Stabilization of chaotic systems via fuzzy time-delayed controller approac

Document Type: Original Manuscript


1 Control and Energy Management Laboratory, National Engineering School of Sfax, University of Sfax, Tunisia

2 ESSTHS, University of Sousse.


In this paper, we investigate the stabilization of unstable periodic orbits of continuous time chaotic systems using
fuzzy time-delayed controllers. For this aim, we present a control method that can achieve stabilization of an unstable
periodic orbit (UPO) without any knowledge of the system model. Our proposal is attained progressively. First, we
combine the input-to-state linearizing controller with the fuzzy method to obtain one that achieves UPO stabilization.
Then, we use reduced order sliding observer to estimate the necessary state for the controller construction. Finally, the
efficiency of the proposed methods is demonstrated using numerical simulations applied to Chua’s system.


[1] M-R. Akbarzadeh-T., S.A. Hosseini, M-B. Naghibi-Sistani, Stable indirect adaptive interval type-2 fuzzy sliding-based control and synchronization of two different chaotic systems, Applied Soft Computing, 55 (2017), 576-587.
[2] M. Ataei, A. Kiyoumarsi, B. Ghorbani ,
Control of chaos in permanent magnet synchronous motor by using optimal lyapunov exponents placement, Physics Letters A, 374(41) (2010), 4226-4230.
[3] S. Beyhan ,
Adaptive fuzzy terminal sliding-mode observer with experimental applications, International Journal of Fuzzy Systems, 18(4) (2016), 585-594.
[4] O. Calvo, J. H. E. Cartwright,
Fuzzy control of chaos, International Journal of Bifurcation and Chaos, 8 (1998), 1743-1747.
[5] Y. C. Chang,
A robust tracking control for chaotic chua’s circuits via fuzzy approach, IEEE Transactions on Circuits and
Systems I: Fundamental Theory and Applications,
48(7) (2001), 889-895.
[6] W. Chang , J. Park, Y. G. C. Joo,
Static output-feedback fuzzy controller for chen’s chaotic system with uncertainties,
Information Sciences,
151 (2003), 227-244.
[7] CL. Chen , G. Feng , D. Sun, X. P. Guan,
H1 output feedback control of discrete-time fuzzy systems with application to chaos control, IEEE Transactions on Fuzzy Systems, 13(4) (2005), 531-543.
[8] G. Chen, X. Yu,
On time-delayed feedback control of chaotic systems, IEEE Transactions on Circuits and Systems-I, 46(6) (1999), 767-772.
[9] M. Feki,
Sliding mode control and synchronization of chaotic systems with parametric uncertainties, Chaos Solitons and Fractals, 41(3) (2009), 1390-1400.
[10] M. Feki,
Synchronization of generalized lorenz system using adaptive controller, IFAC Conference on Analysis and Control of Chaotic Systems CHAOS’06, Reims-France, 2006.
[11] A. F. Filipov,
Differential equations with discontinuous right-hand side, Transactions of the American Mathematical Society, 62 (1960), 199-231.
[12] A. Fourati, M. Feki, N. Derbel,
Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller, Nonlinear Dynamics, 62(3) (2010), 687-704.
[13] S.A. Hosseini, M-R. Akbarzadeh-T, M-B. Naghibi-Sistani,
A synchronizing controller using a direct adaptive interval type-2 fuzzy sliding mode strategy, Proceedings of the IEEE, International Conference on Fuzzy Systems (FUZZ), pp. 1-8, Hyderabad, India, July 7-10, 2013.
[14] A. Isidori,
Nonlinear Control Systems, 3rd ed. United Kingdom: Springer-Verlag, 1995.
[15] H. R. Kobravi, A. Erfanian,
A decentralized adaptive robust method for chaos control, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19(3) (2009), 033111.
[16] K. Konishi, M. Hirai, H. Kokame,
Sliding mode control for a class of chaotic systems, Physics Letters A, 245(6) (1998), 511-517.
[17] TC. Lin , FY. Huang , Z. Du, YC. Lin,
Synchronization of fuzzy modeling chaotic time delay memristor-based chua’s circuits with application to secure communication, International Journal of Fuzzy Systems, 17(2) (2015), 206-214.
[18] R. Luo , The robust adaptive control of chaotic systems with unknown parameters and external disturbance via a scalar input, International Journal of Adaptive Control and Signal Processing, 29(10) (2015), 1296-1307.
