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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Observer-Based Event-Triggered Fuzzy Integral Sliding Mode Control for Hindmarsh Rose Neuronal Model Via T-S Fuzzy systems

Journal of Applied Nonlinear Dynamics 10(1) (2021) 47--63 | DOI:10.5890/JAND.2021.03.003

P. Nirvin , R. Rakkiyappan

Department of Mathematics, Bharathiar University, Coimbatore - 641 046, Tamilnadu, India

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The problem of stability for nonlinear Hindmarsh-Rose (H-R) neuron model with event-triggered fuzzy Integral Sliding Mode Control (ISMC) design is investigated via Takagi-Sugeno (T-S) fuzzy systems. The Event-triggered Communication (ETC) scheme is introduced with a triggered condition of the sampling instant to determine whether the current sampled signal should be transmitted or not. {In order to send and receive the delay measurements for updating the control, the event triggered zero-order-holder (ZOH) is employed. Also, a note observer is designed for the estimation of system state and for the facilitation of the sliding surface design.} Then by analyzing the measured output and observer output, a novel law is presented. Further the stability criterion and the stabilization conditions are estimated based on Lyapunov-Krasovskii functional (LKF) to ensure the asymptotically stability for the considered system. Finally, a numerical example is presented to demonstrate the feasibility of the proposed design scheme.


