  11. Hassim YMM, Ghazali R (2013) Functional link neural network–artificial bee colony for time series temperature prediction. In: International conference on computational science and its applications. Springer, pp 427–437Google Scholar
  12. Haykin S (1999) Neural networks: a comprehensive foundation. Prentice-Hall Inc, Upper Saddle River, pp 161–175zbMATHGoogle Scholar
  13. Hussain AJ, Liatsis P (2003) Recurrent pi-sigma networks for DPCM image coding. Neurocomputing 55(1–2):363–382CrossRefGoogle Scholar
  14. Kaita T, Tomita S, Yamanaka J (2002) On a higher-order neural network for distortion invariant pattern recognition. Pattern Recognit Lett 23(8):977–984CrossRefGoogle Scholar
  15. Lisboa PJ (2002) A review of evidence of health benefit from artificial neural networks in medical intervention. Neural Netw 15(1):11–39CrossRefGoogle Scholar
  16. Lu L, Yu Y, Yang X, Wu W (2019) Time delay chebyshev functional link artificial neural network. Neurocomputing 329:153–164CrossRefGoogle Scholar
  17. Majhi R, Panda G, Sahoo G (2009) Development and performance evaluation of flann based model for forecasting of stock markets. Expert Syst Appl 36(3):6800–6808CrossRefGoogle Scholar
  18. Naderpour H, Mirrashid M (2019) Classification of failure modes in ductile and non-ductile concrete joints. Eng Fail Anal 103:361–375CrossRefGoogle Scholar
  19. Naderpour H, Mirrashid M (2019) Moment capacity estimation of spirally reinforced concrete columns using ANFIS. Complex Intell Syst.
  20. Naderpour H, Mirrashid M, Nagai K (2019) An innovative approach for bond strength modeling in FRP strip-to-concrete joints using adaptive neuro-fuzzy inference system. Eng Comput.
  21. Narendra KS, Parthasarathy K (1990) Identification and control of dynamical systems using neural networks. IEEE Trans Neural Netw 1(1):4–27CrossRefGoogle Scholar
  22. Pao Y (1989) Adaptive pattern recognition and neural networksGoogle Scholar
  23. Patra JC, Pal RN (1995) A functional link artificial neural network for adaptive channel equalization. Signal Process 43(2):181–195CrossRefGoogle Scholar
  24. Patra JC, Pal RN, Chatterji B, Panda G (1999) Identification of nonlinear dynamic systems using functional link artificial neural networks. IEEE Trans Syst Man Cybern Part b (Cybern) 29(2):254–262CrossRefGoogle Scholar
  25. Purwar S, Kar IN, Jha AN (2007) On-line system identification of complex systems using Chebyshev neural networks. Appl Soft Comput 7(1):364–372CrossRefGoogle Scholar
  26. Qi M, Zhang GP (2001) An investigation of model selection criteria for neural network time series forecasting. Eur J Oper Res 132(3):666–680CrossRefGoogle Scholar
  27. Setiono R, Thong JY (2004) An approach to generate rules from neural networks for regression problems. Eur J Oper Res 155(1):239–250CrossRefGoogle Scholar
  28. Shin Y, Ghosh J (1991) The pi-sigma network: an efficient higher-order neural network for pattern classification and function approximation. In: IJCNN-91-Seattle international joint conference on neural networks, vol 1. IEEE, pp 13–18Google Scholar
  29. Shin Y, Ghosh J (1995) Ridge polynomial networks. IEEE Trans Neural Netw 6(3):610–622CrossRefGoogle Scholar
  30. Takagi T, Sugeno M (1993) Fuzzy identification of systems and its applications to modeling and control. In: Readings in fuzzy sets for intelligent systems. Elsevier, pp 387–403Google Scholar
  31. Tawfik H, Liatsis P (1997) Prediction of non-linear time-series using higher-order neural networks. In: Proceeding IWSSIP 97Google Scholar
  32. Teeter J, Chow M-Y (1998) Application of functional link neural network to hvac thermal dynamic system identification. IEEE Trans Ind Electron 45(1):170–176CrossRefGoogle Scholar
  33. Zhang GP (2000) Neural networks for classification: a survey. IEEE Trans Syst Man Cybern Part C (Appl Rev) 30(4):451–462MathSciNetCrossRefGoogle Scholar
  34. Zhou G, Zhou Y, Huang H, Tang Z (2019) Functional networks and applications: a survey. Neurocomputing 335:384–399CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical and Instrumentation EngineeringThapar Institute of Engineering and Technology (Deemed to be University)PatialaIndia
  2. 2.Division of Instrumentation and Control EngineeringNetaji Subhas University of Technology (formerly Netaji Subhas Institute of Technology)New DelhiIndia
  3. 3.Department of Electronics and Communication EngineeringIndraprastha Institute of Information TechnologyNew DelhiIndia

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