Skip to main content
Log in

Event-Triggered Multiple-Model Identifier for a Class of Nonlinear Systems

  • Published:
Journal of Control, Automation and Electrical Systems Aims and scope Submit manuscript

Abstract

This paper treats the event-triggered identification of a class of discrete-time nonlinear dynamical systems. Feedforward neural networks are trained online to behave like the given plant. With event triggering, the neural network parameters are not updated with every available data point. Instead, training takes place as and when required. We show that event-triggered identification requires only a small percentage of data points, resulting in considerably less computational requirement. In addition, we demonstrate that introducing multiple identification models in this scenario requires fewer data points with minimal degradation in the identification performance. We further show that this is better possible with the online sequential learning algorithm for networks with one or two hidden layers than the traditional back propagation algorithm.

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

Access this article

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Åarzén, K. E. (1999). A simple event-based PID controller. In Proceedings of 14th IFAC World Congress, Beijing, China, pp. 8687–8692. IFAC Proceedings Volume 32, Issue 2.

  • Åström, K. J., & Bernhardsson, B. M. (2002). Comparison of Riemann and Lebesgue sampling for first-order stochastic systems. In Proceedings of 41st Conference on Decision and Control (CDC), Las Vegas, NV, USA, pp. 2011–2016.

  • Billings, S. A. (2013). Nonlinear system identification: NARMAX methods in the time, frequency, and Spatio-Temporal domains. London, UK: Wiley.

    Book  MATH  Google Scholar 

  • Bryson, A. E. (1961). A gradient method for optimizing multi-stage allocation processes. In Proceedings of the Harvard University Symposium on Digital Computers and Their Applications, Cambridge, MA, USA.

  • Chen, S., & Billings, S. A. (1989). Representations of nonlinear systems: The NARMAX model. International Journal of Control, 49(3), 1013–1032.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, S., & Billings, S. A. (1992). Neural networks for nonlinear dynamic system modelling and identification. International Journal of Control, 56(2), 319–346.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, S., Billings, S. A., Cowan, C. F. N., & Grant, P. M. (1990). Practical identification of NARMAX models using radial basis functions. International Journal of Control, 52(6), 1327–1350.

    Article  MATH  Google Scholar 

  • Cybenko, G. (1989). Approximation by superpositions of sigmoidal function. Mathematics of Control, Signal, and Systems, 2, 303–314.

    Article  MathSciNet  MATH  Google Scholar 

  • Dreyfus, S. E. (1990). Artificial neural networks, back propagation, and the Kelly–Bryson gradient procedure. Journal of Guidance, Control, and Dynamics, 13(5), 926–928.

    Article  MathSciNet  Google Scholar 

  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). New York, NY, USA: W. W. Norton and Company.

    MATH  Google Scholar 

  • Funahashi, K. (1989). On the approximate realization of continuous mappings by neural networks. Neural Networks, 2, 183–192.

    Article  Google Scholar 

  • George, K., & Mutalik, P. (2019). A multiple model approach to time-series prediction using an online sequential learning algorithm. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(5), 976–990.

    Article  Google Scholar 

  • George, K., & Narendra, K. S. (2019). Multiple models: A survey. In Proceedings of the 19th Yale Workshop on Adaptive and Learning Systems, Yale University, New Haven, Connecticut, USA. Yale University.

  • George, K., Verhaegen, M., & Scherpen, J. M. A. (1999). A systematic and numerically efficient procedure for stable dynamic model inversion of LTI systems. In Procedings of the 38th IEEE Conference on Decision and Control (CDC), Phoenix, Arizona, U.S.A.

  • Girard, A. (2015). Dynamic triggering mechanisms for event-triggered control. IEEE Transactions on Automatic Control, 60(7), 1992–1997.

    Article  MathSciNet  MATH  Google Scholar 

  • Hornik, K., Stinchcombe, M., & White, H. (1989). Multilayer feedforward networks are universal approximators. Neural Networks, 2, 359–366.

    Article  MATH  Google Scholar 

  • Hunt, K. J., & Sbarbaro, D. (1991). Neural networks for nonlinear internal model control. In IEE Proceedings D (Control Theory and Applications), 138(5), pp. 431–438.

  • Hunt, K. J., Sbarbaro, D., Zbikowski, R., & Gawthrop, P. J. (1992). Neural networks for control systems —a survey. Automatica, 28(6), 1083–1112.

    Article  MathSciNet  MATH  Google Scholar 

  • Kelly, H. J. (1960). Gradient theory of optimal flight paths. ARS Journal, 30(10), 947–954.