  1. [1]  Huang, Z., Song, Q., and Feng, C. (2010), Multi-stability in networks with self excitation and high-order synaptic connectivity, IEEE Transactions on Circuits and Systems I, 57(8), 2144-2155.
  2. [2]  Hashimoto, S. and Torikai, H. (2010), A novel hybrid spiking neuron: Bifurcations, responses, and on-chip learning, IEEE Transactions on Circuits and Systems I, 57(8), 2168-2181.
  3. [3]  Massoud, T.M. and Horiuchi, T.K. (2011), A neuromorphic VLSI head direction cell system, IEEE Transactions on Circuits and Systems I, 58(1), 150-163.
  4. [4]  Singer, W. (1999), Neuronal synchrony: A versatile code for the definition of relations, Neuron, 24(1), 49-65.
  5. [5]  Uhlhaas, P.J. and Singer, W. (2006), Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology, Neuron, 52(1), 155-168.
  6. [6]  Neuenschwander, S. and Singer, W. (1996), Long-range synchronization of oscillatory light responses in the cat retina and lateral geniculate nucleus, Nature, 379, 728-732.
  7. [7]  Glass, L. (2001), Synchronization and rhythmic processes in physiology, Nature, 410(6825), 277-284.
  8. [8]  Gray, C.M., Konig, P., Engel, A.K., and Singer, W. (1989), Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties, Nature, 338, 334-337.
  9. [9]  Dayan, P. and Abbott, L.F. (2005), Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, Cambridge, MA, USA: MIT, \underline{Press}, doi: 10.1086/421681.
  10. [10]  FitzHugh, R. (1969), Mathematical models for excitation and propagation in nerve, In: H. P. Schawn (ed.), Biological Engineering, 1(185), McGraw-Hill, New York.
  11. [11]  Hindmarsh J.L. and Rose, R.M. (1984), A model of neuronal bursting using three coupled first order differential equations, Proceeding of the Royal Society of London, Biological Sciences, 221(1222), 87-102.
  12. [12]  Hindmarsh, A.C. (2005), Sundials: Suite of nonlinear and differential/algebraic equation solvers, ACM Transactions Mathematical Software, 31(3), 363-396.
  13. [13]  Lam, H.K., Liu, C., and Wu, L. (2015), Polynomial fuzzy-model-based control systems: stability analysis via approximated membership functions considering sector nonlinearity of control input, IEEE Transactions on Fuzzy Systems, 2202-2214.
  14. [14]  Lam, H.K., Xiao, B., and Yu, Y. (2016), Membership-function-dependent stability analysis and control synthesis of guaranteed cost fuzzy model-based control systems, International Journal of Fuzzy Systems, 537-549.
  15. [15]  Cao, S.G., Pees, N.W., and Feng, G. (1997), Analysis and design for a class of complex control systems - Part II: Fuzzy controller design, Automatica, 1029-1039.
  16. [16]  Feng, G. (2010), Analysis and Synthesis of Fuzzy Control Systems - A Model Based Approach, Boca Raton: CRC, doi: 10.1201/EBK1420092646.
  17. [17]  Tuan, H.D., Apkarian, P., and Narikiyo, T. (2001), Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Transactions on Fuzzy Systems, 324-332.
  18. [18]  Peng, C. and Yang, T.C. (2013), Event-triggered communication and $H_\infty$ control co-design for networked control systems, Automatica, 49(5), 1326-1332.
  19. [19]  Suh, Y.S. (2007), Send-on-delta sensor data transmission with a linear predictor, Sensors, 537-547.
  20. [20]  Liu D. and Yang, G.H. (2019), Dynamic event-triggered control for linear time-invariant systems with $L_2$-gain performance, International Journal of Robust Nonlinear Control, 29(2), 507-518.
  21. [21]  Cassandras, C.G. (2014), The event-driven paradigm for control, communication and optimization, Journal of Control and Decision, 1(1), 3-17.
  22. [22]  Zhang, C., Hu, J., Qiu, J., and Chen, Q. (2017), Reliable output feedback control for T-S fuzzy systems with decentralized event triggering communication and actuator failures, IEEE Transactions on Cybernetics, 47(9), 2592-2602.
  23. [23]  Utkin, V.I. (1977), Variable structure systems with sliding modes, IEEE Transactions on Automatic Control, 22(2), 212-222.
  24. [24]  Edwards, C. and Spurgeon, S.K. (1998), Sliding Mode Control, Theory And Applications, Danville, CA: CRC, doi: 10.1201/9781498701822.
  25. [25]  De Carlo, R.A., Zak, S.H., and Matthews, G.P. (1988), Variable structure control of nonlinear multi-variable systems: A tutorial, Proceeding of theIEEE, 76(3), 212-232.
  26. [26]  Aloui, S., Pages, O., Hajjaji, A. El., Chaari, A., and Koubaa, Y. (2011), Improved fuzzy sliding mode control for a class of MIMO nonlinear uncertain and perturbed systems, Applied Soft Computing, 11(1), 820-826.
  27. [27]  Lee, H., Kim, E., Kang, H.J., and Park, M. (2001), A new sliding-mode control with fuzzy boundary layer, Fuzzy Sets and Systems, 120, 135-143.
  28. [28]  Yu, F.M., Chung, H.Y., and Chen, S.Y. (2003), Fuzzy sliding mode controller design for uncertain time-delayed systems with nonlinear input, Fuzzy Sets and Systems, 140, 359-374.
  29. [29]  Utkin, V.I. and Shi, J. (1996), Integral sliding mode in systems operating under uncertainty conditions, Presented at the 35th Conference Decision Control, Kobe, Japan, Dec, doi: 10.1109/CDC.1996.577594.
  30. [30]  Castanos, F. and Fridman, L. (2006), Analysis and design of integral sliding manifolds for systems with unmatched perturbations, IEEE Transactions Automatic Control, 51(5), 853-858.
  31. [31]  Cao, W.J. and Xu, J.X. (2004), Nonlinear integral-type sliding surface for both matched and unmatched uncertain systems, IEEE Transactions Automatic Control, 49(8), 1355-1360.
  32. [32]  Ho, D.W.C. and Niu, Y. (2007), Robust fuzzy design for nonlinear uncertain stochastic systems via sliding-mode control, IEEE Transactions on Fuzzy Systems, 15(3), 350-358.
  33. [33]  Jiang, B., Karimi, H.R., Kao, Y., and Gao, C. (2018), Takagi-Sugeno model based sliding mode observer design for finite-time synthesis of semi-Markovian jump systems, IEEE Transactions on Systems, Man, Cybernetics: Systems, doi: 10.1109/TSMC.2018.2846656.
  34. [34]  Wang, Y., Karimi, H.R., Shen, H., Fang, Z., and Liu, M. (2017), Fuzzy-model based sliding mode control of nonlinear descriptor systems, IEEE Transactions on Cybernetics, 1-11.
  35. [35]  Jiang, B., Karimi, H.R., Kao, Y., and Gao, C. (2018), A novel robust fuzzy integral sliding mode control for nonlinear semi-Markovian jump T-S fuzzy systems, IEEE Transactions on Fuzzy Systems, 26(6), 3594-3604.
  36. [36]  Beyhan, S., Lendek, Z., Alci, M., and Babuska, R. (2013), Takagi-sugeno fuzzy observer and extended-Kalman filter for adaptive payload estimation, in Proceeding 9th Asian Control Conference (ASCC), Istanbul, Turkey, 1-6.
  37. [37]  Kang, W., Zhong, S., and Shi, K. (2016), Triple integral approach to reachable set bounding for linear singular systems with time-varying delay, Mathematical Method in the Applied Science, 1-12.
  38. [38]  Lee, W.I., Lee, Y.S., and Park, P.G. (2014), Improved criteria on robust stability and $H_1$ performance for linear systems with interval time-varying delays via new triple integral functionals, Applied Mathematics and Computation, 570-577.
  39. [39]  Peng, C., Ma, S., and Xie, X. (2017), Observer-Based Non-PDC Control for Networked T-S Fuzzy Systems With an Event-Triggered Communication, IEEE Transactions on Cybernetics, 2168-2267.
  40. [40]  Beyhan, S. (2017), Affine T-S Fuzzy Model-Based Estimation and Control of Hindmarsh-Rose Neuronal Model, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2168-2216.
  41. [41]  Fridman, L., Strygin, V., and Polyakov, A. (2003), Stabilization of amplitude of oscillations via relay delay control, International Journal of Control, 770-780.
  42. [42]  Sun, J., Liu, G.P., and Chen, J. (2008), Delay-dependent stability and stabilization of neutral time-delay systems, International Journal of Robust and Nonlinear Control, 1364-1375.