[19] H. Medhaffar, T. Damak, N. Derbel,
A decoupled fuzzy indirect adaptive sliding mode controller with application to robot manipulators, International Journal of Modelling, Identification and Control, 1(1) (2006), 23-29.
[20] H. Medhaffar, M. Feki, N. Derbel,
Adaptive Discrete-time Fuzzy Sliding Mode Control For a Class, of Chaotic Systems,
Advances in Science, Technology and Engineering Systems Journal,
2(3) (2017), 395-400.
[21] Y. Miladi , M. Feki, N. Derbel,
Stabilizing the unstable periodic orbits of a hybrid chaotic system using optimal control,Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1043-1056.
[22] Y. Miladi , M. Feki, N. Derbel,
Using unconventional methods to control the chaotic behavior of switched time systems: Application to a stepper motor, Journal of Engineering Science and Technology Review, 6(4) (2013), 81-89.
[23] E. Ott, C. Grebogi, J. A. Yorke,
Controlling chaos, Physical Review Letters, 64 (1990), 1196-1199.
[24] Y. Pan , M.J. Er, T. Sun,
Composite adaptive fuzzy control for synchronizing generalized lorenz systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(2) (2012), 023144.
[25] J. H. Park,
Adaptive modified projective synchronization of a four dimensional chaotic system with uncertain Parameters,Journal of Computational and Applied Mathematics, 213 (2008), 288-293.
[26] J. H. Park,
Adaptive modified projective synchronization of unified chaotic system with uncertain parameter, Chaos, Solitons and Fractals, 34 (2007), 1552-1559.
[27] J. H. Park, O. M. Kwon, S. M. Lee,
LMI optimization approach to stabilization of Genesio-Tesi chaotic system via dynamic controller, Applied Mathematics and Computation, 196(1) (2008), 200-206.
[28] J. H. Park, Z. Tang, J. Fengn,
Pinning cluster synchronization of delay-coupled Lur’e dynamical networks in a convex
, Nonlinear Dynamics, 89(1) (2017), 623-638.
[29] K. Pyragas,
Continuous control of chaos by self-controlling feedback, Phys Lett A, 170 (1992), 421-428.
[30] K. Pyragas,
Delayed feedback control of chaos, Philosophical Transactions of the Royal Society A, 364 (2006), 2309-2334.
[31] W. Shi,
Adaptive fuzzy control for mimo nonlinear systems with nonsymmetric control gain matrix and unknown control direction, IEEE Transactions on Fuzzy Systems, 22(5) (2014), 1288-1300.
[32] P. So, E. Ott, S. J. Schiff, D. T. Kaplan, T. Sauer, C. Grebogi,
Detecting unstable periodic orbits in chaotic experimental
, Physical Review Letters, 76 (1996), 4705-4708.
[33] CH. Wang, CY. Chen,
Intelligent chaos synchronization of fractional order systems via meanbased slide mode controller, International Journal of Fuzzy Systems, 17(2) (2015), 144-157.
[34] L. X. Wang,
Adaptive fuzzy systems and control, Englewood Cliffs, NJ: Prentice-Hall, 1994.
[35] CH. Yang , CL. Wu , YJ. Chen, SH. Shiao,
Reduced fuzzy controllers for lorenz-stenflo system control and synchronization, International Journal of Fuzzy Systems, 17(2) (2015), 158-169.
[36] J. J. Yan, Y. S. Yang, T. Y. Chiang, C. Y. Chen,
Robust synchronization of unified chaotic systems via sliding mode control, Chaos, Solitons and Fractals, 34(3) (2007), 947-954.
[37] H. T. Yau, C. K. Chen, C. Li Chen ,
Sliding mode control of chaotic systems with uncertainties, International Journal of
Bifurcation and Chaos,
10(5) (2000), 1139-1147.
[38] B. Yoo, W. Ham,
Adaptive fuzzy sliding mode control of nonlinear systems, IEEE Transactions Fuzzy systems, 6(2) (1998), 315-321.
[39] L. A. Zadeh,
Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions
Systems, Man and cybernetics,
3(1) (1973), 28-44.
[40] H. Zhang, H. Quin, G. Chen,
Adaptive control of chaotic systems with uncertainties, International Journal of Bifurcation and Chaos, 8(10) (1998), 2041-2046.
[41] L. Zhang, C. Guo, J. Wang, H. Yang,
Design of robust observer-based controller for time-delay switched fuzzy systems with unmeasurable premise variables, Chinese Control And Decision Conference (CCDC), IEEE, 2017.