    Article  MATH  Google Scholar 

  • Leontartis, I. J., & Billings, S. A. (1985). Input-output parametric models for non-linear systems – Part 1: Deterministic non-linear systems Part 2: Stochastic non-linear systems. International Journal of Control, 41(2), 303–328.

    Article  MathSciNet  Google Scholar 

  • Ljung, L. (1999). System identification — theory for the user (2nd ed.). Englewood Cliffs, NJ, USA: Prentice Hall.

    MATH  Google Scholar 

  • Lu, J., Wei, Q., Liu, Y., Zhou, T., & Wang, F. Y. (2022). Event-triggered optimal parallel tracking control for discrete-time nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 52(6), 3772–3784.

    Article  Google Scholar 

  • Massey, F. J., Jr. (1951). The Kolmogorov-Smirnov test for goodness of fit. Journal of the American Statistical Association, 46(253), 68–78.

    Article  MATH  Google Scholar 

  • Merlin, G. B., Moreira, L. G., & Gomes da Silva, J. M., Jr. (2021). Periodic event-triggered control for linear systems in the presence of cone-bounded nonlinear inputs: A discrete-time approach. Journal of Control, Automation and Electrical Systems, 32, 42–56.

    Article  Google Scholar 

  • Mizutani, E., Dreyfus, S. E., & Nishio, K. (2000). On derivation of MLP backpropagation from the Kelley-Bryson optimal-control gradient formula and its application. In Proceedings of the International Joint Conference on Neural Networks, Como, Italy.

  • Narendra, K. S., & Annaswamy, A. M. (1987). Persistent excitation in adaptive systems. International Journal of Control, 45(1), 127–160.

    Article  MathSciNet  MATH  Google Scholar 

  • Narendra, K. S., & Annaswamy, A. M. (1989). Stable Adaptive Systems. Englewood Cliffs, New Jersey, USA: Prentice-Hall.

    MATH  Google Scholar 

  • Narendra, K. S., & Balakrishnan, J. (1994). Improving transient response of adaptive control systems using multiple models and switching. IEEE Transactions on Automatic Control, 39(9), 1861–1866.

  • Narendra, K. S., & Parthasarathy, K. (1990). Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks, 1(1), 4–27.

    Article  Google Scholar 

  • Patel, R., Mandradia, D., & George, K. (2022). A novel training algorithm for neuro-control of discrete-time nonlinear systems. In Proceedings of the 8th IEEE International Conference on Electronics, Computing and Communication Technologies (CONECCT), Bangalore, India.

  • Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323(9), 533–536.

  • Sriram, T. B., Kashi, M. S., Ugarakhod, R., & George, K. (2019). Meta-cognitive neural networks for adaptive identification and control. In Proceedings of the 2019 IEEE TENCON, Kochi, India, pp. 2257–2261.

  • Van Overschee, P., & De Moor, B. (1996). Subspace identification for linear systems: Theory —implementation – Applications. Norwell, MA, USA: Kluwer Academic Publishers.

    Book  MATH  Google Scholar 

  • Verhaegen, M., & Verdult, V. (2007). Filtering and system identification: A least squares approach. Cambridge, U.K.: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Viana, É. S., Gomes da Silva, J. M., Jr., & Moreira, L. G. (2022). Disturbance rejection for uncertain discrete-time linear systems through event-triggered control. Journal of Control, Automation and Electrical Systems, 33, 103–114.

    Article  Google Scholar 

  • Werbos, P. J. (1974). Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. Ph. D. thesis, Harvard University, Cambridge, MA, USA.

  • Wu, L. B., Park, J. J., Xie, X. P., & Li, Y. F. (2022). Adaptive asymptotic tracking control of uncertain nonlinear systems based on Taylor decoupling and event-trigger. IEEE Transactions on Systems, Man and Cybernetics: Systems, 52(4), 2053–2060.

    Article  Google Scholar 

  • Yan, Y., Wu, L., He, X., & Chen, M. (2022). Adaptive fault-tolerant tracking control of uncertain nonlinear systems with event-triggered inputs and full state constraints. Journal of Control, Automation and Electrical Systems. https://doi.org/10.1007/s40313-022-00929-8

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the reviewers for their support through criticisms and comments which helped in improving the quality of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shikha Tripathi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ugarakhod, R., Tripathi, S. & George, K. Event-Triggered Multiple-Model Identifier for a Class of Nonlinear Systems. J Control Autom Electr Syst 34, 971–984 (2023). https://doi.org/10.1007/s40313-023-01015-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40313-023-01015-3

Keywords

Navigation